The Mizar article:

Upper and Lower Sequence on the Cage. Part II

by
Robert Milewski

Received September 28, 2001

Copyright (c) 2001 Association of Mizar Users

MML identifier: JORDAN1G
[ MML identifier index ]


environ

 vocabulary REALSET1, FINSEQ_1, BOOLE, RELAT_1, FINSEQ_5, MATRIX_1, GOBOARD1,
      FINSEQ_4, RFINSEQ, COMPTS_1, RELAT_2, SPPOL_1, EUCLID, JORDAN1E, JORDAN8,
      JORDAN9, FINSEQ_6, PSCOMP_1, TOPREAL1, GOBOARD5, MCART_1, TREES_1,
      FUNCT_1, ARYTM_1, CARD_1, PRE_TOPC, ARYTM_3, GROUP_1, JORDAN1A, SPPOL_2,
      JORDAN3, SPRECT_2, GROUP_2, JORDAN5C, JORDAN6;
 notation TARSKI, XBOOLE_0, ENUMSET1, SUBSET_1, XREAL_0, REAL_1, NAT_1,
      FUNCT_1, PARTFUN1, FINSEQ_1, FINSEQ_4, FINSEQ_5, RFINSEQ, MATRIX_1,
      REALSET1, FINSEQ_6, STRUCT_0, PRE_TOPC, CARD_1, CARD_4, BINARITH,
      CONNSP_1, COMPTS_1, EUCLID, PSCOMP_1, SPRECT_2, GOBOARD1, TOPREAL1,
      GOBOARD5, SPPOL_1, SPPOL_2, JORDAN3, JORDAN8, JORDAN5C, JORDAN6, JORDAN9,
      JORDAN1A, JORDAN1E;
 constructors JORDAN8, REALSET1, JORDAN6, REAL_1, CARD_4, PSCOMP_1, BINARITH,
      CONNSP_1, JORDAN9, JORDAN1A, WSIERP_1, FINSEQ_4, GOBRD13, JORDAN3,
      RFINSEQ, TOPS_2, TOPMETR, TOPREAL2, JORDAN5C, SPRECT_1, FINSEQ_5,
      ENUMSET1, FINSOP_1, JORDAN1E, INT_1;
 clusters XREAL_0, SPRECT_1, TOPREAL6, JORDAN8, REVROT_1, RELSET_1, ARYTM_3,
      SUBSET_1, FINSEQ_6, SPRECT_3, AMISTD_1, GOBOARD1, FINSEQ_1, FINSEQ_5,
      SPPOL_2, JORDAN1A, GOBOARD4, JORDAN1E, BINARITH, REALSET1, MEMBERED,
      ORDINAL2;
 requirements NUMERALS, BOOLE, SUBSET, REAL, ARITHM;
 definitions TARSKI, XBOOLE_0;
 theorems AXIOMS, BINARITH, EUCLID, JORDAN8, PSCOMP_1, JORDAN1A, NAT_1, REAL_1,
      GOBOARD5, REAL_2, FINSEQ_1, FINSEQ_2, JORDAN4, JORDAN6, SPRECT_2,
      FINSEQ_4, FINSEQ_5, FINSEQ_6, GOBOARD7, GOBOARD9, SPPOL_1, SPPOL_2,
      TOPREAL4, REVROT_1, TOPREAL1, SPRECT_3, BOOLMARK, JORDAN5B, AMI_5,
      JORDAN3, JORDAN9, ZFMISC_1, GOBOARD1, SPRECT_1, TARSKI, TOPREAL3,
      FINSEQ_3, RELAT_1, FUNCT_1, TOPREAL5, JORDAN10, GOBOARD2, CARD_1, CARD_2,
      YELLOW_6, PRE_CIRC, ENUMSET1, GROUP_5, SPRECT_4, JORDAN1B, SPRECT_5,
      SCMFSA_7, JORDAN5D, JORDAN1E, PARTFUN2, JORDAN1F, JORDAN5C, REALSET1,
      XBOOLE_0, XBOOLE_1, SQUARE_1, XCMPLX_0, XCMPLX_1, RLVECT_1;
 schemes BINARITH;

begin

  reserve n for Nat;

  definition
   cluster trivial FinSequence;
   existence
   proof
    take {};
    thus thesis;
   end;
  end;

  theorem Th1:
   for f be trivial FinSequence holds
    f is empty or ex x be set st f = <*x*>
   proof
    let f be trivial FinSequence;
    assume f is non empty;
    then consider x be set such that
     A1: f = {x} by REALSET1:def 12;
      x in {x} by TARSKI:def 1;
    then consider y,z be set such that
     A2: x = [y,z] by A1,RELAT_1:def 1;
    take z;
    A3: 1 in dom f by A1,FINSEQ_5:6;
      dom f = {y} by A1,A2,RELAT_1:23;
    then 1 = y by A3,TARSKI:def 1;
    hence f = <*z*> by A1,A2,FINSEQ_1:def 5;
   end;

  definition
   let p be non trivial FinSequence;
   cluster Rev p -> non trivial;
   coherence
   proof
    assume A1: Rev p is trivial;
    per cases by A1,Th1;
     suppose Rev p is empty;
      hence contradiction;
     suppose ex x be set st Rev p = <*x*>;
      then consider x be set such that
       A2: Rev p = <*x*>;
        p = Rev <*x*> by A2,FINSEQ_6:29
         .= <*x*> by FINSEQ_5:63;
      hence contradiction;
   end;
  end;

  theorem Th2:
   for D be non empty set
   for f be FinSequence of D
   for G be Matrix of D
   for p be set holds
    f is_sequence_on G implies f-:p is_sequence_on G
   proof
    let D be non empty set;
    let f be FinSequence of D;
    let G be Matrix of D;
    let p be set;
    assume f is_sequence_on G;
    then f|(p..f) is_sequence_on G by GOBOARD1:38;
    hence f-:p is_sequence_on G by FINSEQ_5:def 1;
   end;

  theorem Th3:
   for D be non empty set
   for f be FinSequence of D
   for G be Matrix of D
   for p be Element of D st p in rng f holds
    f is_sequence_on G implies f:-p is_sequence_on G
   proof
    let D be non empty set;
    let f be FinSequence of D;
    let G be Matrix of D;
    let p be Element of D;
    assume that
     A1: p in rng f and
     A2: f is_sequence_on G;
    consider i be Nat such that
       i+1 = p..f and
     A3: f:-p = f/^i by A1,FINSEQ_5:52;
    thus f:-p is_sequence_on G by A2,A3,JORDAN8:5;
   end;

  theorem Th4:
   for C be compact connected non vertical non horizontal
    Subset of TOP-REAL 2 holds
     Upper_Seq(C,n) is_sequence_on Gauge(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
      Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1;
    then Rotate(Cage(C,n),W-min L~Cage(C,n)) is_sequence_on Gauge(C,n)
                                                              by REVROT_1:34;
    then Rotate(Cage(C,n),W-min L~Cage(C,n))-:E-max L~Cage(C,n)
                                        is_sequence_on Gauge(C,n) by Th2;
    hence thesis by JORDAN1E:def 1;
   end;

  theorem Th5:
   for C be compact connected non vertical non horizontal
    Subset of TOP-REAL 2 holds
     Lower_Seq(C,n) is_sequence_on Gauge(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    A1: E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:50;
      W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:47;
    then A2: E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                           by A1,FINSEQ_6:96;
      Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1;
    then Rotate(Cage(C,n),W-min L~Cage(C,n)) is_sequence_on Gauge(C,n)
                                                              by REVROT_1:34;
    then Rotate(Cage(C,n),W-min L~Cage(C,n)):-E-max L~Cage(C,n)
                                     is_sequence_on Gauge(C,n) by A2,Th3;
    hence thesis by JORDAN1E:def 2;
   end;

  definition
   let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
   let n be Nat;
   cluster Upper_Seq(C,n) -> standard;
   coherence
   proof
      Upper_Seq(C,n) is_sequence_on Gauge(C,n) by Th4;
    hence thesis by JORDAN8:7;
   end;
   cluster Lower_Seq(C,n) -> standard;
   coherence
   proof
      Lower_Seq(C,n) is_sequence_on Gauge(C,n) by Th5;
    hence thesis by JORDAN8:7;
   end;
  end;

  theorem Th6:
   for G be Y_equal-in-column Y_increasing-in-line Matrix of TOP-REAL 2
   for i1,i2,j1,j2 be Nat st [i1,j1] in Indices G & [i2,j2] in Indices G holds
    G*(i1,j1)`2 = G*(i2,j2)`2 implies j1 = j2
   proof
    let G be Y_equal-in-column Y_increasing-in-line Matrix of TOP-REAL 2;
    let i1,i2,j1,j2 be Nat;
    assume that
     A1: [i1,j1] in Indices G and
     A2: [i2,j2] in Indices G and
     A3: G*(i1,j1)`2 = G*(i2,j2)`2 and
     A4: j1 <> j2;
    A5: 1 <= i1 & i1 <= len G & 1 <= j1 & j1 <= width G by A1,GOBOARD5:1;
    A6: 1 <= i2 & i2 <= len G & 1 <= j2 & j2 <= width G by A2,GOBOARD5:1;
    then A7: G*(i1,j2)`2 = G*(1,j2)`2 by A5,GOBOARD5:2
       .= G*(i2,j2)`2 by A6,GOBOARD5:2;
      j1 < j2 or j1 > j2 by A4,REAL_1:def 5;
    hence contradiction by A3,A5,A6,A7,GOBOARD5:5;
   end;

  theorem Th7:
   for G be X_equal-in-line X_increasing-in-column Matrix of TOP-REAL 2
   for i1,i2,j1,j2 be Nat st [i1,j1] in Indices G & [i2,j2] in Indices G holds
    G*(i1,j1)`1 = G*(i2,j2)`1 implies i1 = i2
   proof
    let G be X_equal-in-line X_increasing-in-column Matrix of TOP-REAL 2;
    let i1,i2,j1,j2 be Nat;
    assume that
     A1: [i1,j1] in Indices G and
     A2: [i2,j2] in Indices G and
     A3: G*(i1,j1)`1 = G*(i2,j2)`1 and
     A4: i1 <> i2;
    A5: 1 <= i1 & i1 <= len G & 1 <= j1 & j1 <= width G by A1,GOBOARD5:1;
    A6: 1 <= i2 & i2 <= len G & 1 <= j2 & j2 <= width G by A2,GOBOARD5:1;
    A7: G*(i1,j1)`1 = G*(i1,1)`1 by A5,GOBOARD5:3
       .= G*(i1,j2)`1 by A5,A6,GOBOARD5:3;
      i1 < i2 or i1 > i2 by A4,REAL_1:def 5;
    hence contradiction by A3,A5,A6,A7,GOBOARD5:4;
   end;

canceled 8;

  theorem Th16:
   for f be standard special unfolded (non trivial FinSequence of TOP-REAL 2)
    st (f/.1 <> N-min L~f & f/.len f <> N-min L~f) or
       (f/.1 <> N-max L~f & f/.len f <> N-max L~f) holds
     (N-min L~f)`1 < (N-max L~f)`1
   proof
    let f be standard special unfolded (non trivial FinSequence of TOP-REAL 2);
    set p = N-min L~f;
    set i = p..f;
    assume A1: (f/.1 <> N-min L~f & f/.len f <> N-min L~f) or
               (f/.1 <> N-max L~f & f/.len f <> N-max L~f);
    A2: len f >= 2 by SPPOL_1:2;
    A3: p in rng f by SPRECT_2:43;
    then A4: i in dom f by FINSEQ_4:30;
    A5: p = f.i by A3,FINSEQ_4:29
         .= f/.i by A4,FINSEQ_4:def 4;
    A6: 1 <= i & i <= len f by A4,FINSEQ_3:27;
    A7: p`2 = N-bound L~f by PSCOMP_1:94;
    per cases by A6,REAL_1:def 5;
    suppose A8: i = 1;
       p`2 = (N-max L~f)`2 by PSCOMP_1:95;
     then A9: p`1 <> (N-max L~f)`1 by A1,A5,A8,TOPREAL3:11;
       p`1 <= (N-max L~f)`1 by PSCOMP_1:97;
     hence p`1 < (N-max L~f)`1 by A9,REAL_1:def 5;
    suppose A10: i = len f;
       p`2 = (N-max L~f)`2 by PSCOMP_1:95;
     then A11: p`1 <> (N-max L~f)`1 by A1,A5,A10,TOPREAL3:11;
       p`1 <= (N-max L~f)`1 by PSCOMP_1:97;
     hence p`1 < (N-max L~f)`1 by A11,REAL_1:def 5;
    suppose that
      A12: 1 < i and
      A13: i < len f;
     A14: i-'1+1 = i by A12,AMI_5:4;
     then A15: i-'1 >= 1 by A12,NAT_1:38;
       i-'1 <= i by A14,NAT_1:29;
     then i-'1 <= len f by A13,AXIOMS:22;
     then A16: i-'1 in dom f by A15,FINSEQ_3:27;
     A17: p in LSeg(f,i-'1) by A5,A13,A14,A15,TOPREAL1:27;
     A18: i+1 <= len f by A13,NAT_1:38;
     then A19: p in LSeg(f,i) by A5,A12,TOPREAL1:27;
     A20: p <> f/.(i-'1) by A4,A5,A14,A16,GOBOARD7:31;
       i+1 >= 1 by NAT_1:29;
     then A21: i+1 in dom f by A18,FINSEQ_3:27;
     then A22: p <> f/.(i+1) by A4,A5,GOBOARD7:31;
     A23: f/.(i-'1) in LSeg(f,i-'1) by A13,A14,A15,TOPREAL1:27;
     A24: f/.(i+1) in LSeg(f,i) by A12,A18,TOPREAL1:27;
     A25: f/.(i-'1) in L~f by A2,A16,GOBOARD1:16;
     A26: f/.(i+1) in L~f by A2,A21,GOBOARD1:16;
     A27: not(LSeg(f,i-'1) is vertical & LSeg(f,i) is vertical)
     proof
      assume that
       A28: LSeg(f,i-'1) is vertical and
       A29: LSeg(f,i) is vertical;
      A30: i-'1+1+1 = i-'1+(1+1) by XCMPLX_1:1;
      A31: LSeg(f,i) = LSeg(f/.i,f/.(i+1)) by A12,A18,TOPREAL1:def 5;
      A32: LSeg(f,i-'1) = LSeg(f/.i,f/.(i-'1)) by A13,A14,A15,TOPREAL1:def 5;
      A33: p`1 = (f/.(i+1))`1 & p`1 = (f/.(i-'1))`1
                                   by A17,A19,A23,A24,A28,A29,SPPOL_1:def 3;
      A34: p`2 >= (f/.(i+1))`2 & p`2 >= (f/.(i-'1))`2
                                                   by A7,A25,A26,PSCOMP_1:71;
        (f/.(i+1))`2 <= (f/.(i-'1))`2 or (f/.(i+1))`2 >= (f/.(i-'1))`2;
      then f/.(i-'1) in LSeg(f,i) or f/.(i+1) in LSeg(f,i-'1)
                                            by A5,A31,A32,A33,A34,GOBOARD7:8;
      then f/.(i-'1) in LSeg(f,i-'1) /\ LSeg(f,i) or
           f/.(i+1) in LSeg(f,i-'1) /\ LSeg(f,i) by A23,A24,XBOOLE_0:def 3;
      then LSeg(f,i-'1) /\ LSeg(f,i) <> {f/.i} by A5,A20,A22,TARSKI:def 1;
      hence contradiction by A14,A15,A18,A30,TOPREAL1:def 8;
     end;
       now per cases by A27,SPPOL_1:41;
      suppose LSeg(f,i-'1) is horizontal;
       then A35: p`2 = (f/.(i-'1))`2 by A17,A23,SPPOL_1:def 2;
       then A36: (f/.(i-'1))`1 <> p`1 by A20,TOPREAL3:11;
       A37: f/.(i-'1) in N-most L~f by A7,A25,A35,SPRECT_2:14;
       then (f/.(i-'1))`1 >= p`1 by PSCOMP_1:98;
       then A38: (f/.(i-'1))`1 > p`1 by A36,REAL_1:def 5;
         (f/.(i-'1))`1 <= (N-max L~f)`1 by A37,PSCOMP_1:98;
       hence thesis by A38,AXIOMS:22;
      suppose LSeg(f,i) is horizontal;
       then A39: p`2 = (f/.(i+1))`2 by A19,A24,SPPOL_1:def 2;
       then A40: (f/.(i+1))`1 <> p`1 by A22,TOPREAL3:11;
       A41: f/.(i+1) in N-most L~f by A7,A26,A39,SPRECT_2:14;
       then (f/.(i+1))`1 >= p`1 by PSCOMP_1:98;
       then A42: (f/.(i+1))`1 > p`1 by A40,REAL_1:def 5;
         (f/.(i+1))`1 <= (N-max L~f)`1 by A41,PSCOMP_1:98;
       hence thesis by A42,AXIOMS:22;
     end;
     hence thesis;
   end;

  theorem
     for f be standard special unfolded (non trivial FinSequence of TOP-REAL 2)
    st (f/.1 <> N-min L~f & f/.len f <> N-min L~f) or
       (f/.1 <> N-max L~f & f/.len f <> N-max L~f) holds
     N-min L~f <> N-max L~f
   proof
    let f be standard special unfolded (non trivial FinSequence of TOP-REAL 2);
    assume (f/.1 <> N-min L~f & f/.len f <> N-min L~f) or
           (f/.1 <> N-max L~f & f/.len f <> N-max L~f);
    then (N-min L~f)`1 < (N-max L~f)`1 by Th16;
    hence thesis;
   end;

  theorem Th18:
   for f be standard special unfolded (non trivial FinSequence of TOP-REAL 2)
    st (f/.1 <> S-min L~f & f/.len f <> S-min L~f) or
       (f/.1 <> S-max L~f & f/.len f <> S-max L~f) holds
     (S-min L~f)`1 < (S-max L~f)`1
   proof
    let f be standard special unfolded (non trivial FinSequence of TOP-REAL 2);
    set p = S-min L~f;
    set i = p..f;
    assume A1: (f/.1 <> S-min L~f & f/.len f <> S-min L~f) or
               (f/.1 <> S-max L~f & f/.len f <> S-max L~f);
    A2: len f >= 2 by SPPOL_1:2;
    A3: p in rng f by SPRECT_2:45;
    then A4: i in dom f by FINSEQ_4:30;
    A5: p = f.i by A3,FINSEQ_4:29
         .= f/.i by A4,FINSEQ_4:def 4;
    A6: 1 <= i & i <= len f by A4,FINSEQ_3:27;
    A7: p`2 = S-bound L~f by PSCOMP_1:114;
    per cases by A6,REAL_1:def 5;
    suppose A8: i = 1;
       p`2 = (S-max L~f)`2 by PSCOMP_1:115;
     then A9: p`1 <> (S-max L~f)`1 by A1,A5,A8,TOPREAL3:11;
       p`1 <= (S-max L~f)`1 by PSCOMP_1:117;
     hence p`1 < (S-max L~f)`1 by A9,REAL_1:def 5;
    suppose A10: i = len f;
       p`2 = (S-max L~f)`2 by PSCOMP_1:115;
     then A11: p`1 <> (S-max L~f)`1 by A1,A5,A10,TOPREAL3:11;
       p`1 <= (S-max L~f)`1 by PSCOMP_1:117;
     hence p`1 < (S-max L~f)`1 by A11,REAL_1:def 5;
    suppose that
      A12: 1 < i and
      A13: i < len f;
     A14: i-'1+1 = i by A12,AMI_5:4;
     then A15: i-'1 >= 1 by A12,NAT_1:38;
       i-'1 <= i by A14,NAT_1:29;
     then i-'1 <= len f by A13,AXIOMS:22;
     then A16: i-'1 in dom f by A15,FINSEQ_3:27;
     A17: p in LSeg(f,i-'1) by A5,A13,A14,A15,TOPREAL1:27;
     A18: i+1 <= len f by A13,NAT_1:38;
     then A19: p in LSeg(f,i) by A5,A12,TOPREAL1:27;
     A20: p <> f/.(i-'1) by A4,A5,A14,A16,GOBOARD7:31;
       i+1 >= 1 by NAT_1:29;
     then A21: i+1 in dom f by A18,FINSEQ_3:27;
     then A22: p <> f/.(i+1) by A4,A5,GOBOARD7:31;
     A23: f/.(i-'1) in LSeg(f,i-'1) by A13,A14,A15,TOPREAL1:27;
     A24: f/.(i+1) in LSeg(f,i) by A12,A18,TOPREAL1:27;
     A25: f/.(i-'1) in L~f by A2,A16,GOBOARD1:16;
     A26: f/.(i+1) in L~f by A2,A21,GOBOARD1:16;
     A27: not(LSeg(f,i-'1) is vertical & LSeg(f,i) is vertical)
     proof
      assume that
       A28: LSeg(f,i-'1) is vertical and
       A29: LSeg(f,i) is vertical;
      A30: i-'1+1+1 = i-'1+(1+1) by XCMPLX_1:1;
      A31: LSeg(f,i) = LSeg(f/.i,f/.(i+1)) by A12,A18,TOPREAL1:def 5;
      A32: LSeg(f,i-'1) = LSeg(f/.i,f/.(i-'1)) by A13,A14,A15,TOPREAL1:def 5;
      A33: p`1 = (f/.(i+1))`1 & p`1 = (f/.(i-'1))`1
                                   by A17,A19,A23,A24,A28,A29,SPPOL_1:def 3;
      A34: p`2 <= (f/.(i+1))`2 & p`2 <= (f/.(i-'1))`2
                                                   by A7,A25,A26,PSCOMP_1:71;
        (f/.(i+1))`2 <= (f/.(i-'1))`2 or (f/.(i+1))`2 >= (f/.(i-'1))`2;
      then f/.(i-'1) in LSeg(f,i) or f/.(i+1) in LSeg(f,i-'1)
                                            by A5,A31,A32,A33,A34,GOBOARD7:8;
      then f/.(i-'1) in LSeg(f,i-'1) /\ LSeg(f,i) or
           f/.(i+1) in LSeg(f,i-'1) /\ LSeg(f,i) by A23,A24,XBOOLE_0:def 3;
      then LSeg(f,i-'1) /\ LSeg(f,i) <> {f/.i} by A5,A20,A22,TARSKI:def 1;
      hence contradiction by A14,A15,A18,A30,TOPREAL1:def 8;
     end;
       now per cases by A27,SPPOL_1:41;
      suppose LSeg(f,i-'1) is horizontal;
       then A35: p`2 = (f/.(i-'1))`2 by A17,A23,SPPOL_1:def 2;
       then A36: (f/.(i-'1))`1 <> p`1 by A20,TOPREAL3:11;
       A37: f/.(i-'1) in S-most L~f by A7,A25,A35,SPRECT_2:15;
       then (f/.(i-'1))`1 >= p`1 by PSCOMP_1:118;
       then A38: (f/.(i-'1))`1 > p`1 by A36,REAL_1:def 5;
         (f/.(i-'1))`1 <= (S-max L~f)`1 by A37,PSCOMP_1:118;
       hence thesis by A38,AXIOMS:22;
      suppose LSeg(f,i) is horizontal;
       then A39: p`2 = (f/.(i+1))`2 by A19,A24,SPPOL_1:def 2;
       then A40: (f/.(i+1))`1 <> p`1 by A22,TOPREAL3:11;
       A41: f/.(i+1) in S-most L~f by A7,A26,A39,SPRECT_2:15;
       then (f/.(i+1))`1 >= p`1 by PSCOMP_1:118;
       then A42: (f/.(i+1))`1 > p`1 by A40,REAL_1:def 5;
         (f/.(i+1))`1 <= (S-max L~f)`1 by A41,PSCOMP_1:118;
       hence thesis by A42,AXIOMS:22;
     end;
     hence thesis;
   end;

  theorem
     for f be standard special unfolded (non trivial FinSequence of TOP-REAL 2)
    st (f/.1 <> S-min L~f & f/.len f <> S-min L~f) or
       (f/.1 <> S-max L~f & f/.len f <> S-max L~f) holds
     S-min L~f <> S-max L~f
   proof
    let f be standard special unfolded (non trivial FinSequence of TOP-REAL 2);
    assume (f/.1 <> S-min L~f & f/.len f <> S-min L~f) or
           (f/.1 <> S-max L~f & f/.len f <> S-max L~f);
    then (S-min L~f)`1 < (S-max L~f)`1 by Th18;
    hence thesis;
   end;

  theorem Th20:
   for f be standard special unfolded (non trivial FinSequence of TOP-REAL 2)
    st (f/.1 <> W-min L~f & f/.len f <> W-min L~f) or
       (f/.1 <> W-max L~f & f/.len f <> W-max L~f) holds
     (W-min L~f)`2 < (W-max L~f)`2
   proof
    let f be standard special unfolded (non trivial FinSequence of TOP-REAL 2);
    set p = W-min L~f;
    set i = p..f;
    assume A1: (f/.1 <> W-min L~f & f/.len f <> W-min L~f) or
               (f/.1 <> W-max L~f & f/.len f <> W-max L~f);
    A2: len f >= 2 by SPPOL_1:2;
    A3: p in rng f by SPRECT_2:47;
    then A4: i in dom f by FINSEQ_4:30;
    A5: p = f.i by A3,FINSEQ_4:29
         .= f/.i by A4,FINSEQ_4:def 4;
    A6: 1 <= i & i <= len f by A4,FINSEQ_3:27;
    A7: p`1 = W-bound L~f by PSCOMP_1:84;
    per cases by A6,REAL_1:def 5;
    suppose A8: i = 1;
       p`1 = (W-max L~f)`1 by PSCOMP_1:85;
     then A9: p`2 <> (W-max L~f)`2 by A1,A5,A8,TOPREAL3:11;
       p`2 <= (W-max L~f)`2 by PSCOMP_1:87;
     hence p`2 < (W-max L~f)`2 by A9,REAL_1:def 5;
    suppose A10: i = len f;
       p`1 = (W-max L~f)`1 by PSCOMP_1:85;
     then A11: p`2 <> (W-max L~f)`2 by A1,A5,A10,TOPREAL3:11;
       p`2 <= (W-max L~f)`2 by PSCOMP_1:87;
     hence p`2 < (W-max L~f)`2 by A11,REAL_1:def 5;
    suppose that
      A12: 1 < i and
      A13: i < len f;
     A14: i-'1+1 = i by A12,AMI_5:4;
     then A15: i-'1 >= 1 by A12,NAT_1:38;
       i-'1 <= i by A14,NAT_1:29;
     then i-'1 <= len f by A13,AXIOMS:22;
     then A16: i-'1 in dom f by A15,FINSEQ_3:27;
     A17: p in LSeg(f,i-'1) by A5,A13,A14,A15,TOPREAL1:27;
     A18: i+1 <= len f by A13,NAT_1:38;
     then A19: p in LSeg(f,i) by A5,A12,TOPREAL1:27;
     A20: p <> f/.(i-'1) by A4,A5,A14,A16,GOBOARD7:31;
       i+1 >= 1 by NAT_1:29;
     then A21: i+1 in dom f by A18,FINSEQ_3:27;
     then A22: p <> f/.(i+1) by A4,A5,GOBOARD7:31;
     A23: f/.(i-'1) in LSeg(f,i-'1) by A13,A14,A15,TOPREAL1:27;
     A24: f/.(i+1) in LSeg(f,i) by A12,A18,TOPREAL1:27;
     A25: f/.(i-'1) in L~f by A2,A16,GOBOARD1:16;
     A26: f/.(i+1) in L~f by A2,A21,GOBOARD1:16;
     A27: not(LSeg(f,i-'1) is horizontal & LSeg(f,i) is horizontal)
     proof
      assume that
       A28: LSeg(f,i-'1) is horizontal and
       A29: LSeg(f,i) is horizontal;
      A30: i-'1+1+1 = i-'1+(1+1) by XCMPLX_1:1;
      A31: LSeg(f,i) = LSeg(f/.i,f/.(i+1)) by A12,A18,TOPREAL1:def 5;
      A32: LSeg(f,i-'1) = LSeg(f/.i,f/.(i-'1)) by A13,A14,A15,TOPREAL1:def 5;
      A33: p`2 = (f/.(i+1))`2 & p`2 = (f/.(i-'1))`2
                                   by A17,A19,A23,A24,A28,A29,SPPOL_1:def 2;
      A34: p`1 <= (f/.(i+1))`1 & p`1 <= (f/.(i-'1))`1
                                                   by A7,A25,A26,PSCOMP_1:71;
        (f/.(i+1))`1 <= (f/.(i-'1))`1 or (f/.(i+1))`1 >= (f/.(i-'1))`1;
      then f/.(i-'1) in LSeg(f,i) or f/.(i+1) in LSeg(f,i-'1)
                                            by A5,A31,A32,A33,A34,GOBOARD7:9;
      then f/.(i-'1) in LSeg(f,i-'1) /\ LSeg(f,i) or
           f/.(i+1) in LSeg(f,i-'1) /\ LSeg(f,i) by A23,A24,XBOOLE_0:def 3;
      then LSeg(f,i-'1) /\ LSeg(f,i) <> {f/.i} by A5,A20,A22,TARSKI:def 1;
      hence contradiction by A14,A15,A18,A30,TOPREAL1:def 8;
     end;
       now per cases by A27,SPPOL_1:41;
      suppose LSeg(f,i-'1) is vertical;
       then A35: p`1 = (f/.(i-'1))`1 by A17,A23,SPPOL_1:def 3;
       then A36: (f/.(i-'1))`2 <> p`2 by A20,TOPREAL3:11;
       A37: f/.(i-'1) in W-most L~f by A7,A25,A35,SPRECT_2:16;
       then (f/.(i-'1))`2 >= p`2 by PSCOMP_1:88;
       then A38: (f/.(i-'1))`2 > p`2 by A36,REAL_1:def 5;
         (f/.(i-'1))`2 <= (W-max L~f)`2 by A37,PSCOMP_1:88;
       hence thesis by A38,AXIOMS:22;
      suppose LSeg(f,i) is vertical;
       then A39: p`1 = (f/.(i+1))`1 by A19,A24,SPPOL_1:def 3;
       then A40: (f/.(i+1))`2 <> p`2 by A22,TOPREAL3:11;
       A41: f/.(i+1) in W-most L~f by A7,A26,A39,SPRECT_2:16;
       then (f/.(i+1))`2 >= p`2 by PSCOMP_1:88;
       then A42: (f/.(i+1))`2 > p`2 by A40,REAL_1:def 5;
         (f/.(i+1))`2 <= (W-max L~f)`2 by A41,PSCOMP_1:88;
       hence thesis by A42,AXIOMS:22;
     end;
     hence thesis;
   end;

  theorem
     for f be standard special unfolded (non trivial FinSequence of TOP-REAL 2)
    st (f/.1 <> W-min L~f & f/.len f <> W-min L~f) or
       (f/.1 <> W-max L~f & f/.len f <> W-max L~f) holds
     W-min L~f <> W-max L~f
   proof
    let f be standard special unfolded (non trivial FinSequence of TOP-REAL 2);
    assume (f/.1 <> W-min L~f & f/.len f <> W-min L~f) or
           (f/.1 <> W-max L~f & f/.len f <> W-max L~f);
    then (W-min L~f)`2 < (W-max L~f)`2 by Th20;
    hence thesis;
   end;

  theorem Th22:
   for f be standard special unfolded (non trivial FinSequence of TOP-REAL 2)
    st (f/.1 <> E-min L~f & f/.len f <> E-min L~f) or
       (f/.1 <> E-max L~f & f/.len f <> E-max L~f) holds
     (E-min L~f)`2 < (E-max L~f)`2
   proof
    let f be standard special unfolded (non trivial FinSequence of TOP-REAL 2);
    set p = E-min L~f;
    set i = p..f;
    assume A1: (f/.1 <> E-min L~f & f/.len f <> E-min L~f) or
               (f/.1 <> E-max L~f & f/.len f <> E-max L~f);
    A2: len f >= 2 by SPPOL_1:2;
    A3: p in rng f by SPRECT_2:49;
    then A4: i in dom f by FINSEQ_4:30;
    A5: p = f.i by A3,FINSEQ_4:29
         .= f/.i by A4,FINSEQ_4:def 4;
    A6: 1 <= i & i <= len f by A4,FINSEQ_3:27;
    A7: p`1 = E-bound L~f by PSCOMP_1:104;
    per cases by A6,REAL_1:def 5;
    suppose A8: i = 1;
       p`1 = (E-max L~f)`1 by PSCOMP_1:105;
     then A9: p`2 <> (E-max L~f)`2 by A1,A5,A8,TOPREAL3:11;
       p`2 <= (E-max L~f)`2 by PSCOMP_1:107;
     hence p`2 < (E-max L~f)`2 by A9,REAL_1:def 5;
    suppose A10: i = len f;
       p`1 = (E-max L~f)`1 by PSCOMP_1:105;
     then A11: p`2 <> (E-max L~f)`2 by A1,A5,A10,TOPREAL3:11;
       p`2 <= (E-max L~f)`2 by PSCOMP_1:107;
     hence p`2 < (E-max L~f)`2 by A11,REAL_1:def 5;
    suppose that
      A12: 1 < i and
      A13: i < len f;
     A14: i-'1+1 = i by A12,AMI_5:4;
     then A15: i-'1 >= 1 by A12,NAT_1:38;
       i-'1 <= i by A14,NAT_1:29;
     then i-'1 <= len f by A13,AXIOMS:22;
     then A16: i-'1 in dom f by A15,FINSEQ_3:27;
     A17: p in LSeg(f,i-'1) by A5,A13,A14,A15,TOPREAL1:27;
     A18: i+1 <= len f by A13,NAT_1:38;
     then A19: p in LSeg(f,i) by A5,A12,TOPREAL1:27;
     A20: p <> f/.(i-'1) by A4,A5,A14,A16,GOBOARD7:31;
       i+1 >= 1 by NAT_1:29;
     then A21: i+1 in dom f by A18,FINSEQ_3:27;
     then A22: p <> f/.(i+1) by A4,A5,GOBOARD7:31;
     A23: f/.(i-'1) in LSeg(f,i-'1) by A13,A14,A15,TOPREAL1:27;
     A24: f/.(i+1) in LSeg(f,i) by A12,A18,TOPREAL1:27;
     A25: f/.(i-'1) in L~f by A2,A16,GOBOARD1:16;
     A26: f/.(i+1) in L~f by A2,A21,GOBOARD1:16;
     A27: not(LSeg(f,i-'1) is horizontal & LSeg(f,i) is horizontal)
     proof
      assume that
       A28: LSeg(f,i-'1) is horizontal and
       A29: LSeg(f,i) is horizontal;
      A30: i-'1+1+1 = i-'1+(1+1) by XCMPLX_1:1;
      A31: LSeg(f,i) = LSeg(f/.i,f/.(i+1)) by A12,A18,TOPREAL1:def 5;
      A32: LSeg(f,i-'1) = LSeg(f/.i,f/.(i-'1)) by A13,A14,A15,TOPREAL1:def 5;
      A33: p`2 = (f/.(i+1))`2 & p`2 = (f/.(i-'1))`2
                                   by A17,A19,A23,A24,A28,A29,SPPOL_1:def 2;
      A34: p`1 >= (f/.(i+1))`1 & p`1 >= (f/.(i-'1))`1
                                                   by A7,A25,A26,PSCOMP_1:71;
        (f/.(i+1))`1 <= (f/.(i-'1))`1 or (f/.(i+1))`1 >= (f/.(i-'1))`1;
      then f/.(i-'1) in LSeg(f,i) or f/.(i+1) in LSeg(f,i-'1)
                                            by A5,A31,A32,A33,A34,GOBOARD7:9;
      then f/.(i-'1) in LSeg(f,i-'1) /\ LSeg(f,i) or
           f/.(i+1) in LSeg(f,i-'1) /\ LSeg(f,i) by A23,A24,XBOOLE_0:def 3;
      then LSeg(f,i-'1) /\ LSeg(f,i) <> {f/.i} by A5,A20,A22,TARSKI:def 1;
      hence contradiction by A14,A15,A18,A30,TOPREAL1:def 8;
     end;
       now per cases by A27,SPPOL_1:41;
      suppose LSeg(f,i-'1) is vertical;
       then A35: p`1 = (f/.(i-'1))`1 by A17,A23,SPPOL_1:def 3;
       then A36: (f/.(i-'1))`2 <> p`2 by A20,TOPREAL3:11;
       A37: f/.(i-'1) in E-most L~f by A7,A25,A35,SPRECT_2:17;
       then (f/.(i-'1))`2 >= p`2 by PSCOMP_1:108;
       then A38: (f/.(i-'1))`2 > p`2 by A36,REAL_1:def 5;
         (f/.(i-'1))`2 <= (E-max L~f)`2 by A37,PSCOMP_1:108;
       hence thesis by A38,AXIOMS:22;
      suppose LSeg(f,i) is vertical;
       then A39: p`1 = (f/.(i+1))`1 by A19,A24,SPPOL_1:def 3;
       then A40: (f/.(i+1))`2 <> p`2 by A22,TOPREAL3:11;
       A41: f/.(i+1) in E-most L~f by A7,A26,A39,SPRECT_2:17;
       then (f/.(i+1))`2 >= p`2 by PSCOMP_1:108;
       then A42: (f/.(i+1))`2 > p`2 by A40,REAL_1:def 5;
         (f/.(i+1))`2 <= (E-max L~f)`2 by A41,PSCOMP_1:108;
       hence thesis by A42,AXIOMS:22;
     end;
     hence thesis;
   end;

  theorem
     for f be standard special unfolded (non trivial FinSequence of TOP-REAL 2)
    st (f/.1 <> E-min L~f & f/.len f <> E-min L~f) or
       (f/.1 <> E-max L~f & f/.len f <> E-max L~f) holds
     E-min L~f <> E-max L~f
   proof
    let f be standard special unfolded (non trivial FinSequence of TOP-REAL 2);
    assume (f/.1 <> E-min L~f & f/.len f <> E-min L~f) or
           (f/.1 <> E-max L~f & f/.len f <> E-max L~f);
    then (E-min L~f)`2 < (E-max L~f)`2 by Th22;
    hence thesis;
   end;

  theorem Th24:
   for D be non empty set
   for f be FinSequence of D
   for p,q be Element of D st p in rng f & q in rng f & q..f <= p..f holds
    (f-:p):-q = (f:-q)-:p
   proof
    let D be non empty set;
    let f be FinSequence of D;
    let p,q be Element of D;
    A1: f-:p = f|(p..f) by FINSEQ_5:def 1;
    assume that
     A2: p in rng f and
     A3: q in rng f and
     A4: q..f <= p..f;
    consider i be Nat such that
     A5: i+1 = q..f and
     A6: f:-q = f/^i by A3,FINSEQ_5:52;
      q in rng (f-:p) by A2,A3,A4,FINSEQ_5:49;
    then consider j be Nat such that
     A7: j+1 = q..(f-:p) and
     A8: (f-:p):-q = (f-:p)/^j by FINSEQ_5:52;
    A9: (f:-q)-:p = (f:-q)|(p..(f:-q)) by FINSEQ_5:def 1;
    A10: i < p..f by A4,A5,NAT_1:38;
    then p..f = i + p..(f/^i) by A2,FINSEQ_6:61;
    then A11: p..(f/^i) = p..f-i by XCMPLX_1:26
       .= p..f-'i by A10,SCMFSA_7:3;
      q..(f-:p) = q..f by A2,A3,A4,SPRECT_5:3;
    then i = j by A5,A7,XCMPLX_1:2;
    hence (f-:p):-q = (f:-q)-:p by A1,A6,A8,A9,A11,JORDAN3:21;
   end;

  theorem Th25:
   for C be compact connected non vertical non horizontal Subset of TOP-REAL 2
   for n be Nat holds
    L~(Cage(C,n)-:W-min L~Cage(C,n)) /\ L~(Cage(C,n):-W-min L~Cage(C,n)) =
     {N-min L~Cage(C,n),W-min L~Cage(C,n)}
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let n be Nat;
    set US = Cage(C,n)-:W-min L~Cage(C,n);
    set LS = Cage(C,n):-W-min L~Cage(C,n);
    set f=Cage(C,n);
    set pW=W-min L~Cage(C,n);
    set pN=N-min L~Cage(C,n);
    set pNa=N-max L~Cage(C,n);
    set pSa=S-max L~Cage(C,n);
    set pSi=S-min L~Cage(C,n);
    set pEa=E-max L~Cage(C,n);
    set pEi=E-min L~Cage(C,n);
    A1: N-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:44;
    A2: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:47;
    A3: (W-min L~Cage(C,n))..Cage(C,n) <= (W-min L~Cage(C,n))..Cage(C,n);
      (f:-pW)/.len(f:-pW) = f/.len f by A2,FINSEQ_5:57
       .= f/.1 by FINSEQ_6:def 1
       .= pN by JORDAN9:34;
    then A4: pN in rng (Cage(C,n):-pW) by REVROT_1:3;
    A5: Cage(C,n)-:pW <> {} by A2,FINSEQ_5:50;
      (Cage(C,n):-pW)/.1 = pW by FINSEQ_5:56;
    then A6: W-min L~Cage(C,n) in rng (Cage(C,n):-W-min L~Cage(C,n))
                                                              by FINSEQ_6:46;
      (Cage(C,n)-:pW)/.1 = Cage(C,n)/.1 by A2,FINSEQ_5:47
       .= pN by JORDAN9:34;
    then A7: N-min L~Cage(C,n) in rng (Cage(C,n)-:W-min L~Cage(C,n))
                                                          by A5,FINSEQ_6:46;
    A8: f/.1 = pN by JORDAN9:34;
    then A9: pNa..f <= pEa..f by SPRECT_2:74;
    A10: pEa..f < pEi..f by A8,SPRECT_2:75;
    A11: pEi..f <= pSa..f by A8,SPRECT_2:76;
    A12: pSa..f < pSi..f by A8,SPRECT_2:77;
    A13: pSi..f <= pW..f by A8,SPRECT_2:78;
      pNa..f < pEi..f by A9,A10,AXIOMS:22;
    then pNa..f < pSa..f by A11,AXIOMS:22;
    then pNa..f < pSi..f by A12,AXIOMS:22;
    then pNa..f <= pW..f by A13,AXIOMS:22;
    then A14: pNa in rng (f-:pW) by A1,A2,FINSEQ_5:49;
      len(f-:pW) = pW..f by A2,FINSEQ_5:45;
    then (f-:pW)/.len (f-:pW) = pW by A2,FINSEQ_5:48;
    then A15: pW in rng (Cage(C,n)-:pW) by A5,REVROT_1:3;
      W-max L~f in L~f & pN`2 = N-bound L~f by PSCOMP_1:94,SPRECT_1:15;
    then (W-max L~f)`2 <= pN`2 by PSCOMP_1:71;
    then A16: pN <> pW by SPRECT_2:61;
    then A17: Card {pN,pW} = 2 by CARD_2:76;
      {pN,pW} c= rng LS
    proof
     let x be set;
     assume x in {pN,pW};
     hence x in rng LS by A4,A6,TARSKI:def 2;
    end;
    then A18: Card {pN,pW} c= Card rng LS by CARD_1:27;
      Card rng LS c= Card dom LS by YELLOW_6:3;
    then Card rng LS c= len LS by PRE_CIRC:21;
    then 2 c= len LS by A17,A18,XBOOLE_1:1;
    then len LS >= 2 by CARD_1:56;
    then A19: rng LS c= L~LS by SPPOL_2:18;
    A20: W-min L~Cage(C,n) in rng LS by FINSEQ_6:66;
    A21: pN <> pNa by SPRECT_2:56;
      W-max L~f in L~f & pNa`2 = N-bound L~f by PSCOMP_1:94,SPRECT_1:15;
    then (W-max L~f)`2 <= pNa`2 by PSCOMP_1:71;
    then pNa <> pW by SPRECT_2:61;
    then A22: Card {pN,pNa,pW} = 3 by A16,A21,CARD_2:77;
      {pN,pNa,pW} c= rng US
    proof
     let x be set;
     assume x in {pN,pNa,pW};
     hence x in rng US by A7,A14,A15,ENUMSET1:def 1;
    end;
    then A23: Card {pN,pNa,pW} c= Card rng US by CARD_1:27;
      Card rng US c= Card dom US by YELLOW_6:3;
    then Card rng US c= len US by PRE_CIRC:21;
    then A24: 3 c= len US by A22,A23,XBOOLE_1:1;
    then A25: len US >= 3 by CARD_1:56;
    then len US >= 2 by AXIOMS:22;
    then A26: rng US c= L~US by SPPOL_2:18;
      len US <> 0 by A24,CARD_1:56;
    then A27: US is non empty by FINSEQ_1:25;
    A28: W-min L~Cage(C,n) in rng US by A2,A3,FINSEQ_5:49;
      US/.1 = Cage(C,n)/.1 by A2,FINSEQ_5:47
       .= N-min L~Cage(C,n) by JORDAN9:34;
    then A29: N-min L~Cage(C,n) in rng US by A27,FINSEQ_6:46;
      LS/.(len LS) = Cage(C,n)/.len Cage(C,n) by A2,FINSEQ_5:57
       .= Cage(C,n)/.1 by FINSEQ_6:def 1
       .= N-min L~Cage(C,n) by JORDAN9:34;
    then A30: N-min L~Cage(C,n) in rng LS by REVROT_1:3;
    thus L~US /\ L~LS c=
     {N-min L~Cage(C,n),W-min L~Cage(C,n)}
    proof
     let x be set;
     assume A31: x in L~US /\ L~LS;
     then A32: x in L~US & x in L~LS by XBOOLE_0:def 3;
     reconsider x1=x as Point of TOP-REAL 2 by A31;
     consider i1 be Nat such that
      A33: 1 <= i1 & i1+1 <= len US and
      A34: x1 in LSeg(US,i1) by A32,SPPOL_2:13;
     consider i2 be Nat such that
      A35: 1 <= i2 & i2+1 <= len LS and
      A36: x1 in LSeg(LS,i2) by A32,SPPOL_2:13;
     A37: LSeg(US,i1) = LSeg(f,i1) by A33,SPPOL_2:9;
     set i3=i2-'1;
     A38: i3+1 = i2 by A35,AMI_5:4;
     then A39: LSeg(LS,i2) = LSeg(f,i3+pW..f) by A2,SPPOL_2:10;
     A40: len US = pW..f by A2,FINSEQ_5:45;
     A41: len LS = len f-pW..f+1 by A2,FINSEQ_5:53;
     A42: i2-1 >= 1-1 by A35,REAL_1:49;
       i2 < len f-pW..f+1 by A35,A41,NAT_1:38;
     then i2-1 < len f-pW..f by REAL_1:84;
     then i2-1+pW..f < len f by REAL_1:86;
     then A43: i3+pW..f < len f by A42,BINARITH:def 3;
     A44: i3 >= 0 by NAT_1:18;
     A45: pW..f >= 1 by A2,FINSEQ_4:31;
       i3+1 < len f-pW..f+1 by A35,A38,A41,NAT_1:38;
     then i3 < len f-pW..f by REAL_1:55;
     then A46: i3+pW..f < len f by REAL_1:86;
     A47: i1+1 < pW..f+1 by A33,A40,NAT_1:38;
       1+pW..f <= i3+1+pW..f by A35,A38,REAL_1:55;
     then i1+1 < i3+1+pW..f by A47,AXIOMS:22;
     then i1+1 < i3+pW..f+1 by XCMPLX_1:1;
     then A48: i1+1 <= i3+pW..f by NAT_1:38;
     A49: i3+pW..f+1 <= len f by A46,NAT_1:38;
     A50: len f > 4 by GOBOARD7:36;
     A51: i3+pW..f >= 0 & i1 >= 0 by NAT_1:18;
     A52: pW..f-'1+1 = pW..f by A45,AMI_5:4;
     assume A53: not x in {N-min L~Cage(C,n),W-min L~Cage(C,n)};
       now per cases by A33,A48,REAL_1:def 5;
      suppose i1+1 < i3+pW..f & i1 > 1;
       then LSeg(f,i1) misses LSeg(f,i3+pW..f) by A46,GOBOARD5:def 4;
       then LSeg(f,i1) /\ LSeg(f,i3+pW..f) = {} by XBOOLE_0:def 7;
       hence contradiction by A34,A36,A37,A39,XBOOLE_0:def 3;
      suppose A54: i1 = 1;
         i3+pW..f >= 0+3 by A25,A40,A44,REAL_1:55;
       then A55: i1+1 < i3+pW..f by A54,AXIOMS:22;
         now per cases by A49,REAL_1:def 5;
        suppose i3+pW..f+1 < len f;
         then LSeg(f,i1) misses LSeg(f,i3+pW..f) by A55,GOBOARD5:def 4;
         then LSeg(f,i1) /\ LSeg(f,i3+pW..f) = {} by XBOOLE_0:def 7;
         hence contradiction by A34,A36,A37,A39,XBOOLE_0:def 3;
        suppose i3+pW..f+1 = len f;
         then i3+pW..f = len f-1 by XCMPLX_1:26;
         then i3+pW..f = len f-'1 by A51,BINARITH:def 3;
         then LSeg(f,i1) /\ LSeg(f,i3+pW..f) = {f/.1} by A50,A54,REVROT_1:30;
         then x1 in {f/.1} by A34,A36,A37,A39,XBOOLE_0:def 3;
         then x1 = f/.1 by TARSKI:def 1
            .= pN by JORDAN9:34;
         hence contradiction by A53,TARSKI:def 2;
       end;
       hence contradiction;
      suppose A56: i1+1 = i3+pW..f;
         0+pW..f <= i3+pW..f by A44,REAL_1:55;
       then pW..f = i1+1 by A33,A40,A56,AXIOMS:21;
       then i1 = pW..f-1 by XCMPLX_1:26;
       then A57: i1 = pW..f-'1 by A51,BINARITH:def 3;
         i3+pW..f >= pW..f by NAT_1:29;
       then pW..f < len f by A43,AXIOMS:22;
       then pW..f+1 <= len f by NAT_1:38;
       then pW..f-'1 + (1+1) <= len f by A52,XCMPLX_1:1;
       then LSeg(f,i1) /\ LSeg(f,i3+(pW..f)) = {f/.(pW..f)}
                                          by A33,A52,A56,A57,TOPREAL1:def 8;
       then x1 in {f/.(pW..f)} by A34,A36,A37,A39,XBOOLE_0:def 3;
       then x1 = f/.(pW..f) by TARSKI:def 1
          .= pW by A2,FINSEQ_5:41;
       hence contradiction by A53,TARSKI:def 2;
     end;
     hence contradiction;
    end;
    thus {N-min L~Cage(C,n),W-min L~Cage(C,n)} c= L~US /\ L~LS
    proof
     let x be set;
     assume A58: x in {N-min L~Cage(C,n),W-min L~Cage(C,n)};
     per cases by A58,TARSKI:def 2;
      suppose x = N-min L~Cage(C,n);
       hence x in L~US /\ L~LS by A19,A26,A29,A30,XBOOLE_0:def 3;
      suppose x = W-min L~Cage(C,n);
       hence x in L~US /\ L~LS by A19,A20,A26,A28,XBOOLE_0:def 3;
    end;
   end;

  theorem Th26:
   for C be compact connected non vertical non horizontal
    Subset of TOP-REAL 2 holds
     Lower_Seq(C,n) = Rotate(Cage(C,n),E-max L~Cage(C,n))-:W-min L~Cage(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    set Nmi = N-min L~Cage(C,n);
    set Nma = N-max L~Cage(C,n);
    set Wmi = W-min L~Cage(C,n);
    set Wma = W-max L~Cage(C,n);
    set Ema = E-max L~Cage(C,n);
    set Emi = E-min L~Cage(C,n);
    set Sma = S-max L~Cage(C,n);
    set Smi = S-min L~Cage(C,n);
    set RotWmi = Rotate(Cage(C,n),Wmi);
    set RotEma = Rotate(Cage(C,n),Ema);
    A1: Cage(C,n)/.1 = Nmi by JORDAN9:34;
    A2: Nmi in rng Cage(C,n) by SPRECT_2:43;
    A3: Wmi in rng Cage(C,n) by SPRECT_2:47;
    A4: Ema in rng Cage(C,n) by SPRECT_2:50;
    A5: Nmi..Cage(C,n) < Nma..Cage(C,n) by A1,SPRECT_2:72;
    A6: Nma..Cage(C,n) <= Ema..Cage(C,n) by A1,SPRECT_2:74;
    then A7: Nmi..Cage(C,n) < Ema..Cage(C,n) by A5,AXIOMS:22;
    A8: Ema..Cage(C,n) < Emi..Cage(C,n) by A1,SPRECT_2:75;
    A9: Emi..Cage(C,n) <= Sma..Cage(C,n) by A1,SPRECT_2:76;
    A10: Sma..Cage(C,n) < Smi..Cage(C,n) by A1,SPRECT_2:77;
    A11: Smi..Cage(C,n) <= Wmi..Cage(C,n) by A1,SPRECT_2:78;
      Ema..Cage(C,n) < Sma..Cage(C,n) by A8,A9,AXIOMS:22;
    then Ema..Cage(C,n) < Smi..Cage(C,n) by A10,AXIOMS:22;
    then A12: Ema..Cage(C,n) < Wmi..Cage(C,n) by A11,AXIOMS:22;
    then A13: Ema in rng (Cage(C,n)-:Wmi) by A3,A4,FINSEQ_5:49;
    A14: Cage(C,n)-:Wmi <> {} by A3,FINSEQ_5:50;
      (Cage(C,n)-:Wmi)/.1 = Cage(C,n)/.1 by A3,FINSEQ_5:47
       .= Nmi by JORDAN9:34;
    then A15: Nmi in rng (Cage(C,n)-:Wmi) by A14,FINSEQ_6:46;
      len(Cage(C,n)-:Wmi) = Wmi..Cage(C,n) by A3,FINSEQ_5:45;
    then (Cage(C,n)-:Wmi)/.len (Cage(C,n)-:Wmi) = Wmi by A3,FINSEQ_5:48;
    then A16: Wmi in rng (Cage(C,n)-:Wmi) by A14,REVROT_1:3;
      Wma in L~Cage(C,n) & Nmi`2 = N-bound L~Cage(C,n)
                                                  by PSCOMP_1:94,SPRECT_1:15;
    then Wma`2 <= Nmi`2 by PSCOMP_1:71;
    then Nmi <> Wmi by SPRECT_2:61;
    then A17: Card {Nmi,Wmi} = 2 by CARD_2:76;
      {Nmi,Wmi} c= rng (Cage(C,n)-:Wmi)
    proof
     let x be set;
     assume x in {Nmi,Wmi};
     hence x in rng (Cage(C,n)-:Wmi) by A15,A16,TARSKI:def 2;
    end;
    then A18: Card {Nmi,Wmi} c= Card rng (Cage(C,n)-:Wmi) by CARD_1:27;
      Card rng (Cage(C,n)-:Wmi) c= Card dom (Cage(C,n)-:Wmi) by YELLOW_6:3;
    then Card rng (Cage(C,n)-:Wmi) c= len (Cage(C,n)-:Wmi) by PRE_CIRC:21;
    then 2 c= len (Cage(C,n)-:Wmi) by A17,A18,XBOOLE_1:1;
    then len (Cage(C,n)-:Wmi) >= 2 by CARD_1:56;
    then A19: rng (Cage(C,n)-:Wmi) c= L~(Cage(C,n)-:Wmi) by SPPOL_2:18;
    A20: Nmi..Cage(C,n) < Wmi..Cage(C,n) by A7,A12,AXIOMS:22;
    then A21: Nmi in rng (Cage(C,n)-:Wmi) by A2,A3,FINSEQ_5:49;
    A22: Ema..(Cage(C,n)-:Wmi) <> 1
    proof
     assume A23: Ema..(Cage(C,n)-:Wmi) = 1;
       Nmi..(Cage(C,n)-:Wmi) = Nmi..Cage(C,n) by A2,A3,A20,SPRECT_5:3
        .= 1 by A1,FINSEQ_6:47;
     hence contradiction by A5,A6,A13,A21,A23,FINSEQ_5:10;
    end;
    then A24: Ema in rng (Cage(C,n)-:Wmi/^1) by A13,FINSEQ_6:83;
    A25: Nmi`1 < Nma`1 by SPRECT_2:55;
      Nma`1 <= (NE-corner L~Cage(C,n))`1 by PSCOMP_1:97;
    then Nma`1 <= E-bound L~Cage(C,n) by PSCOMP_1:76;
    then A26: Nmi <> Ema by A25,PSCOMP_1:104;
      not Ema in rng (Cage(C,n):-Wmi)
    proof
     assume A27: Ema in rng (Cage(C,n):-Wmi);
       (Cage(C,n):-Wmi)/.len(Cage(C,n):-Wmi) = Cage(C,n)/.len Cage(C,n)
                                                            by A3,FINSEQ_5:57
        .= Cage(C,n)/.1 by FINSEQ_6:def 1
        .= Nmi by JORDAN9:34;
     then A28: Nmi in rng (Cage(C,n):-Wmi) by REVROT_1:3;
       (Cage(C,n):-Wmi)/.1 = Wmi by FINSEQ_5:56;
     then A29: Wmi in rng (Cage(C,n):-Wmi) by FINSEQ_6:46;
       Wma in L~Cage(C,n) & Nmi`2 = N-bound L~Cage(C,n)
                                                  by PSCOMP_1:94,SPRECT_1:15;
     then Wma`2 <= Nmi`2 by PSCOMP_1:71;
     then Nmi <> Wmi by SPRECT_2:61;
     then A30: Card {Nmi,Wmi} = 2 by CARD_2:76;
       {Nmi,Wmi} c= rng (Cage(C,n):-Wmi)
     proof
      let x be set;
      assume x in {Nmi,Wmi};
      hence x in rng (Cage(C,n):-Wmi) by A28,A29,TARSKI:def 2;
     end;
     then A31: Card {Nmi,Wmi} c= Card rng (Cage(C,n):-Wmi) by CARD_1:27;
       Card rng (Cage(C,n):-Wmi) c= Card dom (Cage(C,n):-Wmi) by YELLOW_6:3;
     then Card rng (Cage(C,n):-Wmi) c= len (Cage(C,n):-Wmi) by PRE_CIRC:21;
     then 2 c= len (Cage(C,n):-Wmi) by A30,A31,XBOOLE_1:1;
     then len (Cage(C,n):-Wmi) >= 2 by CARD_1:56;
     then rng (Cage(C,n):-Wmi) c= L~(Cage(C,n):-Wmi) by SPPOL_2:18;
     then Ema in L~(Cage(C,n)-:Wmi) /\ L~(Cage(C,n):-Wmi)
                                                   by A13,A19,A27,XBOOLE_0:def
3;
     then Ema in {Nmi,Wmi} by Th25;
     then Ema = Wmi by A26,TARSKI:def 2;
     hence contradiction by TOPREAL5:25;
    end;
    then A32: Ema in rng (Cage(C,n)-:Wmi/^1) \ rng (Cage(C,n):-Wmi)
                                                          by A24,XBOOLE_0:def 4
;
    A33: Wmi in rng (Cage(C,n):-Ema) by A3,A4,A12,FINSEQ_6:67;
      RotWmi:-Ema = ((Cage(C,n):-Wmi)^((Cage(C,n)-:Wmi)/^1)):-Ema
                                                        by A3,FINSEQ_6:def 2
       .= ((Cage(C,n)-:Wmi)/^1):-Ema by A32,FINSEQ_6:70
       .= (Cage(C,n)-:Wmi):-Ema by A13,A22,FINSEQ_6:89
       .= (Cage(C,n):-Ema)-:Wmi by A3,A4,A12,Th24
       .= ((Cage(C,n):-Ema)^((Cage(C,n)-:Ema)/^1))-:Wmi by A33,FINSEQ_6:71
       .= RotEma-:Wmi by A4,FINSEQ_6:def 2;
    hence Lower_Seq(C,n) = Rotate(Cage(C,n),E-max L~Cage(C,n))-:
                                        W-min L~Cage(C,n) by JORDAN1E:def 2;
   end;

  theorem Th27:
   for C be compact non vertical non horizontal Subset of TOP-REAL 2 holds
    (W-min L~Cage(C,n))..Upper_Seq(C,n) = 1
   proof
    let C be compact non vertical non horizontal Subset of TOP-REAL 2;
      Upper_Seq(C,n)/.1 = W-min L~Cage(C,n) by JORDAN1F:5;
    hence (W-min L~Cage(C,n))..Upper_Seq(C,n) = 1 by FINSEQ_6:47;
   end;

  theorem Th28:
   for C be compact non vertical non horizontal Subset of TOP-REAL 2 holds
    (W-min L~Cage(C,n))..Upper_Seq(C,n) < (W-max L~Cage(C,n))..Upper_Seq(C,n)
   proof
    let C be compact non vertical non horizontal Subset of TOP-REAL 2;
    set Wmi = W-min L~Cage(C,n);
    set Wma = W-max L~Cage(C,n);
    set Nmi = N-min L~Cage(C,n);
    set Nma = N-max L~Cage(C,n);
    set Ema = E-max L~Cage(C,n);
    set Rot = Rotate(Cage(C,n),Wmi);
    A1: Upper_Seq(C,n) = Rot-:Ema by JORDAN1E:def 1;
    A2: L~Rot = L~Cage(C,n) by REVROT_1:33;
      Wmi in rng Cage(C,n) by SPRECT_2:47;
    then A3: Rot/.1 = Wmi by FINSEQ_6:98;
    A4: Wmi in rng Rot by A2,SPRECT_2:47;
    A5: Ema in rng Rot by A2,SPRECT_2:50;
    A6: Wmi..Rot < Wma..Rot by A2,A3,SPRECT_5:22;
    A7: Wma..Rot <= Nmi..Rot by A2,A3,SPRECT_5:24;
    A8: Nmi..Rot < Nma..Rot by A2,A3,SPRECT_5:25;
    A9: Nma..Rot <= Ema..Rot by A2,A3,SPRECT_5:26;
      Wma..Rot < Nma..Rot by A7,A8,AXIOMS:22;
    then A10: Wma..Rot < Ema..Rot by A9,AXIOMS:22;
    then Wmi..Rot < Ema..Rot by A6,AXIOMS:22;
    then A11: Wmi..(Rot-:Ema) = Wmi..Rot by A4,A5,SPRECT_5:3;
      Wma in rng Rot by A2,SPRECT_2:48;
    hence (W-min L~Cage(C,n))..Upper_Seq(C,n) <
            (W-max L~Cage(C,n))..Upper_Seq(C,n) by A1,A5,A6,A10,A11,SPRECT_5:3;
   end;

  theorem Th29:
   for C be compact non vertical non horizontal Subset of TOP-REAL 2 holds
    (W-max L~Cage(C,n))..Upper_Seq(C,n) <= (N-min L~Cage(C,n))..Upper_Seq(C,n)
   proof
    let C be compact non vertical non horizontal Subset of TOP-REAL 2;
    set Wmi = W-min L~Cage(C,n);
    set Wma = W-max L~Cage(C,n);
    set Nmi = N-min L~Cage(C,n);
    set Nma = N-max L~Cage(C,n);
    set Ema = E-max L~Cage(C,n);
    set Rot = Rotate(Cage(C,n),Wmi);
    A1: Upper_Seq(C,n) = Rot-:Ema by JORDAN1E:def 1;
    A2: L~Rot = L~Cage(C,n) by REVROT_1:33;
      Wmi in rng Cage(C,n) by SPRECT_2:47;
    then A3: Rot/.1 = Wmi by FINSEQ_6:98;
    A4: Wma in rng Rot by A2,SPRECT_2:48;
    A5: Ema in rng Rot by A2,SPRECT_2:50;
    A6: Wma..Rot <= Nmi..Rot by A2,A3,SPRECT_5:24;
    A7: Nmi..Rot < Nma..Rot by A2,A3,SPRECT_5:25;
      Nma..Rot <= Ema..Rot by A2,A3,SPRECT_5:26;
    then A8: Nmi..Rot < Ema..Rot by A7,AXIOMS:22;
    then Wma..Rot < Ema..Rot by A6,AXIOMS:22;
    then A9: Wma..(Rot-:Ema) = Wma..Rot by A4,A5,SPRECT_5:3;
      Nmi in rng Rot by A2,SPRECT_2:43;
    hence (W-max L~Cage(C,n))..Upper_Seq(C,n) <=
            (N-min L~Cage(C,n))..Upper_Seq(C,n) by A1,A5,A6,A8,A9,SPRECT_5:3;
   end;

  theorem Th30:
   for C be compact non vertical non horizontal Subset of TOP-REAL 2 holds
    (N-min L~Cage(C,n))..Upper_Seq(C,n) < (N-max L~Cage(C,n))..Upper_Seq(C,n)
   proof
    let C be compact non vertical non horizontal Subset of TOP-REAL 2;
    set Wmi = W-min L~Cage(C,n);
    set Nmi = N-min L~Cage(C,n);
    set Nma = N-max L~Cage(C,n);
    set Ema = E-max L~Cage(C,n);
    set Rot = Rotate(Cage(C,n),Wmi);
    A1: Upper_Seq(C,n) = Rot-:Ema by JORDAN1E:def 1;
    A2: L~Rot = L~Cage(C,n) by REVROT_1:33;
      Wmi in rng Cage(C,n) by SPRECT_2:47;
    then A3: Rot/.1 = Wmi by FINSEQ_6:98;
    A4: Nmi in rng Rot by A2,SPRECT_2:43;
    A5: Ema in rng Rot by A2,SPRECT_2:50;
    A6: Nmi..Rot < Nma..Rot by A2,A3,SPRECT_5:25;
    A7: Nma..Rot <= Ema..Rot by A2,A3,SPRECT_5:26;
    then Nmi..Rot < Ema..Rot by A6,AXIOMS:22;
    then A8: Nmi..(Rot-:Ema) = Nmi..Rot by A4,A5,SPRECT_5:3;
      Nma in rng Rot by A2,SPRECT_2:44;
    hence (N-min L~Cage(C,n))..Upper_Seq(C,n) <
          (N-max L~Cage(C,n))..Upper_Seq(C,n) by A1,A5,A6,A7,A8,SPRECT_5:3;
   end;

  theorem Th31:
   for C be compact non vertical non horizontal Subset of TOP-REAL 2 holds
    (N-max L~Cage(C,n))..Upper_Seq(C,n) <= (E-max L~Cage(C,n))..Upper_Seq(C,n)
   proof
    let C be compact non vertical non horizontal Subset of TOP-REAL 2;
    set Wmi = W-min L~Cage(C,n);
    set Nma = N-max L~Cage(C,n);
    set Ema = E-max L~Cage(C,n);
    set Rot = Rotate(Cage(C,n),Wmi);
    A1: Upper_Seq(C,n) = Rot-:Ema by JORDAN1E:def 1;
    A2: L~Rot = L~Cage(C,n) by REVROT_1:33;
      Wmi in rng Cage(C,n) by SPRECT_2:47;
    then A3: Rot/.1 = Wmi by FINSEQ_6:98;
    A4: Nma in rng Rot by A2,SPRECT_2:44;
    A5: Ema in rng Rot by A2,SPRECT_2:50;
    A6: Nma..Rot <= Ema..Rot by A2,A3,SPRECT_5:26;
    then Nma..(Rot-:Ema) = Nma..Rot by A4,A5,SPRECT_5:3;
    hence thesis by A1,A5,A6,SPRECT_5:3;
   end;

  theorem Th32:
   for C be compact non vertical non horizontal Subset of TOP-REAL 2 holds
    (E-max L~Cage(C,n))..Upper_Seq(C,n) = len Upper_Seq(C,n)
   proof
    let C be compact non vertical non horizontal Subset of TOP-REAL 2;
    A1: Upper_Seq(C,n) = Rotate(Cage(C,n),W-min L~Cage(C,n))-:E-max L~Cage(C,n)
                                                          by JORDAN1E:def 1;
    A2: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:47;
      E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:50;
    then A3: E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                           by A2,FINSEQ_6:96;
      (E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)) <=
     (E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n));
    then E-max L~Cage(C,n) in rng Upper_Seq(C,n) by A1,A3,FINSEQ_5:49;
    then A4: Upper_Seq(C,n) just_once_values E-max L~Cage(C,n) by FINSEQ_4:10;
      Upper_Seq(C,n)/.len Upper_Seq(C,n) = E-max L~Cage(C,n) by JORDAN1F:7;
    hence (E-max L~Cage(C,n))..Upper_Seq(C,n) = len Upper_Seq(C,n)
                                                            by A4,REVROT_1:1;
   end;

  theorem Th33:
   for C be compact non vertical non horizontal Subset of TOP-REAL 2 holds
    (E-max L~Cage(C,n))..Lower_Seq(C,n) = 1
   proof
    let C be compact non vertical non horizontal Subset of TOP-REAL 2;
      Lower_Seq(C,n)/.1 = E-max L~Cage(C,n) by JORDAN1F:6;
    hence (E-max L~Cage(C,n))..Lower_Seq(C,n) = 1 by FINSEQ_6:47;
   end;

  theorem Th34:
   for C be compact connected non vertical non horizontal
    Subset of TOP-REAL 2 holds
     (E-max L~Cage(C,n))..Lower_Seq(C,n) < (E-min L~Cage(C,n))..Lower_Seq(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    set Ema = E-max L~Cage(C,n);
    set Emi = E-min L~Cage(C,n);
    set Sma = S-max L~Cage(C,n);
    set Smi = S-min L~Cage(C,n);
    set Wmi = W-min L~Cage(C,n);
    set Rot = Rotate(Cage(C,n),Ema);
    A1: Lower_Seq(C,n) = Rot-:Wmi by Th26;
    A2: L~Rot = L~Cage(C,n) by REVROT_1:33;
      Ema in rng Cage(C,n) by SPRECT_2:50;
    then A3: Rot/.1 = Ema by FINSEQ_6:98;
    A4: Ema in rng Rot by A2,SPRECT_2:50;
    A5: Wmi in rng Rot by A2,SPRECT_2:47;
    A6: Ema..Rot < Emi..Rot by A2,A3,SPRECT_5:38;
    A7: Emi..Rot <= Sma..Rot by A2,A3,SPRECT_5:40;
    A8: Sma..Rot < Smi..Rot by A2,A3,SPRECT_5:41;
    A9: Smi..Rot <= Wmi..Rot by A2,A3,SPRECT_5:42;
      Emi..Rot < Smi..Rot by A7,A8,AXIOMS:22;
    then A10: Emi..Rot < Wmi..Rot by A9,AXIOMS:22;
    then Ema..Rot < Wmi..Rot by A6,AXIOMS:22;
    then A11: Ema..(Rot-:Wmi) = Ema..Rot by A4,A5,SPRECT_5:3;
      Emi in rng Rot by A2,SPRECT_2:49;
    hence (E-max L~Cage(C,n))..Lower_Seq(C,n) <
            (E-min L~Cage(C,n))..Lower_Seq(C,n) by A1,A5,A6,A10,A11,SPRECT_5:3;
   end;

  theorem Th35:
   for C be compact connected non vertical non horizontal
    Subset of TOP-REAL 2 holds
    (E-min L~Cage(C,n))..Lower_Seq(C,n) <= (S-max L~Cage(C,n))..Lower_Seq(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    set Ema = E-max L~Cage(C,n);
    set Emi = E-min L~Cage(C,n);
    set Sma = S-max L~Cage(C,n);
    set Smi = S-min L~Cage(C,n);
    set Wmi = W-min L~Cage(C,n);
    set Rot = Rotate(Cage(C,n),Ema);
    A1: Lower_Seq(C,n) = Rot-:Wmi by Th26;
    A2: L~Rot = L~Cage(C,n) by REVROT_1:33;
      Ema in rng Cage(C,n) by SPRECT_2:50;
    then A3: Rot/.1 = Ema by FINSEQ_6:98;
    A4: Emi in rng Rot by A2,SPRECT_2:49;
    A5: Wmi in rng Rot by A2,SPRECT_2:47;
    A6: Emi..Rot <= Sma..Rot by A2,A3,SPRECT_5:40;
    A7: Sma..Rot < Smi..Rot by A2,A3,SPRECT_5:41;
      Smi..Rot <= Wmi..Rot by A2,A3,SPRECT_5:42;
    then A8: Sma..Rot < Wmi..Rot by A7,AXIOMS:22;
    then Emi..Rot < Wmi..Rot by A6,AXIOMS:22;
    then A9: Emi..(Rot-:Wmi) = Emi..Rot by A4,A5,SPRECT_5:3;
      Sma in rng Rot by A2,SPRECT_2:46;
    hence (E-min L~Cage(C,n))..Lower_Seq(C,n) <=
            (S-max L~Cage(C,n))..Lower_Seq(C,n) by A1,A5,A6,A8,A9,SPRECT_5:3;
   end;

  theorem Th36:
   for C be compact connected non vertical non horizontal
    Subset of TOP-REAL 2 holds
     (S-max L~Cage(C,n))..Lower_Seq(C,n) < (S-min L~Cage(C,n))..Lower_Seq(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    set Ema = E-max L~Cage(C,n);
    set Sma = S-max L~Cage(C,n);
    set Smi = S-min L~Cage(C,n);
    set Wmi = W-min L~Cage(C,n);
    set Rot = Rotate(Cage(C,n),Ema);
    A1: Lower_Seq(C,n) = Rot-:Wmi by Th26;
    A2: L~Rot = L~Cage(C,n) by REVROT_1:33;
      Ema in rng Cage(C,n) by SPRECT_2:50;
    then A3: Rot/.1 = Ema by FINSEQ_6:98;
    A4: Sma in rng Rot by A2,SPRECT_2:46;
    A5: Wmi in rng Rot by A2,SPRECT_2:47;
    A6: Sma..Rot < Smi..Rot by A2,A3,SPRECT_5:41;
    A7: Smi..Rot <= Wmi..Rot by A2,A3,SPRECT_5:42;
    then Sma..Rot < Wmi..Rot by A6,AXIOMS:22;
    then A8: Sma..(Rot-:Wmi) = Sma..Rot by A4,A5,SPRECT_5:3;
      Smi in rng Rot by A2,SPRECT_2:45;
    hence thesis by A1,A5,A6,A7,A8,SPRECT_5:3;
   end;

  theorem Th37:
   for C be compact connected non vertical non horizontal
    Subset of TOP-REAL 2 holds
    (S-min L~Cage(C,n))..Lower_Seq(C,n) <= (W-min L~Cage(C,n))..Lower_Seq(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    set Ema = E-max L~Cage(C,n);
    set Smi = S-min L~Cage(C,n);
    set Wmi = W-min L~Cage(C,n);
    set Rot = Rotate(Cage(C,n),Ema);
    A1: Lower_Seq(C,n) = Rot-:Wmi by Th26;
    A2: L~Rot = L~Cage(C,n) by REVROT_1:33;
      Ema in rng Cage(C,n) by SPRECT_2:50;
    then A3: Rot/.1 = Ema by FINSEQ_6:98;
    A4: Smi in rng Rot by A2,SPRECT_2:45;
    A5: Wmi in rng Rot by A2,SPRECT_2:47;
    A6: Smi..Rot <= Wmi..Rot by A2,A3,SPRECT_5:42;
    then Smi..(Rot-:Wmi) = Smi..Rot by A4,A5,SPRECT_5:3;
    hence thesis by A1,A5,A6,SPRECT_5:3;
   end;

  theorem Th38:
   for C be compact connected non vertical non horizontal
    Subset of TOP-REAL 2 holds
     (W-min L~Cage(C,n))..Lower_Seq(C,n) = len Lower_Seq(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    A1: Lower_Seq(C,n) = Rotate(Cage(C,n),E-max L~Cage(C,n))-:W-min L~Cage(C,n)
                                                                   by Th26;
    A2: E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:50;
      W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:47;
    then A3: W-min L~Cage(C,n) in rng Rotate(Cage(C,n),E-max L~Cage(C,n))
                                                           by A2,FINSEQ_6:96;
      (W-min L~Cage(C,n))..Rotate(Cage(C,n),E-max L~Cage(C,n)) <=
     (W-min L~Cage(C,n))..Rotate(Cage(C,n),E-max L~Cage(C,n));
    then W-min L~Cage(C,n) in rng Lower_Seq(C,n) by A1,A3,FINSEQ_5:49;
    then A4: Lower_Seq(C,n) just_once_values W-min L~Cage(C,n) by FINSEQ_4:10;
      Lower_Seq(C,n)/.len Lower_Seq(C,n) = W-min L~Cage(C,n) by JORDAN1F:8;
    hence thesis by A4,REVROT_1:1;
   end;

  theorem Th39:
   for C be compact connected non vertical non horizontal
    Subset of TOP-REAL 2 holds
     (Upper_Seq(C,n)/.2)`1 = W-bound L~Cage(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    set Ca = Cage(C,n);
    set US = Upper_Seq(C,n);
    set Wmin = W-min L~Cage(C,n);
    set Emax = E-max L~Cage(C,n);
    set Nmin = N-min L~Cage(C,n);
    A1: Emax in rng Ca by SPRECT_2:50;
    A2: Wmin in rng Ca by SPRECT_2:47;
    then A3: Emax in rng Rotate(Ca,Wmin) by A1,FINSEQ_6:96;
      (Ca:-Wmin)/.len(Ca:-Wmin) = Ca/.len Ca by A2,FINSEQ_5:57
       .= Ca/.1 by FINSEQ_6:def 1
       .= Nmin by JORDAN9:34;
    then A4: Nmin in rng (Ca:-Wmin) by REVROT_1:3;
      (Ca:-Wmin)/.1 = Wmin by FINSEQ_5:56;
    then A5: Wmin in rng (Ca:-Wmin) by FINSEQ_6:46;
      W-max L~Ca in L~Ca & Nmin`2 = N-bound L~Ca by PSCOMP_1:94,SPRECT_1:15;
    then (W-max L~Ca)`2 <= Nmin`2 by PSCOMP_1:71;
    then Nmin <> Wmin by SPRECT_2:61;
    then A6: Card {Nmin,Wmin} = 2 by CARD_2:76;
      {Nmin,Wmin} c= rng (Ca:-Wmin)
    proof
     let x be set;
     assume x in {Nmin,Wmin};
     hence x in rng (Ca:-Wmin) by A4,A5,TARSKI:def 2;
    end;
    then A7: Card {Nmin,Wmin} c= Card rng (Ca:-Wmin) by CARD_1:27;
      Card rng (Ca:-Wmin) c= Card dom (Ca:-Wmin) by YELLOW_6:3;
    then Card rng (Ca:-Wmin) c= len (Ca:-Wmin) by PRE_CIRC:21;
    then 2 c= len (Ca:-Wmin) by A6,A7,XBOOLE_1:1;
    then A8: len (Ca:-Wmin) >= 2 by CARD_1:56;
    then A9: len(Ca:-Wmin) >= 1 by AXIOMS:22;
      len US >= 3 by JORDAN1E:19;
    then len US >= 2 by AXIOMS:22;
    then 2 in Seg len US by FINSEQ_1:3;
    then A10: 2 in Seg(Emax..Rotate(Ca,Wmin)) by JORDAN1E:12;
    A11: 1 <= Wmin..Ca by A2,FINSEQ_4:31;
      Ca/.1 = Nmin by JORDAN9:34;
    then Wmin..Ca < len Ca by SPRECT_2:80;
    then A12: Wmin..Ca+1 <= len Ca by NAT_1:38;
    A13: US/.2 = (Rotate(Ca,Wmin)-:Emax)/.2 by JORDAN1E:def 1
       .= Rotate(Ca,Wmin)/.2 by A3,A10,FINSEQ_5:46
       .= Ca/.(2-'1+Wmin..Ca) by A2,A8,REVROT_1:9
       .= Ca/.(2-1+Wmin..Ca) by BINARITH:def 3;
      US/.1 = (Rotate(Ca,Wmin)-:Emax)/.1 by JORDAN1E:def 1
       .= Rotate(Ca,Wmin)/.1 by A3,FINSEQ_5:47
       .= Ca/.(1-'1+Wmin..Ca) by A2,A9,REVROT_1:9
       .= Ca/.(0+Wmin..Ca) by GOBOARD9:1;
    then Ca/.(Wmin..Ca) = Wmin by JORDAN1F:5;
    hence thesis by A11,A12,A13,JORDAN1E:26;
   end;

  theorem
     for C be compact connected non vertical non horizontal
    Subset of TOP-REAL 2 holds
     (Lower_Seq(C,n)/.2)`1 = E-bound L~Cage(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    set Ca = Cage(C,n);
    set LS = Lower_Seq(C,n);
    set Emax = E-max L~Cage(C,n);
    set Emin = E-min L~Cage(C,n);
    set Smax = S-max L~Cage(C,n);
    set Smin = S-min L~Cage(C,n);
    set Wmin = W-min L~Cage(C,n);
    set Nmin = N-min L~Cage(C,n);
    A1: Wmin in rng Ca by SPRECT_2:47;
    A2: Emax in rng Ca by SPRECT_2:50;
    then A3: Wmin in rng Rotate(Ca,Emax) by A1,FINSEQ_6:96;
      (Ca:-Emax)/.len(Ca:-Emax) = Ca/.len Ca by A2,FINSEQ_5:57
       .= Ca/.1 by FINSEQ_6:def 1
       .= Nmin by JORDAN9:34;
    then A4: Nmin in rng (Ca:-Emax) by REVROT_1:3;
      (Ca:-Emax)/.1 = Emax by FINSEQ_5:56;
    then A5: Emax in rng (Ca:-Emax) by FINSEQ_6:46;
      N-max L~Ca in L~Ca & Emax`1 = E-bound L~Ca by PSCOMP_1:104,SPRECT_1:13;
    then (N-max L~Ca)`1 <= Emax`1 by PSCOMP_1:71;
    then Nmin <> Emax by SPRECT_2:55;
    then A6: Card {Nmin,Emax} = 2 by CARD_2:76;
      {Nmin,Emax} c= rng (Ca:-Emax)
    proof
     let x be set;
     assume x in {Nmin,Emax};
     hence x in rng (Ca:-Emax) by A4,A5,TARSKI:def 2;
    end;
    then A7: Card {Nmin,Emax} c= Card rng (Ca:-Emax) by CARD_1:27;
      Card rng (Ca:-Emax) c= Card dom (Ca:-Emax) by YELLOW_6:3;
    then Card rng (Ca:-Emax) c= len (Ca:-Emax) by PRE_CIRC:21;
    then 2 c= len (Ca:-Emax) by A6,A7,XBOOLE_1:1;
    then A8: len (Ca:-Emax) >= 2 by CARD_1:56;
    then A9: len(Ca:-Emax) >= 1 by AXIOMS:22;
      len LS >= 3 by JORDAN1E:19;
    then len LS >= 2 by AXIOMS:22;
    then 2 <= Wmin..LS by Th38;
    then 2 <= Wmin..(Rotate(Ca,Emax)-:Wmin) by Th26;
    then 2 <= Wmin..Rotate(Ca,Emax) by A3,FINSEQ_6:77;
    then A10: 2 in Seg (Wmin..Rotate(Ca,Emax)) by FINSEQ_1:3;
    A11: 1 <= Emax..Ca by A2,FINSEQ_4:31;
    A12: Ca/.1 = Nmin by JORDAN9:34;
    then A13: Emax..Ca < Emin..Ca by SPRECT_2:75;
      Emin..Ca <= Smax..Ca by A12,SPRECT_2:76;
    then A14: Emax..Ca < Smax..Ca by A13,AXIOMS:22;
      Smax..Ca < Smin..Ca by A12,SPRECT_2:77;
    then A15: Emax..Ca < Smin..Ca by A14,AXIOMS:22;
      Smin..Ca <= Wmin..Ca by A12,SPRECT_2:78;
    then A16: Emax..Ca < Wmin..Ca by A15,AXIOMS:22;
      Wmin..Ca < len Ca by A12,SPRECT_2:80;
    then Emax..Ca < len Ca by A16,AXIOMS:22;
    then A17: Emax..Ca+1 <= len Ca by NAT_1:38;
    A18: LS/.2 = (Rotate(Ca,Emax)-:Wmin)/.2 by Th26
       .= Rotate(Ca,Emax)/.2 by A3,A10,FINSEQ_5:46
       .= Ca/.(2-'1+Emax..Ca) by A2,A8,REVROT_1:9
       .= Ca/.(2-1+Emax..Ca) by BINARITH:def 3;
      LS/.1 = (Rotate(Ca,Emax)-:Wmin)/.1 by Th26
       .= Rotate(Ca,Emax)/.1 by A3,FINSEQ_5:47
       .= Ca/.(1-'1+Emax..Ca) by A2,A9,REVROT_1:9
       .= Ca/.(0+Emax..Ca) by GOBOARD9:1;
    then Ca/.(Emax..Ca) = Emax by JORDAN1F:6;
    hence thesis by A11,A17,A18,JORDAN1E:24;
   end;

  theorem Th41:
   for C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2 holds
    W-bound L~Cage(C,n) + E-bound L~Cage(C,n) = W-bound C + E-bound C
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    thus W-bound L~Cage(C,n)+E-bound L~Cage(C,n) = W-bound L~Cage(C,n)+
          (E-bound C + (E-bound C - W-bound C)/(2|^n)) by JORDAN1A:85
       .= (W-bound C - (E-bound C - W-bound C)/(2|^n))+
          (E-bound C + (E-bound C - W-bound C)/(2|^n)) by JORDAN1A:83
       .= W-bound C + E-bound C by XCMPLX_1:28;
   end;

  theorem
     for C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2 holds
    S-bound L~Cage(C,n) + N-bound L~Cage(C,n) = S-bound C + N-bound C
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    thus S-bound L~Cage(C,n)+N-bound L~Cage(C,n) = S-bound L~Cage(C,n)+
          (N-bound C + (N-bound C - S-bound C)/(2|^n)) by JORDAN10:6
       .= (S-bound C - (N-bound C - S-bound C)/(2|^n))+
          (N-bound C + (N-bound C - S-bound C)/(2|^n)) by JORDAN1A:84
       .= S-bound C + N-bound C by XCMPLX_1:28;
   end;

  theorem Th43:
   for C be compact connected non vertical non horizontal Subset of TOP-REAL 2
   for n,i be Nat st 1 <= i & i <= width Gauge(C,n) & n > 0 holds
    Gauge(C,n)*(Center Gauge(C,n),i)`1 = (W-bound C + E-bound C) / 2
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let n,i be Nat such that
     A1: 1 <= i & i <= width Gauge(C,n);
    A2: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
      len Gauge(C,1) >= 4 by JORDAN8:13;
    then A3: len Gauge(C,1) >= 1 by AXIOMS:22;
    assume n > 0;
    hence Gauge(C,n)*(Center Gauge(C,n),i)`1 =
          Gauge(C,1)*(Center Gauge(C,1),1)`1 by A1,A2,A3,JORDAN1A:57
       .= (W-bound C + E-bound C) / 2 by A3,JORDAN1A:59;
   end;

  theorem
     for C be compact connected non vertical non horizontal Subset of TOP-REAL
2
   for n,i be Nat st 1 <= i & i <= len Gauge(C,n) & n > 0 holds
    Gauge(C,n)*(i,Center Gauge(C,n))`2 = (S-bound C + N-bound C) / 2
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let n,i be Nat such that
     A1: 1 <= i & i <= len Gauge(C,n);
      len Gauge(C,1) >= 4 by JORDAN8:13;
    then A2: len Gauge(C,1) >= 1 by AXIOMS:22;
    assume n > 0;
    hence Gauge(C,n)*(i,Center Gauge(C,n))`2 =
          Gauge(C,1)*(1,Center Gauge(C,1))`2 by A1,A2,JORDAN1A:58
       .= (S-bound C + N-bound C) / 2 by A2,JORDAN1A:60;
   end;

  theorem Th45:
   for f be S-Sequence_in_R2
   for k1,k2 be Nat st 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f &
    f/.1 in L~mid(f,k1,k2) holds k1 = 1 or k2 = 1
   proof
    let f be S-Sequence_in_R2;
    let k1,k2 be Nat;
    assume that
     A1: 1 <= k1 and
     A2: k1 <= len f and
     A3: 1 <= k2 and
     A4: k2 <= len f and
     A5: f/.1 in L~mid(f,k1,k2);
    assume A6: k1 <> 1 & k2 <> 1;
    consider j be Nat such that
     A7: 1 <= j and
     A8: j+1 <= len mid(f,k1,k2) and
     A9: f/.1 in LSeg(mid(f,k1,k2),j) by A5,SPPOL_2:13;
    A10: len f >= 2 by TOPREAL1:def 10;
    per cases by REAL_1:def 5;
     suppose A11: k1 < k2;
      then len mid(f,k1,k2) = k2-'k1 + 1 by A1,A2,A3,A4,JORDAN3:27;
      then j < k2-'k1 + 1 by A8,NAT_1:38;
      then LSeg(mid(f,k1,k2),j) = LSeg(f,j+k1-'1) by A1,A4,A7,A11,JORDAN4:31;
      then A12: j+k1-'1 = 1 by A9,A10,JORDAN5B:33;
        j+k1 >= 1+1 by A1,A7,REAL_1:55;
      then j+k1-1 >= 1+1-1 by REAL_1:49;
      then j+k1-1 >= 0 by AXIOMS:22;
      then j+k1-'1 = j+k1-1 by BINARITH:def 3;
      then j+(k1-1) = 1 by A12,XCMPLX_1:29;
      then k1-1 = 1-j by XCMPLX_1:26;
      then k1-1 <= 0 by A7,REAL_2:106;
      then k1-1 = 0 by A1,SQUARE_1:12;
      hence contradiction by A6,XCMPLX_1:15;
     suppose A13: k1 > k2;
      then len mid(f,k1,k2) = k1-'k2 + 1 by A1,A2,A3,A4,JORDAN3:27;
      then A14: j < k1-'k2 + 1 by A8,NAT_1:38;
      then LSeg(mid(f,k1,k2),j) = LSeg(f,k1-'j) by A2,A3,A7,A13,JORDAN4:32;
      then A15: k1-'j = 1 by A9,A10,JORDAN5B:33;
        k1-k2 > 0 by A13,SQUARE_1:11;
      then k1-'k2 = k1-k2 by BINARITH:def 3;
      then j-1 < k1-k2 by A14,REAL_1:84;
      then j-1+k2 < k1 by REAL_1:86;
      then j-(1-k2) < k1 by XCMPLX_1:37;
      then j+-(1-k2) < k1 by XCMPLX_0:def 8;
      then -(1-k2) < k1-j by REAL_1:86;
      then A16: k2-1 < k1-j by XCMPLX_1:143;
      then k1-j > 0 by A3,SQUARE_1:12;
      then k1-j = 1 by A15,BINARITH:def 3;
      then k2 < 1+1 by A16,REAL_1:84;
      then k2 <= 1 by NAT_1:38;
      hence contradiction by A3,A6,AXIOMS:21;
     suppose k1 = k2;
      then mid(f,k1,k2) = <*f/.k1*> by A1,A2,JORDAN4:27;
      hence contradiction by A5,SPPOL_2:12;
   end;

  theorem Th46:
   for f be S-Sequence_in_R2
   for k1,k2 be Nat st 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f &
    f/.(len f) in L~mid(f,k1,k2) holds k1 = len f or k2 = len f
   proof
    let f be S-Sequence_in_R2;
    let k1,k2 be Nat;
    assume that
     A1: 1 <= k1 and
     A2: k1 <= len f and
     A3: 1 <= k2 and
     A4: k2 <= len f and
     A5: f/.len f in L~mid(f,k1,k2);
    assume A6: k1 <> len f & k2 <> len f;
    consider j be Nat such that
     A7: 1 <= j and
     A8: j+1 <= len mid(f,k1,k2) and
     A9: f/.len f in LSeg(mid(f,k1,k2),j) by A5,SPPOL_2:13;
    per cases by REAL_1:def 5;
     suppose A10: k1 < k2;
      then k2-k1 > 0 by SQUARE_1:11;
      then A11: k2-'k1 = k2-k1 by BINARITH:def 3;
      A12: j+k1 >= 1+1 by A1,A7,REAL_1:55;
      then A13: j+k1-1 >= 1+1-1 by REAL_1:49;
      then A14: j+k1-1 >= 0 by AXIOMS:22;
      then A15: j+k1-'1 = j+k1-1 by BINARITH:def 3;
        len mid(f,k1,k2) = k2-'k1 + 1 by A1,A2,A3,A4,A10,JORDAN3:27;
      then A16: j < k2-'k1 + 1 by A8,NAT_1:38;
      then j-1 < k2-k1 by A11,REAL_1:84;
      then j-1+k1 < k2 by REAL_1:86;
      then j+k1-1 < k2 by XCMPLX_1:29;
      then A17: j+k1-1 < len f by A4,AXIOMS:22;
      then A18: j+k1-'1 in dom f by A13,A15,FINSEQ_3:27;
      A19: j+k1 >= 1 by A12,AXIOMS:22;
        j+k1-1+1 <= len f by A15,A17,NAT_1:38;
      then j+k1 <= len f by XCMPLX_1:27;
      then j+k1 in Seg len f by A19,FINSEQ_1:3;
      then j+k1-1+1 in Seg len f by XCMPLX_1:27;
      then A20: j+k1-'1+1 in dom f by A15,FINSEQ_1:def 3;
        LSeg(mid(f,k1,k2),j) = LSeg(f,j+k1-'1) by A1,A4,A7,A10,A16,JORDAN4:31;
      then A21: j+k1-'1+1=len f by A9,A18,A20,GOBOARD2:7;
        j+k1-'1 = j+k1-1 by A14,BINARITH:def 3;
      then A22: len f = j+k1 by A21,XCMPLX_1:27;
        j < k2+1-k1 by A11,A16,XCMPLX_1:29;
      then len f < k2+1 by A22,REAL_1:86;
      then len f <= k2 by NAT_1:38;
      hence contradiction by A4,A6,AXIOMS:21;
     suppose A23: k1 > k2;
      then len mid(f,k1,k2) = k1-'k2 + 1 by A1,A2,A3,A4,JORDAN3:27;
      then A24: j < k1-'k2 + 1 by A8,NAT_1:38;
        k1-k2 > 0 by A23,SQUARE_1:11;
      then k1-'k2 = k1-k2 by BINARITH:def 3;
      then j-1 < k1-k2 by A24,REAL_1:84;
      then j-1+k2 < k1 by REAL_1:86;
      then j-(1-k2) < k1 by XCMPLX_1:37;
      then j+-(1-k2) < k1 by XCMPLX_0:def 8;
      then -(1-k2) < k1-j by REAL_1:86;
      then A25: k2-1 < k1-j by XCMPLX_1:143;
       A26: k2-1 >= 0 by A3,SQUARE_1:12;
      A27: k1-j > 0 by A3,A25,SQUARE_1:12;
      then A28: k1-'j = k1-j by BINARITH:def 3;
      then A29: k1-j >= 0+1 by A25,A26,NAT_1:38;
        k1-j <= k1-1 by A7,REAL_2:106;
      then k1-j+1 <= k1-1+1 by REAL_1:55;
      then k1-j+1 <= k1 by XCMPLX_1:27;
      then k1-j < k1 by A28,NAT_1:38;
      then A30: k1-j < len f by A2,AXIOMS:22;
      then A31: k1-'j in dom f by A28,A29,FINSEQ_3:27;
      A32: k1-j+1 > 0+1 by A25,A26,REAL_1:53;
        k1-j+1 <= len f by A28,A30,NAT_1:38;
      then A33: k1-'j+1 in dom f by A28,A32,FINSEQ_3:27;
        LSeg(mid(f,k1,k2),j) = LSeg(f,k1-'j) by A2,A3,A7,A23,A24,JORDAN4:32;
      then k1-'j+1=len f by A9,A31,A33,GOBOARD2:7;
      then A34: k1-j+1=len f by A27,BINARITH:def 3;
        k1-j <= k1-1 by A7,REAL_2:106;
      then len f <= k1-1+1 by A34,REAL_1:55;
      then len f <= k1 by XCMPLX_1:27;
      hence contradiction by A2,A6,AXIOMS:21;
     suppose k1 = k2;
      then mid(f,k1,k2) = <*f/.k1*> by A1,A2,JORDAN4:27;
      hence contradiction by A5,SPPOL_2:12;
   end;

  theorem Th47:
   for C be compact non vertical non horizontal Subset of TOP-REAL 2
   for n be Nat holds
    rng Upper_Seq(C,n) c= rng Cage(C,n) &
    rng Lower_Seq(C,n) c= rng Cage(C,n)
   proof
    let C be compact non vertical non horizontal Subset of TOP-REAL 2;
    let n be Nat;
    A1: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:47;
      E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:50;
    then A2: E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                           by A1,FINSEQ_6:96;
      Upper_Seq(C,n) = Rotate(Cage(C,n),W-min L~Cage(C,n))-:E-max L~Cage(C,n)
                                                          by JORDAN1E:def 1;
    then rng Upper_Seq(C,n) c= rng Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                              by FINSEQ_5:51;
    hence rng Upper_Seq(C,n) c= rng Cage(C,n) by A1,FINSEQ_6:96;
      Lower_Seq(C,n) = Rotate(Cage(C,n),W-min L~Cage(C,n)):-E-max L~Cage(C,n)
                                                          by JORDAN1E:def 2;
    then rng Lower_Seq(C,n) c= rng Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                           by A2,FINSEQ_5:58;
    hence rng Lower_Seq(C,n) c= rng Cage(C,n) by A1,FINSEQ_6:96;
   end;

  theorem Th48:
   for C be compact non vertical non horizontal Subset of TOP-REAL 2 holds
    Upper_Seq(C,n) is_a_h.c._for Cage(C,n)
   proof
    let C be compact non vertical non horizontal Subset of TOP-REAL 2;
    A1: Upper_Seq(C,n) is_in_the_area_of Cage(C,n) by JORDAN1E:21;
    A2: (Upper_Seq(C,n)/.1)`1 = (W-min L~Cage(C,n))`1 by JORDAN1F:5
       .= W-bound L~Cage(C,n) by PSCOMP_1:84;
      (Upper_Seq(C,n)/.len Upper_Seq(C,n))`1 = (E-max L~Cage(C,n))`1
                                                                by JORDAN1F:7
       .= E-bound L~Cage(C,n) by PSCOMP_1:104;
    hence Upper_Seq(C,n) is_a_h.c._for Cage(C,n) by A1,A2,SPRECT_2:def 2;
   end;

  theorem Th49:
   for C be compact non vertical non horizontal Subset of TOP-REAL 2 holds
    Rev Lower_Seq(C,n) is_a_h.c._for Cage(C,n)
   proof
    let C be compact non vertical non horizontal Subset of TOP-REAL 2;
      Lower_Seq(C,n) is_in_the_area_of Cage(C,n) by JORDAN1E:22;
    then A1: Rev Lower_Seq(C,n) is_in_the_area_of Cage(C,n) by SPRECT_3:68;
    A2: (Rev Lower_Seq(C,n)/.1)`1 = (Lower_Seq(C,n)/.len Lower_Seq(C,n))`1
                                                               by FINSEQ_5:68
       .= (W-min L~Cage(C,n))`1 by JORDAN1F:8
       .= W-bound L~Cage(C,n) by PSCOMP_1:84;
      (Rev Lower_Seq(C,n)/.len Rev Lower_Seq(C,n))`1 =
          (Rev Lower_Seq(C,n)/.len Lower_Seq(C,n))`1 by FINSEQ_5:def 3
       .= (Lower_Seq(C,n)/.1)`1 by FINSEQ_5:68
       .= (E-max L~Cage(C,n))`1 by JORDAN1F:6
       .= E-bound L~Cage(C,n) by PSCOMP_1:104;
    hence Rev Lower_Seq(C,n) is_a_h.c._for Cage(C,n) by A1,A2,SPRECT_2:def 2;
   end;

  theorem Th50:
   for C be compact connected non vertical non horizontal Subset of TOP-REAL 2
   for i be Nat st 1 < i & i <= len Gauge(C,n) holds
    not Gauge(C,n)*(i,1) in rng Upper_Seq(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let i be Nat;
    assume that
     A1: 1 < i and
     A2: i <= len Gauge(C,n) and
     A3: Gauge(C,n)*(i,1) in rng Upper_Seq(C,n);
    consider i2 be Nat such that
     A4: i2 in dom Upper_Seq(C,n) and
     A5: Upper_Seq(C,n).i2 = Gauge(C,n)*(i,1) by A3,FINSEQ_2:11;
    set i1 = (N-min L~Cage(C,n))..Upper_Seq(C,n);
      (W-min L~Cage(C,n))..Upper_Seq(C,n) = 1 by Th27;
    then A6: 1 < (W-max L~Cage(C,n))..Upper_Seq(C,n) by Th28;
      (W-max L~Cage(C,n))..Upper_Seq(C,n) <= i1 by Th29;
    then A7: i1 > 1 by A6,AXIOMS:22;
    A8: i1 < (N-max L~Cage(C,n))..Upper_Seq(C,n) by Th30;
      (E-max L~Cage(C,n))..Upper_Seq(C,n) = len Upper_Seq(C,n) by Th32;
    then (N-max L~Cage(C,n))..Upper_Seq(C,n) <= len Upper_Seq(C,n) by Th31;
    then A9: i1 < len Upper_Seq(C,n) by A8,AXIOMS:22;
    A10: 1 <= i2 & i2 <= len Upper_Seq(C,n) by A4,FINSEQ_3:27;
    A11: i1 in dom Upper_Seq(C,n) by A7,A9,FINSEQ_3:27;
    set f = Rotate(Cage(C,n),W-min L~Cage(C,n));
    A12: Upper_Seq(C,n) = f-:E-max L~Cage(C,n) by JORDAN1E:def 1;
    A13: N-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:43;
    A14: E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:50;
    A15: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:47;
    then A16: rng f = rng Cage(C,n) by FINSEQ_6:96;
    A17: f/.1 = W-min L~Cage(C,n) by A15,FINSEQ_6:98;
    A18: L~Cage(C,n) = L~f by REVROT_1:33;
    then A19: (N-min L~Cage(C,n))..f < (N-max L~Cage(C,n))..f
                                                          by A17,SPRECT_5:25;
      (N-max L~Cage(C,n))..f <= (E-max L~Cage(C,n))..f by A17,A18,SPRECT_5:26;
    then (N-min L~Cage(C,n))..f < (E-max L~Cage(C,n))..f by A19,AXIOMS:22;
    then A20: N-min L~Cage(C,n) in rng Upper_Seq(C,n)
                                              by A12,A13,A14,A16,FINSEQ_5:49;
    then A21: Upper_Seq(C,n)/.i1 = N-min L~Cage(C,n) by FINSEQ_5:41;
    A22: i1 <> i2
    proof
     assume i1 = i2;
     then Gauge(C,n)*(i,1) = N-min L~Cage(C,n) by A5,A11,A21,FINSEQ_4:def 4;
     then (Gauge(C,n)*(i,1))`2 = N-bound L~Cage(C,n) by PSCOMP_1:94;
     then S-bound L~Cage(C,n) = N-bound L~Cage(C,n) by A1,A2,JORDAN1A:93;
     hence contradiction by SPRECT_1:18;
    end;
    then mid(Upper_Seq(C,n),i1,i2) is_S-Seq by A7,A9,A10,JORDAN3:39;
    then reconsider h1 = mid(Upper_Seq(C,n),i1,i2) as
            one-to-one special FinSequence of TOP-REAL 2 by TOPREAL1:def 10;
    set h = Rev h1;
      3 <= len Lower_Seq(C,n) by JORDAN1E:19;
    then A23: 2 <= len Lower_Seq(C,n) by AXIOMS:22;
    A24: len Lower_Seq(C,n) = len Rev Lower_Seq(C,n) by FINSEQ_5:def 3;
    A25: len h1 = len h by FINSEQ_5:def 3;
    then A26: len h <> 1 by A4,A11,A22,SPRECT_2:10;
    A27: len h >= 1 by A4,A11,A25,SPRECT_2:9;
    then len h > 1 by A26,REAL_1:def 5;
    then A28: 1+1 <= len h by NAT_1:38;
    A29: Rev Lower_Seq(C,n) is_a_h.c._for Cage(C,n) by Th49;
    A30: h is special by SPPOL_2:42;
      Upper_Seq(C,n) is_in_the_area_of Cage(C,n) by JORDAN1E:21;
    then h1 is_in_the_area_of Cage(C,n) by A4,A11,SPRECT_2:26;
    then A31: h is_in_the_area_of Cage(C,n) by SPRECT_3:68;
      len h > 0 by A27,AXIOMS:22;
    then A32: h1 is non empty by A25,FINSEQ_1:25;
    A33: Rev Lower_Seq(C,n) is special by SPPOL_2:42;
    A34: L~Rev Lower_Seq(C,n) = L~Lower_Seq(C,n) by SPPOL_2:22;
    A35: (h/.1)`2 = (h1/.len h1)`2 by A32,FINSEQ_5:68
       .= (Upper_Seq(C,n)/.i2)`2 by A4,A11,SPRECT_2:13
       .= (Gauge(C,n)*(i,1))`2 by A4,A5,FINSEQ_4:def 4
       .= S-bound L~Cage(C,n) by A1,A2,JORDAN1A:93;
      (h/.len h)`2 = (h1/.1)`2 by A25,A32,FINSEQ_5:68
       .= (Upper_Seq(C,n)/.i1)`2 by A4,A11,SPRECT_2:12
       .= (N-min L~Cage(C,n))`2 by A20,FINSEQ_5:41
       .= N-bound L~Cage(C,n) by PSCOMP_1:94;
    then h is_a_v.c._for Cage(C,n) by A31,A35,SPRECT_2:def 3;
    then L~Rev Lower_Seq(C,n) meets L~h by A23,A24,A28,A29,A30,A33,SPRECT_2:33
;
    then consider x be set such that
     A36: x in L~Lower_Seq(C,n) and
     A37: x in L~h by A34,XBOOLE_0:3;
    A38: L~h = L~h1 by SPPOL_2:22;
      L~mid(Upper_Seq(C,n),i1,i2) c= L~Upper_Seq(C,n) by A7,A9,A10,JORDAN4:47;
    then x in L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n) by A36,A37,A38,XBOOLE_0:def
3;
    then A39: x in {W-min L~Cage(C,n),E-max L~Cage(C,n)} by JORDAN1E:20;
      4 <= len Gauge(C,n) by JORDAN8:13;
    then A40: 1 <= len Gauge(C,n) by AXIOMS:22;
    A41: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
    per cases by A39,TARSKI:def 2;
     suppose x = W-min L~Cage(C,n);
      then x = Upper_Seq(C,n)/.1 by JORDAN1F:5;
      then i2 = 1 by A7,A9,A10,A37,A38,Th45;
      then Upper_Seq(C,n)/.1 = Gauge(C,n)*(i,1) by A4,A5,FINSEQ_4:def 4;
      then W-min(L~Cage(C,n)) = Gauge(C,n)*(i,1) by JORDAN1F:5;
      then Gauge(C,n)*(i,1)`1 = W-bound(L~Cage(C,n)) by PSCOMP_1:84
         .= Gauge(C,n)*(1,1)`1 by A40,JORDAN1A:94;
      hence contradiction by A1,A2,A40,A41,GOBOARD5:4;
     suppose x = E-max L~Cage(C,n);
      then x = Upper_Seq(C,n)/.len Upper_Seq(C,n) by JORDAN1F:7;
      then i2 = len Upper_Seq(C,n) by A7,A9,A10,A37,A38,Th46;
      then Upper_Seq(C,n)/.len Upper_Seq(C,n) = Gauge(C,n)*(i,1)
                                                    by A4,A5,FINSEQ_4:def 4;
      then A42: E-max(L~Cage(C,n)) = Gauge(C,n)*(i,1) by JORDAN1F:7;
      A43: (E-min L~Cage(C,n))`2 < (E-max L~Cage(C,n))`2 by SPRECT_2:57;
        (SE-corner L~Cage(C,n))`2 <= (E-min L~Cage(C,n))`2 by PSCOMP_1:107;
      then (SE-corner L~Cage(C,n))`2 < (E-max L~Cage(C,n))`2 by A43,AXIOMS:22;
      then S-bound L~Cage(C,n) < (Gauge(C,n)*(i,1))`2 by A42,PSCOMP_1:79;
      hence contradiction by A1,A2,JORDAN1A:93;
   end;

  theorem Th51:
   for C be compact connected non vertical non horizontal Subset of TOP-REAL 2
   for i be Nat st 1 <= i & i < len Gauge(C,n) holds
    not Gauge(C,n)*(i,width Gauge(C,n)) in rng Lower_Seq(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let i be Nat;
    set wi = width Gauge(C,n);
    assume that
     A1: 1 <= i and
     A2: i < len Gauge(C,n) and
     A3: Gauge(C,n)*(i,wi) in rng Lower_Seq(C,n);
    consider i2 be Nat such that
     A4: i2 in dom Lower_Seq(C,n) and
     A5: Lower_Seq(C,n).i2 = Gauge(C,n)*(i,wi) by A3,FINSEQ_2:11;
    set i1 = (S-max L~Cage(C,n))..Lower_Seq(C,n);
    A6: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
      (E-max L~Cage(C,n))..Lower_Seq(C,n) = 1 by Th33;
    then A7: 1 < (E-min L~Cage(C,n))..Lower_Seq(C,n) by Th34;
      (E-min L~Cage(C,n))..Lower_Seq(C,n) <= i1 by Th35;
    then A8: i1 > 1 by A7,AXIOMS:22;
    A9: i1 < (S-min L~Cage(C,n))..Lower_Seq(C,n) by Th36;
      (W-min L~Cage(C,n))..Lower_Seq(C,n) = len Lower_Seq(C,n) by Th38;
    then (S-min L~Cage(C,n))..Lower_Seq(C,n) <= len Lower_Seq(C,n) by Th37;
    then A10: i1 < len Lower_Seq(C,n) by A9,AXIOMS:22;
    A11: 1 <= i2 & i2 <= len Lower_Seq(C,n) by A4,FINSEQ_3:27;
    A12: i1 in dom Lower_Seq(C,n) by A8,A10,FINSEQ_3:27;
    set f = Rotate(Cage(C,n),E-max L~Cage(C,n));
    A13: Lower_Seq(C,n) = f-:W-min L~Cage(C,n) by Th26;
    A14: S-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46;
    A15: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:47;
    A16: E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:50;
    then A17: rng f = rng Cage(C,n) by FINSEQ_6:96;
    A18: f/.1 = E-max L~Cage(C,n) by A16,FINSEQ_6:98;
    A19: L~Cage(C,n) = L~f by REVROT_1:33;
    then A20: (S-max L~Cage(C,n))..f < (S-min L~Cage(C,n))..f
                                                          by A18,SPRECT_5:41;
      (S-min L~Cage(C,n))..f <= (W-min L~Cage(C,n))..f by A18,A19,SPRECT_5:42;
    then (S-max L~Cage(C,n))..f < (W-min L~Cage(C,n))..f by A20,AXIOMS:22;
    then A21: S-max L~Cage(C,n) in rng Lower_Seq(C,n)
                                              by A13,A14,A15,A17,FINSEQ_5:49;
    then A22: Lower_Seq(C,n)/.i1 = S-max L~Cage(C,n) by FINSEQ_5:41;
    A23: i1 <> i2
    proof
     assume i1 = i2;
     then Gauge(C,n)*(i,wi) = S-max L~Cage(C,n) by A5,A12,A22,FINSEQ_4:def 4;
     then (Gauge(C,n)*(i,wi))`2 = S-bound L~Cage(C,n) by PSCOMP_1:114;
     then N-bound L~Cage(C,n) = S-bound L~Cage(C,n) by A1,A2,A6,JORDAN1A:91;
     hence contradiction by SPRECT_1:18;
    end;
    then mid(Lower_Seq(C,n),i1,i2) is_S-Seq by A8,A10,A11,JORDAN3:39;
    then reconsider h = mid(Lower_Seq(C,n),i1,i2) as
            one-to-one special FinSequence of TOP-REAL 2 by TOPREAL1:def 10;
      3 <= len Upper_Seq(C,n) by JORDAN1E:19;
    then A24: 2 <= len Upper_Seq(C,n) by AXIOMS:22;
    A25: len h <> 1 by A4,A12,A23,SPRECT_2:10;
      len h >= 1 by A4,A12,SPRECT_2:9;
    then len h > 1 by A25,REAL_1:def 5;
    then A26: 1+1 <= len h by NAT_1:38;
    A27: Upper_Seq(C,n) is_a_h.c._for Cage(C,n) by Th48;
      Lower_Seq(C,n) is_in_the_area_of Cage(C,n) by JORDAN1E:22;
    then A28: h is_in_the_area_of Cage(C,n) by A4,A12,SPRECT_2:26;
    A29: (h/.len h)`2 = (Lower_Seq(C,n)/.i2)`2 by A4,A12,SPRECT_2:13
       .= (Gauge(C,n)*(i,wi))`2 by A4,A5,FINSEQ_4:def 4
       .= N-bound L~Cage(C,n) by A1,A2,A6,JORDAN1A:91;
      (h/.1)`2 = (Lower_Seq(C,n)/.i1)`2 by A4,A12,SPRECT_2:12
       .= (S-max L~Cage(C,n))`2 by A21,FINSEQ_5:41
       .= S-bound L~Cage(C,n) by PSCOMP_1:114;
    then h is_a_v.c._for Cage(C,n) by A28,A29,SPRECT_2:def 3;
    then L~Upper_Seq(C,n) meets L~h by A24,A26,A27,SPRECT_2:33;
    then consider x be set such that
     A30: x in L~Upper_Seq(C,n) and
     A31: x in L~h by XBOOLE_0:3;
      L~mid(Lower_Seq(C,n),i1,i2) c= L~Lower_Seq(C,n) by A8,A10,A11,JORDAN4:47;
    then x in L~Lower_Seq(C,n) /\ L~Upper_Seq(C,n) by A30,A31,XBOOLE_0:def 3;
    then A32: x in {W-min L~Cage(C,n),E-max L~Cage(C,n)} by JORDAN1E:20;
      4 <= len Gauge(C,n) by JORDAN8:13;
    then A33: 1 <= len Gauge(C,n) by AXIOMS:22;
    per cases by A32,TARSKI:def 2;
     suppose x = E-max L~Cage(C,n);
      then x = Lower_Seq(C,n)/.1 by JORDAN1F:6;
      then i2 = 1 by A8,A10,A11,A31,Th45;
      then Lower_Seq(C,n)/.1 = Gauge(C,n)*(i,wi) by A4,A5,FINSEQ_4:def 4;
      then E-max(L~Cage(C,n)) = Gauge(C,n)*(i,wi) by JORDAN1F:6;
      then Gauge(C,n)*(i,wi)`1 = E-bound(L~Cage(C,n)) by PSCOMP_1:104
         .= Gauge(C,n)*(len Gauge(C,n),wi)`1 by A6,A33,JORDAN1A:92;
      hence contradiction by A1,A2,A6,A33,GOBOARD5:4;
     suppose x = W-min L~Cage(C,n);
      then x = Lower_Seq(C,n)/.len Lower_Seq(C,n) by JORDAN1F:8;
      then i2 = len Lower_Seq(C,n) by A8,A10,A11,A31,Th46;
      then Lower_Seq(C,n)/.len Lower_Seq(C,n) = Gauge(C,n)*(i,wi)
                                                    by A4,A5,FINSEQ_4:def 4;
      then A34: W-min(L~Cage(C,n)) = Gauge(C,n)*(i,wi) by JORDAN1F:8;
      A35: (W-max L~Cage(C,n))`2 > (W-min L~Cage(C,n))`2 by SPRECT_2:61;
        (NW-corner L~Cage(C,n))`2 >= (W-max L~Cage(C,n))`2 by PSCOMP_1:87;
      then (NW-corner L~Cage(C,n))`2 > (W-min L~Cage(C,n))`2 by A35,AXIOMS:22;
      then N-bound L~Cage(C,n) > (Gauge(C,n)*(i,wi))`2 by A34,PSCOMP_1:75;
      hence contradiction by A1,A2,A6,JORDAN1A:91;
   end;

  theorem Th52:
   for C be compact connected non vertical non horizontal Subset of TOP-REAL 2
   for i be Nat st 1 < i & i <= len Gauge(C,n) holds
    not Gauge(C,n)*(i,1) in L~Upper_Seq(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let i be Nat such that
     A1: 1 < i & i <= len Gauge(C,n) and
     A2: Gauge(C,n)*(i,1) in L~Upper_Seq(C,n);
    set Gi1 = Gauge(C,n)*(i,1);
    consider ii be Nat such that
     A3: 1 <= ii & ii+1 <= len Upper_Seq(C,n) and
     A4: Gi1 in LSeg(Upper_Seq(C,n),ii) by A2,SPPOL_2:13;
    A5: LSeg(Upper_Seq(C,n),ii) =
       LSeg(Upper_Seq(C,n)/.ii,Upper_Seq(C,n)/.(ii+1)) by A3,TOPREAL1:def 5;
      Upper_Seq(C,n) is_sequence_on Gauge(C,n) by Th4;
    then consider i1,j1,i2,j2 being Nat such that
     A6: [i1,j1] in Indices Gauge(C,n) and
     A7: Upper_Seq(C,n)/.ii = Gauge(C,n)*(i1,j1) and
     A8: [i2,j2] in Indices Gauge(C,n) and
     A9: Upper_Seq(C,n)/.(ii+1) = Gauge(C,n)*(i2,j2) and
     A10: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or
          i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A3,JORDAN8:6;
    A11: 1 <= i1 & i1 <= len Gauge(C,n) & 1 <= j1 & j1 <= width Gauge(C,n)
                                                            by A6,GOBOARD5:1;
    A12: 1 <= i2 & i2 <= len Gauge(C,n) & 1 <= j2 & j2 <= width Gauge(C,n)
                                                            by A8,GOBOARD5:1;
    A13: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
      len Gauge(C,n) >= 4 by JORDAN8:13;
    then len Gauge(C,n) > 1 by AXIOMS:22;
    then A14: [i,1] in Indices Gauge(C,n) by A1,A13,GOBOARD7:10;
      ii+1 >= 1 by NAT_1:29;
    then A15: ii+1 in dom Upper_Seq(C,n) by A3,FINSEQ_3:27;
      ii < len Upper_Seq(C,n) by A3,NAT_1:38;
    then A16: ii in dom Upper_Seq(C,n) by A3,FINSEQ_3:27;
    A17: not Gi1 in rng Upper_Seq(C,n) by A1,Th50;
    per cases by A10;
     suppose A18: i1 = i2 & j1+1 = j2;
      then Gauge(C,n)*(i1,j1)`1 = Gauge(C,n)*(i2,1)`1 by A11,GOBOARD5:3
         .= Gauge(C,n)*(i2,j2)`1 by A12,GOBOARD5:3;
      then LSeg(Upper_Seq(C,n)/.ii,Upper_Seq(C,n)/.(ii+1)) is vertical
                                                         by A7,A9,SPPOL_1:37;
      then Gi1`1 = Gauge(C,n)*(i1,j1)`1 by A4,A5,A7,SPRECT_3:20;
      then A19: i1 = i by A6,A14,Th7;
        j1 <= j2 by A18,NAT_1:29;
      then Gauge(C,n)*(i1,j1)`2 <= Gauge(C,n)*(i2,j2)`2
                                                  by A11,A12,A18,SPRECT_3:24;
      then A20: Gauge(C,n)*(i1,j1)`2 <= Gi1`2 by A4,A5,A7,A9,TOPREAL1:10;
        Gi1`2 <= Gauge(C,n)*(i1,j1)`2 by A11,A19,SPRECT_3:24;
      then Gi1`2 = Gauge(C,n)*(i1,j1)`2 by A20,AXIOMS:21;
      then j1 = 1 by A6,A14,Th6;
      hence contradiction by A7,A16,A17,A19,PARTFUN2:4;
     suppose A21: i1+1 = i2 & j1 = j2;
      then Gauge(C,n)*(i1,j1)`2 = Gauge(C,n)*(1,j2)`2 by A11,GOBOARD5:2
         .= Gauge(C,n)*(i2,j2)`2 by A12,GOBOARD5:2;
      then LSeg(Upper_Seq(C,n)/.ii,Upper_Seq(C,n)/.(ii+1)) is horizontal
                                                         by A7,A9,SPPOL_1:36;
      then Gi1`2 = Gauge(C,n)*(i1,j1)`2 by A4,A5,A7,SPRECT_3:19;
      then A22: j1 = 1 by A6,A14,Th6;
        i2 > 1 by A11,A21,NAT_1:38;
      then not Upper_Seq(C,n)/.(ii+1) in rng Upper_Seq(C,n)
                                                    by A9,A12,A21,A22,Th50;
      hence contradiction by A15,PARTFUN2:4;
     suppose A23: i1 = i2+1 & j1 = j2;
      then Gauge(C,n)*(i1,j1)`2 = Gauge(C,n)*(1,j2)`2 by A11,GOBOARD5:2
         .= Gauge(C,n)*(i2,j2)`2 by A12,GOBOARD5:2;
      then LSeg(Upper_Seq(C,n)/.ii,Upper_Seq(C,n)/.(ii+1)) is horizontal
                                                         by A7,A9,SPPOL_1:36;
      then Gi1`2 = Gauge(C,n)*(i1,j1)`2 by A4,A5,A7,SPRECT_3:19;
      then A24: j1 = 1 by A6,A14,Th6;
        i1 > 1 by A12,A23,NAT_1:38;
      then not Upper_Seq(C,n)/.ii in rng Upper_Seq(C,n) by A7,A11,A24,Th50;
      hence contradiction by A16,PARTFUN2:4;
     suppose A25: i1 = i2 & j1 = j2+1;
      then Gauge(C,n)*(i1,j1)`1 = Gauge(C,n)*(i2,1)`1 by A11,GOBOARD5:3
         .= Gauge(C,n)*(i2,j2)`1 by A12,GOBOARD5:3;
      then LSeg(Upper_Seq(C,n)/.ii,Upper_Seq(C,n)/.(ii+1)) is vertical
                                                         by A7,A9,SPPOL_1:37;
      then Gi1`1 = Gauge(C,n)*(i1,j1)`1 by A4,A5,A7,SPRECT_3:20;
      then A26: i1 = i by A6,A14,Th7;
        j2 <= j1 by A25,NAT_1:29;
      then Gauge(C,n)*(i2,j2)`2 <= Gauge(C,n)*(i1,j1)`2
                                                  by A11,A12,A25,SPRECT_3:24;
      then A27: Gauge(C,n)*(i2,j2)`2 <= Gi1`2 by A4,A5,A7,A9,TOPREAL1:10;
        Gi1`2 <= Gauge(C,n)*(i2,j2)`2 by A12,A25,A26,SPRECT_3:24;
      then Gi1`2 = Gauge(C,n)*(i2,j2)`2 by A27,AXIOMS:21;
      then j2 = 1 by A8,A14,Th6;
      hence contradiction by A9,A15,A17,A25,A26,PARTFUN2:4;
   end;

  theorem
     for C be compact connected non vertical non horizontal Subset of TOP-REAL
2
   for i be Nat st 1 <= i & i < len Gauge(C,n) holds
    not Gauge(C,n)*(i,width Gauge(C,n)) in L~Lower_Seq(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    set wi = width Gauge(C,n);
    let i be Nat such that
     A1: 1 <= i & i < len Gauge(C,n) and
     A2: Gauge(C,n)*(i,wi) in L~Lower_Seq(C,n);
    set Gi1 = Gauge(C,n)*(i,wi);
    consider ii be Nat such that
     A3: 1 <= ii & ii+1 <= len Lower_Seq(C,n) and
     A4: Gi1 in LSeg(Lower_Seq(C,n),ii) by A2,SPPOL_2:13;
    A5: LSeg(Lower_Seq(C,n),ii) =
       LSeg(Lower_Seq(C,n)/.ii,Lower_Seq(C,n)/.(ii+1)) by A3,TOPREAL1:def 5;
      Lower_Seq(C,n) is_sequence_on Gauge(C,n) by Th5;
    then consider i1,j1,i2,j2 being Nat such that
     A6: [i1,j1] in Indices Gauge(C,n) and
     A7: Lower_Seq(C,n)/.ii = Gauge(C,n)*(i1,j1) and
     A8: [i2,j2] in Indices Gauge(C,n) and
     A9: Lower_Seq(C,n)/.(ii+1) = Gauge(C,n)*(i2,j2) and
     A10: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or
          i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A3,JORDAN8:6;
    A11: 1 <= i1 & i1 <= len Gauge(C,n) & 1 <= j1 & j1 <= width Gauge(C,n)
                                                            by A6,GOBOARD5:1;
    A12: 1 <= i2 & i2 <= len Gauge(C,n) & 1 <= j2 & j2 <= width Gauge(C,n)
                                                            by A8,GOBOARD5:1;
    A13: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
      len Gauge(C,n) >= 4 by JORDAN8:13;
    then len Gauge(C,n) > 1 by AXIOMS:22;
    then A14: [i,wi] in Indices Gauge(C,n) by A1,A13,GOBOARD7:10;
      ii+1 >= 1 by NAT_1:29;
    then A15: ii+1 in dom Lower_Seq(C,n) by A3,FINSEQ_3:27;
      ii < len Lower_Seq(C,n) by A3,NAT_1:38;
    then A16: ii in dom Lower_Seq(C,n) by A3,FINSEQ_3:27;
    A17: not Gi1 in rng Lower_Seq(C,n) by A1,Th51;
    per cases by A10;
     suppose A18: i1 = i2 & j2+1 = j1;
      then Gauge(C,n)*(i1,j1)`1 = Gauge(C,n)*(i2,1)`1 by A11,GOBOARD5:3
         .= Gauge(C,n)*(i2,j2)`1 by A12,GOBOARD5:3;
      then LSeg(Lower_Seq(C,n)/.ii,Lower_Seq(C,n)/.(ii+1)) is vertical
                                                         by A7,A9,SPPOL_1:37;
      then Gi1`1 = Gauge(C,n)*(i1,j1)`1 by A4,A5,A7,SPRECT_3:20;
      then A19: i1 = i by A6,A14,Th7;
        j1 >= j2 by A18,NAT_1:29;
      then Gauge(C,n)*(i1,j1)`2 >= Gauge(C,n)*(i2,j2)`2
                                                  by A11,A12,A18,SPRECT_3:24;
      then A20: Gauge(C,n)*(i1,j1)`2 >= Gi1`2 by A4,A5,A7,A9,TOPREAL1:10;
        Gi1`2 >= Gauge(C,n)*(i1,j1)`2 by A11,A19,SPRECT_3:24;
      then Gi1`2 = Gauge(C,n)*(i1,j1)`2 by A20,AXIOMS:21;
      then j1 = wi by A6,A14,Th6;
      hence contradiction by A7,A16,A17,A19,PARTFUN2:4;
     suppose A21: i2+1 = i1 & j1 = j2;
      then Gauge(C,n)*(i1,j1)`2 = Gauge(C,n)*(1,j2)`2 by A11,GOBOARD5:2
         .= Gauge(C,n)*(i2,j2)`2 by A12,GOBOARD5:2;
      then LSeg(Lower_Seq(C,n)/.ii,Lower_Seq(C,n)/.(ii+1)) is horizontal
                                                         by A7,A9,SPPOL_1:36;
      then Gi1`2 = Gauge(C,n)*(i1,j1)`2 by A4,A5,A7,SPRECT_3:19;
      then A22: j1 = wi by A6,A14,Th6;
        i2 < len Gauge(C,n) by A11,A21,NAT_1:38;
      then not Lower_Seq(C,n)/.(ii+1) in rng Lower_Seq(C,n)
                                                   by A9,A12,A21,A22,Th51;
      hence contradiction by A15,PARTFUN2:4;
     suppose A23: i2 = i1+1 & j1 = j2;
      then Gauge(C,n)*(i1,j1)`2 = Gauge(C,n)*(1,j2)`2 by A11,GOBOARD5:2
         .= Gauge(C,n)*(i2,j2)`2 by A12,GOBOARD5:2;
      then LSeg(Lower_Seq(C,n)/.ii,Lower_Seq(C,n)/.(ii+1)) is horizontal
                                                         by A7,A9,SPPOL_1:36;
      then Gi1`2 = Gauge(C,n)*(i1,j1)`2 by A4,A5,A7,SPRECT_3:19;
      then A24: j1 = wi by A6,A14,Th6;
        i1 < len Gauge(C,n) by A12,A23,NAT_1:38;
      then not Lower_Seq(C,n)/.ii in rng Lower_Seq(C,n) by A7,A11,A24,Th51;
      hence contradiction by A16,PARTFUN2:4;
     suppose A25: i1 = i2 & j2 = j1+1;
      then Gauge(C,n)*(i1,j1)`1 = Gauge(C,n)*(i2,1)`1 by A11,GOBOARD5:3
         .= Gauge(C,n)*(i2,j2)`1 by A12,GOBOARD5:3;
      then LSeg(Lower_Seq(C,n)/.ii,Lower_Seq(C,n)/.(ii+1)) is vertical
                                                         by A7,A9,SPPOL_1:37;
      then Gi1`1 = Gauge(C,n)*(i1,j1)`1 by A4,A5,A7,SPRECT_3:20;
      then A26: i1 = i by A6,A14,Th7;
        j2 >= j1 by A25,NAT_1:29;
      then Gauge(C,n)*(i2,j2)`2 >= Gauge(C,n)*(i1,j1)`2
                                                  by A11,A12,A25,SPRECT_3:24;
      then A27: Gauge(C,n)*(i2,j2)`2 >= Gi1`2 by A4,A5,A7,A9,TOPREAL1:10;
        Gi1`2 >= Gauge(C,n)*(i2,j2)`2 by A12,A25,A26,SPRECT_3:24;
      then Gi1`2 = Gauge(C,n)*(i2,j2)`2 by A27,AXIOMS:21;
      then j2 = wi by A8,A14,Th6;
      hence contradiction by A9,A15,A17,A25,A26,PARTFUN2:4;
   end;

  theorem Th54:
   for C be compact connected non vertical non horizontal Subset of TOP-REAL 2
   for i,j be Nat st 1 <= i & i <= len Gauge(C,n) &
    1 <= j & j <= width Gauge(C,n) & Gauge(C,n)*(i,j) in L~Cage(C,n) holds
     LSeg(Gauge(C,n)*(i,1),Gauge(C,n)*(i,j)) meets L~Lower_Seq(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let i,j be Nat;
    set Gij = Gauge(C,n)*(i,j);
    assume that
     A1: 1 <= i & i <= len Gauge(C,n) and
     A2: 1 <= j & j <= width Gauge(C,n) and
     A3: Gij in L~Cage(C,n);
    set NE = NE-corner L~Cage(C,n);
    set v1 = L_Cut(Upper_Seq(C,n),Gij);
    set Gv1 = <*Gauge(C,n)*(i,1)*>^v1;
    set v = Gv1^<*NE*>;
    set h = mid(Lower_Seq(C,n),2,len Lower_Seq(C,n));
    A4: L~Cage(C,n) = L~Upper_Seq(C,n) \/ L~Lower_Seq(C,n) by JORDAN1E:17;
    A5: Lower_Seq(C,n) = Rotate(Cage(C,n),W-min L~Cage(C,n)):-
                                        E-max L~Cage(C,n) by JORDAN1E:def 2;
    A6: Upper_Seq(C,n) = Rotate(Cage(C,n),W-min L~Cage(C,n))-:
                                        E-max L~Cage(C,n) by JORDAN1E:def 1;
      len Lower_Seq(C,n) >= 3 & len Upper_Seq(C,n) >= 3 by JORDAN1E:19;
    then A7: len Lower_Seq(C,n) >= 2 & len Lower_Seq(C,n) >= 1 &
     len Upper_Seq(C,n) >= 2 & len Upper_Seq(C,n) >= 1 by AXIOMS:22;
    A8: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
    A9: Gauge(C,n)*(i,1)`2 = S-bound L~Cage(C,n) by A1,JORDAN1A:93;
    set Wmi = W-min L~Cage(C,n);
      now per cases by A1,A3,A4,REAL_1:def 5,XBOOLE_0:def 2;
     suppose A10: Gij in L~Upper_Seq(C,n) & i = 1;
      set G11 = Gauge(C,n)*(1,1);
      A11: G11`1 = W-bound L~Cage(C,n) by A1,A10,JORDAN1A:94;
      A12: Wmi`1 = W-bound L~Cage(C,n) by PSCOMP_1:84;
      A13: S-bound L~Cage(C,n) = G11`2 by A1,A10,JORDAN1A:93;
        Wmi in L~Cage(C,n) by SPRECT_1:15;
      then A14: G11`2 <= Wmi`2 by A13,PSCOMP_1:71;
      A15: Gij`1 = W-bound L~Cage(C,n) by A2,A8,A10,JORDAN1A:94;
      then Gij in W-most L~Cage(C,n) by A3,SPRECT_2:16;
      then Wmi`2 <= Gij`2 by PSCOMP_1:88;
      then A16: Wmi in LSeg(Gauge(C,n)*(1,1),Gauge(C,n)*(1,j))
                                             by A10,A11,A12,A14,A15,GOBOARD7:8;
      A17: rng Lower_Seq(C,n) c= L~Lower_Seq(C,n) by A7,SPPOL_2:18;
        Lower_Seq(C,n)/.(len Lower_Seq(C,n)) = Wmi by JORDAN1F:8;
      then Wmi in rng Lower_Seq(C,n) by REVROT_1:3;
      hence LSeg(Gauge(C,n)*(i,1),Gij) meets L~Lower_Seq(C,n)
                                                   by A10,A16,A17,XBOOLE_0:3;
     suppose A18: Gij in L~Upper_Seq(C,n) &
      Gij <> Upper_Seq(C,n).len Upper_Seq(C,n)
      & E-max L~Cage(C,n) <> NE & i > 1;
      then A19: v1 is non empty by JORDAN1E:7;
      then len v1 <> 0 by FINSEQ_1:25;
      then len v1 > 0 by NAT_1:19;
      then A20: 0+1 <= len v1 by NAT_1:38;
      then A21: 1 in dom v1 by FINSEQ_3:27;
      A22: len v1 in dom v1 & len Upper_Seq(C,n) in dom Upper_Seq(C,n)
                                                 by A7,A20,FINSEQ_3:27;
      A23: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:47;
        E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:50;
      then A24: E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                           by A23,FINSEQ_6:96;
      A25: v1/.(len v1) = v1.(len v1) by A22,FINSEQ_4:def 4
         .= Upper_Seq(C,n).len Upper_Seq(C,n) by A18,JORDAN1B:5
         .= Upper_Seq(C,n)/.len Upper_Seq(C,n) by A22,FINSEQ_4:def 4
         .= (Rotate(Cage(C,n),W-min L~Cage(C,n))-:E-max L~Cage(C,n))/.
            ((E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)))
                                                       by A6,A24,FINSEQ_5:45
         .= E-max L~Cage(C,n) by A24,FINSEQ_5:48;
      then A26: Gv1/.len Gv1 = E-max L~Cage(C,n) by A19,SPRECT_3:11;
      A27: v1/.1 = v1.1 by A21,FINSEQ_4:def 4
         .= Gij by A18,JORDAN3:58;
      then A28: (v1^<*NE*>)/.1 = Gij by A20,BOOLMARK:8;
      A29: 1+len v1 >= 1+1 by A20,REAL_1:55;
        len v = len Gv1 + 1 by FINSEQ_2:19
         .= 1 + len v1 + 1 by FINSEQ_5:8;
      then A30: 2 < len v by A29,NAT_1:38;
        S-bound L~Cage(C,n) < N-bound L~Cage(C,n) by SPRECT_1:34;
      then NE <> Gauge(C,n)*(i,1) by A9,PSCOMP_1:77;
      then not NE in {Gauge(C,n)*(i,1)} by TARSKI:def 1;
      then A31: not NE in rng <*Gauge(C,n)*(i,1)*> by FINSEQ_1:56;
        len Cage(C,n) > 4 by GOBOARD7:36;
      then len Cage(C,n) >= 2 by AXIOMS:22;
      then A32: rng Cage(C,n) c= L~Cage(C,n) by SPPOL_2:18;
      A33: not NE in rng Cage(C,n)
      proof
       assume A34: NE in rng Cage(C,n);
       A35: NE`1 = E-bound L~Cage(C,n) & NE`2 = N-bound L~Cage(C,n)
                                                  by PSCOMP_1:76,77;
       then NE`2 >= S-bound L~Cage(C,n) by SPRECT_1:24;
       then NE in { p where p is Point of TOP-REAL 2 :
        p`1 = E-bound L~Cage(C,n) & p`2 <= N-bound L~Cage(C,n) &
        p`2 >= S-bound L~Cage(C,n) } by A35;
       then NE in LSeg(SE-corner L~Cage(C,n), NE-corner L~Cage(C,n))
                                                              by SPRECT_1:25;
       then NE in LSeg(SE-corner L~Cage(C,n), NE-corner L~Cage(C,n)) /\
        L~Cage(C,n) by A32,A34,XBOOLE_0:def 3;
       then NE in E-most L~Cage(C,n) by PSCOMP_1:def 40;
       then A36: NE`2 <= (E-max L~Cage(C,n))`2 by PSCOMP_1:108;
         (E-max L~Cage(C,n))`2 <= NE`2 by PSCOMP_1:107;
       then A37: (E-max L~Cage(C,n))`2 = NE`2 by A36,AXIOMS:21;
         (E-max L~Cage(C,n))`1 = NE`1 by PSCOMP_1:105;
       hence contradiction by A18,A37,TOPREAL3:11;
      end;
        now per cases;
       suppose Gij <> Upper_Seq(C,n).(Index(Gij,Upper_Seq(C,n))+1);
        then v1 = <*Gij*>^mid(Upper_Seq(C,n),
         Index(Gij,Upper_Seq(C,n))+1,len Upper_Seq(C,n)) by JORDAN3:def 4;
        then rng v1 = rng <*Gij*> \/ rng mid(Upper_Seq(C,n),
         Index(Gij,Upper_Seq(C,n))+1,len Upper_Seq(C,n)) by FINSEQ_1:44;
        then A38: rng v1 = {Gij} \/ rng mid(Upper_Seq(C,n),
         Index(Gij,Upper_Seq(C,n))+1,len Upper_Seq(C,n)) by FINSEQ_1:55;
          not NE in L~Cage(C,n)
        proof
         assume NE in L~Cage(C,n);
         then consider i be Nat such that
          A39: 1 <= i and
          A40: i+1 <= len Cage(C,n) and
          A41: NE in LSeg(Cage(C,n)/.i,Cage(C,n)/.(i+1)) by SPPOL_2:14;
         per cases by A39,A40,TOPREAL1:def 7;
          suppose A42: (Cage(C,n)/.i)`1 = (Cage(C,n)/.(i+1))`1;
           then A43: NE`1 = (Cage(C,n)/.i)`1 by A41,GOBOARD7:5;
           A44: NE`2 = N-bound L~Cage(C,n) by PSCOMP_1:77;
           A45: i < len Cage(C,n) by A40,NAT_1:38;
           then A46: (Cage(C,n)/.i)`2 <= NE`2 by A39,A44,JORDAN5D:13;
           A47: 1 <= i+1 by NAT_1:29;
           then A48: (Cage(C,n)/.(i+1))`2 <= NE`2 by A40,A44,JORDAN5D:13;
           A49: i in dom Cage(C,n) & i+1 in dom Cage(C,n)
                                          by A39,A40,A45,A47,FINSEQ_3:27;
             (Cage(C,n)/.i)`2 <= (Cage(C,n)/.(i+1))`2 or
            (Cage(C,n)/.i)`2 >= (Cage(C,n)/.(i+1))`2;
           then NE`2 <= (Cage(C,n)/.(i+1))`2 or NE`2 <= (Cage(C,n)/.i)`2
                                                          by A41,TOPREAL1:10;
           then NE`2 = (Cage(C,n)/.(i+1))`2 or NE`2 = (Cage(C,n)/.i)`2
                                                        by A46,A48,AXIOMS:21;
           then NE = (Cage(C,n)/.(i+1)) or NE = (Cage(C,n)/.i)
                                                      by A42,A43,TOPREAL3:11;
           hence contradiction by A33,A49,PARTFUN2:4;
          suppose A50: (Cage(C,n)/.i)`2 = (Cage(C,n)/.(i+1))`2;
           then A51: NE`2 = (Cage(C,n)/.i)`2 by A41,GOBOARD7:6;
           A52: NE`1 = E-bound L~Cage(C,n) by PSCOMP_1:76;
           A53: i < len Cage(C,n) by A40,NAT_1:38;
           then A54: (Cage(C,n)/.i)`1 <= NE`1 by A39,A52,JORDAN5D:14;
           A55: 1 <= i+1 by NAT_1:29;
           then A56: (Cage(C,n)/.(i+1))`1 <= NE`1 by A40,A52,JORDAN5D:14;
           A57: i in dom Cage(C,n) & i+1 in dom Cage(C,n)
                                          by A39,A40,A53,A55,FINSEQ_3:27;
             (Cage(C,n)/.i)`1 <= (Cage(C,n)/.(i+1))`1 or
            (Cage(C,n)/.i)`1 >= (Cage(C,n)/.(i+1))`1;
           then NE`1 <= (Cage(C,n)/.(i+1))`1 or NE`1 <= (Cage(C,n)/.i)`1
                                                           by A41,TOPREAL1:9;
           then NE`1 = (Cage(C,n)/.(i+1))`1 or NE`1 = (Cage(C,n)/.i)`1
                                                        by A54,A56,AXIOMS:21;
           then NE = (Cage(C,n)/.(i+1)) or NE = (Cage(C,n)/.i)
                                                      by A50,A51,TOPREAL3:11;
           hence contradiction by A33,A57,PARTFUN2:4;
        end;
        then A58: not NE in {Gij} by A3,TARSKI:def 1;
        A59: rng mid(Upper_Seq(C,n),Index(Gij,Upper_Seq(C,n))+1,
         len Upper_Seq(C,n)) c= rng Upper_Seq(C,n) by JORDAN3:28;
          rng Upper_Seq(C,n) c= rng Cage(C,n) by Th47;
        then rng mid(Upper_Seq(C,n),Index(Gij,Upper_Seq(C,n))+1,
         len Upper_Seq(C,n)) c= rng Cage(C,n) by A59,XBOOLE_1:1;
        then not NE in rng mid(Upper_Seq(C,n),Index(Gij,Upper_Seq(C,n))+1,
         len Upper_Seq(C,n)) by A33;
        hence not NE in rng v1 by A38,A58,XBOOLE_0:def 2;
       suppose Gij = Upper_Seq(C,n).(Index(Gij,Upper_Seq(C,n))+1);
        then v1 = mid(Upper_Seq(C,n),
         Index(Gij,Upper_Seq(C,n))+1,len Upper_Seq(C,n)) by JORDAN3:def 4;
        then A60: rng v1 c= rng Upper_Seq(C,n) by JORDAN3:28;
          rng Upper_Seq(C,n) c= rng Cage(C,n) by Th47;
        then rng v1 c= rng Cage(C,n) by A60,XBOOLE_1:1;
        hence not NE in rng v1 by A33;
      end;
      then not NE in rng <*Gauge(C,n)*(i,1)*> \/ rng v1 by A31,XBOOLE_0:def 2;
      then not NE in rng Gv1 by FINSEQ_1:44;
      then rng Gv1 misses {NE} by ZFMISC_1:56;
      then A61: rng Gv1 misses rng <*NE*> by FINSEQ_1:55;
      A62: not Gauge(C,n)*(i,1) in L~Upper_Seq(C,n) by A1,A18,Th52;
        rng Upper_Seq(C,n) c= L~Upper_Seq(C,n) by A7,SPPOL_2:18;
      then A63: not Gauge(C,n)*(i,1) in rng Upper_Seq(C,n) by A1,A18,Th52;
        not Gauge(C,n)*(i,1) in {Gij} by A18,A62,TARSKI:def 1;
      then A64: not Gauge(C,n)*(i,1) in rng <*Gij*> by FINSEQ_1:55;
      set ci = mid(Upper_Seq(C,n),Index(Gij,Upper_Seq(C,n))+1,
       len Upper_Seq(C,n));
        now per cases;
       suppose A65: Gij <> Upper_Seq(C,n).(Index(Gij,Upper_Seq(C,n))+1);
          rng ci c= rng Upper_Seq(C,n) by JORDAN3:28;
        then not Gauge(C,n)*(i,1) in rng ci by A63;
        then not Gauge(C,n)*(i,1) in rng <*Gij*> \/ rng ci by A64,XBOOLE_0:def
2;
        then not Gauge(C,n)*(i,1) in rng(<*Gij*>^ci) by FINSEQ_1:44;
        hence not Gauge(C,n)*(i,1) in rng v1 by A65,JORDAN3:def 4;
       suppose Gij = Upper_Seq(C,n).(Index(Gij,Upper_Seq(C,n))+1);
        then v1 = ci by JORDAN3:def 4;
        then rng v1 c= rng Upper_Seq(C,n) by JORDAN3:28;
        hence not Gauge(C,n)*(i,1) in rng v1 by A63;
      end;
      then {Gauge(C,n)*(i,1)} misses rng v1 by ZFMISC_1:56;
      then A66: rng <*Gauge(C,n)*(i,1)*> misses rng v1 by FINSEQ_1:55;
      A67: <*Gauge(C,n)*(i,1)*> is one-to-one by FINSEQ_3:102;
      A68: v1 is_S-Seq by A18,JORDAN3:69;
      then v1 is one-to-one by TOPREAL1:def 10;
      then A69: Gv1 is one-to-one by A66,A67,FINSEQ_3:98;
        <*NE*> is one-to-one by FINSEQ_3:102;
      then A70: v is one-to-one by A61,A69,FINSEQ_3:98;
      A71: <*Gauge(C,n)*(i,1)*> is special by SPPOL_2:39;
      A72: v1 is special by A68,TOPREAL1:def 10;
        (<*Gauge(C,n)*(i,1)*>/.len <*Gauge(C,n)*(i,1)*>)`1 =
            (<*Gauge(C,n)*(i,1)*>/.1)`1 by FINSEQ_1:56
         .= Gauge(C,n)*(i,1)`1 by FINSEQ_4:25
         .= (v1/.1)`1 by A1,A2,A27,GOBOARD5:3;
      then A73: Gv1 is special by A71,A72,GOBOARD2:13;
      A74: <*NE*> is special by SPPOL_2:39;
        (Gv1/.len Gv1)`1 = NE`1 by A26,PSCOMP_1:105
         .= (<*NE*>/.1)`1 by FINSEQ_4:25;
      then A75: v is special by A73,A74,GOBOARD2:13;
      A76: len Lower_Seq(C,n) >= 2+1 by JORDAN1E:19;
      then A77: len Lower_Seq(C,n) > 2 by NAT_1:38;
      A78: len Lower_Seq(C,n) > 1 by A76,AXIOMS:22;
      then h is S-Sequence_in_R2 by A77,JORDAN3:39;
      then Rev h is S-Sequence_in_R2 by SPPOL_2:47;
      then A79: 2 <= len Rev h & Rev h is one-to-one & Rev h is special
                                                         by TOPREAL1:def 10;
      A80: Lower_Seq(C,n) is_in_the_area_of Cage(C,n) by JORDAN1E:22;
        3 <= len Lower_Seq(C,n) by JORDAN1E:19;
      then 2 <= len Lower_Seq(C,n) by AXIOMS:22;
      then A81: 2 in dom Lower_Seq(C,n) by FINSEQ_3:27;
      A82: len Lower_Seq(C,n) in dom Lower_Seq(C,n) by SCMFSA_7:12;
      then h is_in_the_area_of Cage(C,n) by A80,A81,SPRECT_2:26;
      then A83: Rev h is_in_the_area_of Cage(C,n) by SPRECT_3:68;
      A84: h is non empty by A77,A78,JORDAN1B:3;
      A85: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:47;
      A86: E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:50;
      then A87: E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                          by A85,FINSEQ_6:96;
      then Lower_Seq(C,n)/.(len Lower_Seq(C,n)) =
            Rotate(Cage(C,n),W-min L~Cage(C,n))/.
             (len Rotate(Cage(C,n),W-min L~Cage(C,n))) by A5,FINSEQ_5:57
         .= Rotate(Cage(C,n),W-min L~Cage(C,n))/.1 by FINSEQ_6:def 1
         .= W-min L~Cage(C,n) by A85,FINSEQ_6:98;
      then (Lower_Seq(C,n)/.len Lower_Seq(C,n))`1 = W-bound L~Cage(C,n)
                                                              by PSCOMP_1:84;
      then (h/.len h)`1 = W-bound L~Cage(C,n) by A81,A82,SPRECT_2:13;
      then A88: (Rev h/.1)`1 = W-bound L~Cage(C,n) by A84,FINSEQ_5:68;
      A89: Cage(C,n)/.1 = N-min L~Cage(C,n) by JORDAN9:34;
      then A90: (E-max L~Cage(C,n))..Cage(C,n) < (E-min L~Cage(C,n))..Cage(C,n)
                                                           by SPRECT_2:75;
      A91: (E-min L~Cage(C,n))..Cage(C,n) <= (S-max L~Cage(C,n))..Cage(C,n)
                                                           by A89,SPRECT_2:76;
      A92: (S-max L~Cage(C,n))..Cage(C,n) < (S-min L~Cage(C,n))..Cage(C,n)
                                                           by A89,SPRECT_2:77;
      A93: (S-min L~Cage(C,n))..Cage(C,n) <= (W-min L~Cage(C,n))..Cage(C,n)
                                                           by A89,SPRECT_2:78;
        (E-max L~Cage(C,n))..Cage(C,n) < (S-max L~Cage(C,n))..Cage(C,n)
                                                          by A90,A91,AXIOMS:22;
      then (E-max L~Cage(C,n))..Cage(C,n) < (S-min L~Cage(C,n))..Cage(C,n)
                                                             by A92,AXIOMS:22;
      then A94: (E-max L~Cage(C,n))..Cage(C,n) < (W-min L~Cage(C,n))..Cage(C,n)
                                                             by A93,AXIOMS:22;
      then A95: (E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)) =
       len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)-
        (W-min L~Cage(C,n))..Cage(C,n) by A85,A86,SPRECT_5:10;
        0 <= (E-max L~Cage(C,n))..Cage(C,n) by NAT_1:18;
      then len Cage(C,n)+0 <= len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)
                                                                by AXIOMS:24;
      then len Cage(C,n)-(W-min L~Cage(C,n))..Cage(C,n) <=
       (E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                             by A95,REAL_1:49;
      then len Cage(C,n)-(W-min L~Cage(C,n))..Cage(C,n)+1 <=
       1+(E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)) by AXIOMS:24;
      then A96: len(Cage(C,n):-W-min L~Cage(C,n)) <=
       1+(E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                          by A85,FINSEQ_5:53;
        1+(E-max L~Cage(C,n))..Cage(C,n)<=0+(W-min L~Cage(C,n))..Cage(C,n)
                                                              by A94,NAT_1:38;
      then 1+(E-max L~Cage(C,n))..Cage(C,n)-(W-min L~Cage(C,n))..Cage(C,n)<=0
                                                                by REAL_1:86;
      then len Cage(C,n)+(1+(E-max L~Cage(C,n))..Cage(C,n)-
       (W-min L~Cage(C,n))..Cage(C,n)) <= len Cage(C,n)+0 by AXIOMS:24;
      then len Cage(C,n)+(1+(E-max L~Cage(C,n))..Cage(C,n))-
       (W-min L~Cage(C,n))..Cage(C,n) <= len Cage(C,n) by XCMPLX_1:29;
      then 1+(len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n))-
       (W-min L~Cage(C,n))..Cage(C,n) <= len Cage(C,n) by XCMPLX_1:1;
      then A97: 1+(E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)) <=
       len Cage(C,n) by A95,XCMPLX_1:29;
      A98: 1+(len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)-
       (W-min L~Cage(C,n))..Cage(C,n))+(W-min L~Cage(C,n))..Cage(C,n)-
        len Cage(C,n) = 1+(len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n))-
            (W-min L~Cage(C,n))..Cage(C,n)+(W-min L~Cage(C,n))..Cage(C,n)-
            len Cage(C,n) by XCMPLX_1:29
         .= 1+(len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n))+(-
            (W-min L~Cage(C,n))..Cage(C,n))+(W-min L~Cage(C,n))..Cage(C,n)-
            len Cage(C,n) by XCMPLX_0:def 8
         .= 1+(len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n))-len Cage(C,n)
                                                                 by XCMPLX_1:
139
         .= 1+len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)-len Cage(C,n)
                                                                 by XCMPLX_1:1
         .= 1+len Cage(C,n)-len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)
                                                                 by XCMPLX_1:29

         .= 1+(E-max L~Cage(C,n))..Cage(C,n) by XCMPLX_1:26;
      then A99: 1+(len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)-
       (W-min L~Cage(C,n))..Cage(C,n))+(W-min L~Cage(C,n))..Cage(C,n)-
        len Cage(C,n) >= 0 by NAT_1:18;
      A100: Lower_Seq(C,n)/.(1+1) = Rotate(Cage(C,n),W-min L~Cage(C,n))/.
       (1+(E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)))
                                                    by A5,A81,A87,FINSEQ_5:55
         .= Cage(C,n)/.(1+(E-max L~Cage(C,n))..Rotate(Cage(C,n),
             W-min L~Cage(C,n))+(W-min L~Cage(C,n))..Cage(C,n)-'len Cage(C,n))
                                                     by A85,A96,A97,REVROT_1:17
         .= Cage(C,n)/.((E-max L~Cage(C,n))..Cage(C,n)+1)
                                               by A95,A98,A99,BINARITH:def 3;
      A101: (N-max L~Cage(C,n))..Cage(C,n) <= (E-max L~Cage(C,n))..Cage(C,n)
                                                           by A89,SPRECT_2:74;
        (N-max L~Cage(C,n))..Cage(C,n) > 1 by A89,SPRECT_2:73;
      then A102: (E-max L~Cage(C,n))..Cage(C,n) > 1 by A101,AXIOMS:22;
        (W-min L~Cage(C,n))..Cage(C,n) < len Cage(C,n) by A89,SPRECT_2:80;
      then (E-max L~Cage(C,n))..Cage(C,n) < len Cage(C,n) by A94,AXIOMS:22;
      then A103: (E-max L~Cage(C,n))..Cage(C,n)+1 <= len Cage(C,n) by NAT_1:38;
        Cage(C,n)/.((E-max L~Cage(C,n))..Cage(C,n)) = E-max L~Cage(C,n)
                                                         by A86,FINSEQ_5:41;
      then (Cage(C,n)/.((E-max L~Cage(C,n))..Cage(C,n)+1))`1 =
       E-bound L~Cage(C,n) by A102,A103,JORDAN1E:24;
      then (h/.1)`1 = E-bound L~Cage(C,n) by A81,A82,A100,SPRECT_2:12;
      then (Rev h/.len h)`1 = E-bound L~Cage(C,n) by A84,FINSEQ_5:68;
      then (Rev h/.len Rev h)`1 = E-bound L~Cage(C,n) by FINSEQ_5:def 3;
      then A104: Rev h is_a_h.c._for Cage(C,n) by A83,A88,SPRECT_2:def 2;
        now let m be Nat;
       assume A105: m in dom <*Gauge(C,n)*(i,1)*>;
       then m in Seg 1 by FINSEQ_1:55;
       then m = 1 by FINSEQ_1:4,TARSKI:def 1;
       then <*Gauge(C,n)*(i,1)*>.m = Gauge(C,n)*(i,1) by FINSEQ_1:57;
       then A106: <*Gauge(C,n)*(i,1)*>/.m = Gauge(C,n)*(i,1)
                                                      by A105,FINSEQ_4:def 4;
         width Gauge(C,n) >= 4 by A8,JORDAN8:13;
       then A107: 1 <= width Gauge(C,n) by AXIOMS:22;
       then Gauge(C,n)*(1,1)`1 <= Gauge(C,n)*(i,1)`1 by A1,SPRECT_3:25;
       hence W-bound L~Cage(C,n) <= (<*Gauge(C,n)*(i,1)*>/.m)`1
                                                  by A8,A106,A107,JORDAN1A:94;
         (Gauge(C,n)*(i,1))`1 <= Gauge(C,n)*(len Gauge(C,n),1)`1
                                                       by A1,A107,SPRECT_3:25;
       hence (<*Gauge(C,n)*(i,1)*>/.m)`1 <= E-bound L~Cage(C,n)
                                                  by A8,A106,A107,JORDAN1A:92;
       thus S-bound L~Cage(C,n) <= (<*Gauge(C,n)*(i,1)*>/.m)`2
                                                       by A1,A106,JORDAN1A:93;
         S-bound L~Cage(C,n) = Gauge(C,n)*(i,1)`2 by A1,JORDAN1A:93;
       hence (<*Gauge(C,n)*(i,1)*>/.m)`2 <= N-bound L~Cage(C,n)
                                                          by A106,SPRECT_1:24;
      end;
      then A108: <*Gauge(C,n)*(i,1)*> is_in_the_area_of Cage(C,n)
                                                          by SPRECT_2:def 1;
      A109: Upper_Seq(C,n) is_in_the_area_of Cage(C,n) by JORDAN1E:21;
      then <*Gij*> is_in_the_area_of Cage(C,n) by A18,SPRECT_3:63;
      then v1 is_in_the_area_of Cage(C,n) by A18,A109,SPRECT_3:73;
      then A110: Gv1 is_in_the_area_of Cage(C,n) by A108,SPRECT_2:28;
        <*NE*> is_in_the_area_of Cage(C,n) by SPRECT_2:29;
      then A111: v is_in_the_area_of Cage(C,n) by A110,SPRECT_2:28;
        v = <*Gauge(C,n)*(i,1)*>^(v1^<*NE*>) by FINSEQ_1:45;
      then v/.1 = Gauge(C,n)*(i,1) by FINSEQ_5:16;
      then A112: (v/.1)`2 = S-bound L~Cage(C,n) by A1,JORDAN1A:93;
        len v = len Gv1 + 1 by FINSEQ_2:19;
      then v/.(len v) = NE by TOPREAL4:1;
      then (v/.len v)`2 = N-bound L~Cage(C,n) by PSCOMP_1:77;
      then v is_a_v.c._for Cage(C,n) by A111,A112,SPRECT_2:def 3;
      then L~Rev h meets L~v by A30,A70,A75,A79,A104,SPRECT_2:33;
      then L~h meets L~v by SPPOL_2:22;
      then consider x be set such that
       A113: x in L~h and
       A114: x in L~v by XBOOLE_0:3;
      A115: L~h c= L~Lower_Seq(C,n) by A7,JORDAN4:47;
      A116: L~v1 c= L~Upper_Seq(C,n) by A18,JORDAN3:77;
        L~v = L~(<*Gauge(C,n)*(i,1)*>^(v1^<*NE*>)) by FINSEQ_1:45
         .= LSeg(Gauge(C,n)*(i,1),(v1^<*NE*>)/.1) \/ L~(v1^<*NE*>)
                                                                by SPPOL_2:20
         .= LSeg(Gauge(C,n)*(i,1),(v1^<*NE*>)/.1) \/
            (L~v1 \/ LSeg(v1/.(len v1),NE)) by A19,SPPOL_2:19;
      then A117: x in LSeg(Gauge(C,n)*(i,1),(v1^<*NE*>)/.1) or
       x in L~v1 \/ LSeg(v1/.(len v1),NE) by A114,XBOOLE_0:def 2;
        Lower_Seq(C,n)/.1 = (Rotate(Cage(C,n),W-min L~Cage(C,n)):-
                                     E-max L~Cage(C,n))/.1 by JORDAN1E:def 2
         .= E-max L~Cage(C,n) by FINSEQ_5:56;
      then A118: not E-max L~Cage(C,n) in L~h by A77,JORDAN5B:16;
        now per cases by A117,XBOOLE_0:def 2;
       suppose x in LSeg(Gauge(C,n)*(i,1),(v1^<*NE*>)/.1);
        then x in L~<*Gauge(C,n)*(i,1),Gij*> by A28,SPPOL_2:21;
        hence L~Lower_Seq(C,n) meets L~<*Gauge(C,n)*(i,1),Gij*>
                                                 by A113,A115,XBOOLE_0:3;
       suppose A119: x in L~v1;
        then x in L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n)
                                           by A113,A115,A116,XBOOLE_0:def 3;
        then x in {W-min L~Cage(C,n),E-max L~Cage(C,n)} by JORDAN1E:20;
        then A120: x = W-min L~Cage(C,n) by A113,A118,TARSKI:def 2;
          1 in dom Upper_Seq(C,n) by A7,FINSEQ_3:27;
        then Upper_Seq(C,n).1 = Upper_Seq(C,n)/.1 by FINSEQ_4:def 4
           .= Rotate(Cage(C,n),W-min L~Cage(C,n))/.1 by A6,A24,FINSEQ_5:47
           .= W-min L~Cage(C,n) by A23,FINSEQ_6:98;
        then x = Gij by A18,A119,A120,JORDAN1E:11;
        then x in LSeg(Gauge(C,n)*(i,1),Gij) by TOPREAL1:6;
        then x in L~<*Gauge(C,n)*(i,1),Gij*> by SPPOL_2:21;
        hence L~Lower_Seq(C,n) meets L~<*Gauge(C,n)*(i,1),Gij*>
                                              by A113,A115,XBOOLE_0:3;
       suppose A121: x in LSeg(v1/.(len v1),NE);
          x in L~Cage(C,n) by A4,A113,A115,XBOOLE_0:def 2;
        then x in LSeg(E-max L~Cage(C,n), NE) /\ L~Cage(C,n)
                                              by A25,A121,XBOOLE_0:def 3;
        then x in {E-max L~Cage(C,n)} by PSCOMP_1:112;
        hence L~Lower_Seq(C,n) meets L~<*Gauge(C,n)*(i,1),Gij*>
                                              by A113,A118,TARSKI:def 1;
      end;
      then L~<*Gauge(C,n)*(i,1),Gij*> meets L~Lower_Seq(C,n);
      hence LSeg(Gauge(C,n)*(i,1),Gij) meets L~Lower_Seq(C,n) by SPPOL_2:21;
     suppose A122: Gij in L~Upper_Seq(C,n) &
      Gij <> Upper_Seq(C,n).len Upper_Seq(C,n)
      & E-max L~Cage(C,n) = NE & i > 1;
      then A123: v1 is non empty by JORDAN1E:7;
      then len v1 <> 0 by FINSEQ_1:25;
      then len v1 > 0 by NAT_1:19;
      then A124: 0+1 <= len v1 by NAT_1:38;
      then A125: 1 in dom v1 by FINSEQ_3:27;
      A126: len v1 in dom v1 & len Upper_Seq(C,n) in dom Upper_Seq(C,n)
                                                    by A7,A124,FINSEQ_3:27;
      A127: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:47;
        E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:50;
      then A128: E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                           by A127,FINSEQ_6:96;
        v1/.(len v1) = v1.(len v1) by A126,FINSEQ_4:def 4
         .= Upper_Seq(C,n).len Upper_Seq(C,n) by A122,JORDAN1B:5
         .= Upper_Seq(C,n)/.len Upper_Seq(C,n) by A126,FINSEQ_4:def 4
         .= (Rotate(Cage(C,n),W-min L~Cage(C,n))-:E-max L~Cage(C,n))/.
            ((E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)))
                                                       by A6,A128,FINSEQ_5:45
         .= E-max L~Cage(C,n) by A128,FINSEQ_5:48;
      then A129: Gv1/.len Gv1 = E-max L~Cage(C,n) by A123,SPRECT_3:11;
      A130: v1/.1 = v1.1 by A125,FINSEQ_4:def 4
         .= Gij by A122,JORDAN3:58;
        1+len v1 >= 1+1 by A124,REAL_1:55;
      then A131: len Gv1 >= 2 by FINSEQ_5:8;
      A132: not Gauge(C,n)*(i,1) in L~Upper_Seq(C,n) by A1,A122,Th52;
        rng Upper_Seq(C,n) c= L~Upper_Seq(C,n) by A7,SPPOL_2:18;
      then A133: not Gauge(C,n)*(i,1) in rng Upper_Seq(C,n) by A1,A122,Th52;
        not Gauge(C,n)*(i,1) in {Gij} by A122,A132,TARSKI:def 1;
      then A134: not Gauge(C,n)*(i,1) in rng <*Gij*> by FINSEQ_1:55;
      set ci = mid(Upper_Seq(C,n),Index(Gij,Upper_Seq(C,n))+1,
       len Upper_Seq(C,n));
        now per cases;
       suppose A135: Gij <> Upper_Seq(C,n).(Index(Gij,Upper_Seq(C,n))+1);
          rng ci c= rng Upper_Seq(C,n) by JORDAN3:28;
        then not Gauge(C,n)*(i,1) in rng ci by A133;
        then not Gauge(C,n)*(i,1) in rng <*Gij*> \/ rng ci by A134,XBOOLE_0:def
2;
        then not Gauge(C,n)*(i,1) in rng(<*Gij*>^ci) by FINSEQ_1:44;
        hence not Gauge(C,n)*(i,1) in rng v1 by A135,JORDAN3:def 4;
       suppose Gij = Upper_Seq(C,n).(Index(Gij,Upper_Seq(C,n))+1);
        then v1 = ci by JORDAN3:def 4;
        then rng v1 c= rng Upper_Seq(C,n) by JORDAN3:28;
        hence not Gauge(C,n)*(i,1) in rng v1 by A133;
      end;
      then {Gauge(C,n)*(i,1)} misses rng v1 by ZFMISC_1:56;
      then A136: rng <*Gauge(C,n)*(i,1)*> misses rng v1 by FINSEQ_1:55;
      A137: <*Gauge(C,n)*(i,1)*> is one-to-one by FINSEQ_3:102;
      A138: v1 is_S-Seq by A122,JORDAN3:69;
      then v1 is one-to-one by TOPREAL1:def 10;
      then A139: Gv1 is one-to-one by A136,A137,FINSEQ_3:98;
      A140: <*Gauge(C,n)*(i,1)*> is special by SPPOL_2:39;
      A141: v1 is special by A138,TOPREAL1:def 10;
        (<*Gauge(C,n)*(i,1)*>/.len <*Gauge(C,n)*(i,1)*>)`1 =
            (<*Gauge(C,n)*(i,1)*>/.1)`1 by FINSEQ_1:56
         .= Gauge(C,n)*(i,1)`1 by FINSEQ_4:25
         .= (v1/.1)`1 by A1,A2,A130,GOBOARD5:3;
      then A142: Gv1 is special by A140,A141,GOBOARD2:13;
      A143: len Lower_Seq(C,n) >= 2+1 by JORDAN1E:19;
      then A144: len Lower_Seq(C,n) > 2 by NAT_1:38;
      A145: len Lower_Seq(C,n) > 1 by A143,AXIOMS:22;
      then h is S-Sequence_in_R2 by A144,JORDAN3:39;
      then Rev h is S-Sequence_in_R2 by SPPOL_2:47;
      then A146: 2 <= len Rev h & Rev h is one-to-one & Rev h is special
                                                         by TOPREAL1:def 10;
      A147: Lower_Seq(C,n) is_in_the_area_of Cage(C,n) by JORDAN1E:22;
        3 <= len Lower_Seq(C,n) by JORDAN1E:19;
      then 2 <= len Lower_Seq(C,n) by AXIOMS:22;
      then A148: 2 in dom Lower_Seq(C,n) by FINSEQ_3:27;
      A149: len Lower_Seq(C,n) in dom Lower_Seq(C,n) by SCMFSA_7:12;
      then h is_in_the_area_of Cage(C,n) by A147,A148,SPRECT_2:26;
      then A150: Rev h is_in_the_area_of Cage(C,n) by SPRECT_3:68;
      A151: h is non empty by A144,A145,JORDAN1B:3;
      A152: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:47;
      A153: E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:50;
      then A154: E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                          by A152,FINSEQ_6:96;
      then Lower_Seq(C,n)/.(len Lower_Seq(C,n)) =
            Rotate(Cage(C,n),W-min L~Cage(C,n))/.
             (len Rotate(Cage(C,n),W-min L~Cage(C,n))) by A5,FINSEQ_5:57
         .= Rotate(Cage(C,n),W-min L~Cage(C,n))/.1 by FINSEQ_6:def 1
         .= W-min L~Cage(C,n) by A152,FINSEQ_6:98;
      then (Lower_Seq(C,n)/.len Lower_Seq(C,n))`1 = W-bound L~Cage(C,n)
                                                              by PSCOMP_1:84;
      then (h/.len h)`1 = W-bound L~Cage(C,n) by A148,A149,SPRECT_2:13;
      then A155: (Rev h/.1)`1 = W-bound L~Cage(C,n) by A151,FINSEQ_5:68;
      A156: Cage(C,n)/.1 = N-min L~Cage(C,n) by JORDAN9:34;
      then A157: (E-max L~Cage(C,n))..Cage(C,n) < (E-min L~Cage(C,n))..Cage(C,n
)
                                                           by SPRECT_2:75;
      A158: (E-min L~Cage(C,n))..Cage(C,n) <= (S-max L~Cage(C,n))..Cage(C,n)
                                                           by A156,SPRECT_2:76;
      A159: (S-max L~Cage(C,n))..Cage(C,n) < (S-min L~Cage(C,n))..Cage(C,n)
                                                           by A156,SPRECT_2:77;
      A160: (S-min L~Cage(C,n))..Cage(C,n) <= (W-min L~Cage(C,n))..Cage(C,n)
                                                           by A156,SPRECT_2:78;
        (E-max L~Cage(C,n))..Cage(C,n) < (S-max L~Cage(C,n))..Cage(C,n)
                                                          by A157,A158,AXIOMS:
22;
      then (E-max L~Cage(C,n))..Cage(C,n) < (S-min L~Cage(C,n))..Cage(C,n)
                                                             by A159,AXIOMS:22;
      then A161: (E-max L~Cage(C,n))..Cage(C,n) < (W-min L~Cage(C,n))..Cage(C,n
)
                                                             by A160,AXIOMS:22;
      then A162: (E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)) =
       len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)-
        (W-min L~Cage(C,n))..Cage(C,n) by A152,A153,SPRECT_5:10;
        0 <= (E-max L~Cage(C,n))..Cage(C,n) by NAT_1:18;
      then len Cage(C,n)+0 <= len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)
                                                                by AXIOMS:24;
      then len Cage(C,n)-(W-min L~Cage(C,n))..Cage(C,n) <=
       (E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                             by A162,REAL_1:49;
      then len Cage(C,n)-(W-min L~Cage(C,n))..Cage(C,n)+1 <=
       1+(E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)) by AXIOMS:24;
      then A163: len(Cage(C,n):-W-min L~Cage(C,n)) <=
       1+(E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                          by A152,FINSEQ_5:53;
        1+(E-max L~Cage(C,n))..Cage(C,n)<=0+(W-min L~Cage(C,n))..Cage(C,n)
                                                              by A161,NAT_1:38;
      then 1+(E-max L~Cage(C,n))..Cage(C,n)-(W-min L~Cage(C,n))..Cage(C,n)<=0
                                                                by REAL_1:86;
      then len Cage(C,n)+(1+(E-max L~Cage(C,n))..Cage(C,n)-
       (W-min L~Cage(C,n))..Cage(C,n)) <= len Cage(C,n)+0 by AXIOMS:24;
      then len Cage(C,n)+(1+(E-max L~Cage(C,n))..Cage(C,n))-
       (W-min L~Cage(C,n))..Cage(C,n) <= len Cage(C,n) by XCMPLX_1:29;
      then 1+(len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n))-
       (W-min L~Cage(C,n))..Cage(C,n) <= len Cage(C,n) by XCMPLX_1:1;
      then A164: 1+(E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)) <=
       len Cage(C,n) by A162,XCMPLX_1:29;
      A165: 1+(len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)-
       (W-min L~Cage(C,n))..Cage(C,n))+(W-min L~Cage(C,n))..Cage(C,n)-
        len Cage(C,n) = 1+(len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n))-
            (W-min L~Cage(C,n))..Cage(C,n)+(W-min L~Cage(C,n))..Cage(C,n)-
            len Cage(C,n) by XCMPLX_1:29
         .= 1+(len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n))+(-
            (W-min L~Cage(C,n))..Cage(C,n))+(W-min L~Cage(C,n))..Cage(C,n)-
            len Cage(C,n) by XCMPLX_0:def 8
         .= 1+(len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n))-len Cage(C,n)
                                                                 by XCMPLX_1:
139
         .= 1+len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)-len Cage(C,n)
                                                                 by XCMPLX_1:1
         .= 1+len Cage(C,n)-len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)
                                                                 by XCMPLX_1:29

         .= 1+(E-max L~Cage(C,n))..Cage(C,n) by XCMPLX_1:26;
      then A166: 1+(len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)-
       (W-min L~Cage(C,n))..Cage(C,n))+(W-min L~Cage(C,n))..Cage(C,n)-
        len Cage(C,n) >= 0 by NAT_1:18;
      A167: Lower_Seq(C,n)/.(1+1) = Rotate(Cage(C,n),W-min L~Cage(C,n))/.
       (1+(E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)))
                                                    by A5,A148,A154,FINSEQ_5:55
         .= Cage(C,n)/.(1+(E-max L~Cage(C,n))..Rotate(Cage(C,n),
             W-min L~Cage(C,n))+(W-min L~Cage(C,n))..Cage(C,n)-'len Cage(C,n))
                                     by A152,A163,A164,REVROT_1:17
         .= Cage(C,n)/.((E-max L~Cage(C,n))..Cage(C,n)+1)
                                     by A162,A165,A166,BINARITH:def 3;
      A168: (N-max L~Cage(C,n))..Cage(C,n) <= (E-max L~Cage(C,n))..Cage(C,n)
                                                           by A156,SPRECT_2:74;
        (N-max L~Cage(C,n))..Cage(C,n) > 1 by A156,SPRECT_2:73;
      then A169: (E-max L~Cage(C,n))..Cage(C,n) > 1 by A168,AXIOMS:22;
        (W-min L~Cage(C,n))..Cage(C,n) < len Cage(C,n) by A156,SPRECT_2:80;
      then (E-max L~Cage(C,n))..Cage(C,n) < len Cage(C,n) by A161,AXIOMS:22;
      then A170: (E-max L~Cage(C,n))..Cage(C,n)+1 <= len Cage(C,n) by NAT_1:38;
        Cage(C,n)/.((E-max L~Cage(C,n))..Cage(C,n)) = E-max L~Cage(C,n)
                                                         by A153,FINSEQ_5:41;
      then (Cage(C,n)/.((E-max L~Cage(C,n))..Cage(C,n)+1))`1 =
       E-bound L~Cage(C,n) by A169,A170,JORDAN1E:24;
      then (h/.1)`1 = E-bound L~Cage(C,n) by A148,A149,A167,SPRECT_2:12;
      then (Rev h/.len h)`1 = E-bound L~Cage(C,n) by A151,FINSEQ_5:68;
      then (Rev h/.len Rev h)`1 = E-bound L~Cage(C,n) by FINSEQ_5:def 3;
      then A171: Rev h is_a_h.c._for Cage(C,n) by A150,A155,SPRECT_2:def 2;
        now let m be Nat;
       assume A172: m in dom <*Gauge(C,n)*(i,1)*>;
       then m in Seg 1 by FINSEQ_1:55;
       then m = 1 by FINSEQ_1:4,TARSKI:def 1;
       then <*Gauge(C,n)*(i,1)*>.m = Gauge(C,n)*(i,1) by FINSEQ_1:57;
       then A173: <*Gauge(C,n)*(i,1)*>/.m = Gauge(C,n)*(i,1)
                                                      by A172,FINSEQ_4:def 4;
         width Gauge(C,n) >= 4 by A8,JORDAN8:13;
       then A174: 1 <= width Gauge(C,n) by AXIOMS:22;
       then Gauge(C,n)*(1,1)`1 <= Gauge(C,n)*(i,1)`1 by A1,SPRECT_3:25;
       hence W-bound L~Cage(C,n) <= (<*Gauge(C,n)*(i,1)*>/.m)`1
                                                  by A8,A173,A174,JORDAN1A:94;
         (Gauge(C,n)*(i,1))`1 <= Gauge(C,n)*(len Gauge(C,n),1)`1
                                                       by A1,A174,SPRECT_3:25;
       hence (<*Gauge(C,n)*(i,1)*>/.m)`1 <= E-bound L~Cage(C,n)
                                                  by A8,A173,A174,JORDAN1A:92;
       thus S-bound L~Cage(C,n) <= (<*Gauge(C,n)*(i,1)*>/.m)`2
                                                       by A1,A173,JORDAN1A:93;
         S-bound L~Cage(C,n) = Gauge(C,n)*(i,1)`2 by A1,JORDAN1A:93;
       hence (<*Gauge(C,n)*(i,1)*>/.m)`2 <= N-bound L~Cage(C,n)
                                                          by A173,SPRECT_1:24;
      end;
      then A175: <*Gauge(C,n)*(i,1)*> is_in_the_area_of Cage(C,n)
                                                          by SPRECT_2:def 1;
      A176: Upper_Seq(C,n) is_in_the_area_of Cage(C,n) by JORDAN1E:21;
      then <*Gij*> is_in_the_area_of Cage(C,n) by A122,SPRECT_3:63;
      then v1 is_in_the_area_of Cage(C,n) by A122,A176,SPRECT_3:73;
      then A177: Gv1 is_in_the_area_of Cage(C,n) by A175,SPRECT_2:28;
        Gv1/.1 = Gauge(C,n)*(i,1) by FINSEQ_5:16;
      then A178: (Gv1/.1)`2 = S-bound L~Cage(C,n) by A1,JORDAN1A:93;
        (Gv1/.len Gv1)`2 = N-bound L~Cage(C,n) by A122,A129,PSCOMP_1:77;
      then Gv1 is_a_v.c._for Cage(C,n) by A177,A178,SPRECT_2:def 3;
      then L~Rev h meets L~Gv1 by A131,A139,A142,A146,A171,SPRECT_2:33;
      then L~h meets L~Gv1 by SPPOL_2:22;
      then consider x be set such that
       A179: x in L~h and
       A180: x in L~Gv1 by XBOOLE_0:3;
      A181: L~h c= L~Lower_Seq(C,n) by A7,JORDAN4:47;
      A182: L~v1 c= L~Upper_Seq(C,n) by A122,JORDAN3:77;
      A183: L~Gv1 = LSeg(Gauge(C,n)*(i,1),v1/.1) \/ L~v1 by A123,SPPOL_2:20;
        Lower_Seq(C,n)/.1 = (Rotate(Cage(C,n),W-min L~Cage(C,n)):-
                                     E-max L~Cage(C,n))/.1 by JORDAN1E:def 2
         .= E-max L~Cage(C,n) by FINSEQ_5:56;
      then A184: not E-max L~Cage(C,n) in L~h by A144,JORDAN5B:16;
        now per cases by A180,A183,XBOOLE_0:def 2;
       suppose x in LSeg(Gauge(C,n)*(i,1),v1/.1);
        then x in L~<*Gauge(C,n)*(i,1),Gij*> by A130,SPPOL_2:21;
        hence L~Lower_Seq(C,n) meets L~<*Gauge(C,n)*(i,1),Gij*>
                                                by A179,A181,XBOOLE_0:3;
       suppose A185: x in L~v1;
        then x in L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n)
                                           by A179,A181,A182,XBOOLE_0:def 3;
        then x in {W-min L~Cage(C,n),E-max L~Cage(C,n)} by JORDAN1E:20;
        then A186: x = W-min L~Cage(C,n) by A179,A184,TARSKI:def 2;
          1 in dom Upper_Seq(C,n) by A7,FINSEQ_3:27;
        then Upper_Seq(C,n).1 = Upper_Seq(C,n)/.1 by FINSEQ_4:def 4
           .= Rotate(Cage(C,n),W-min L~Cage(C,n))/.1 by A6,A128,FINSEQ_5:47
           .= W-min L~Cage(C,n) by A127,FINSEQ_6:98;
        then x = Gij by A122,A185,A186,JORDAN1E:11;
        then x in LSeg(Gauge(C,n)*(i,1),Gij) by TOPREAL1:6;
        then x in L~<*Gauge(C,n)*(i,1),Gij*> by SPPOL_2:21;
        hence L~Lower_Seq(C,n) meets L~<*Gauge(C,n)*(i,1),Gij*>
                                              by A179,A181,XBOOLE_0:3;
      end;
      then L~<*Gauge(C,n)*(i,1),Gij*> meets L~Lower_Seq(C,n);
      hence LSeg(Gauge(C,n)*(i,1),Gij) meets L~Lower_Seq(C,n) by SPPOL_2:21;
     suppose A187: Gij in L~Lower_Seq(C,n);
        Gij in LSeg(Gauge(C,n)*(i,1),Gij) by TOPREAL1:6;
      hence LSeg(Gauge(C,n)*(i,1),Gij) meets L~Lower_Seq(C,n)
                                                         by A187,XBOOLE_0:3;
     suppose A188: Gij in L~Upper_Seq(C,n) &
       Gij = Upper_Seq(C,n).len Upper_Seq(C,n);
      then v1 is non empty by JORDAN1E:7;
      then len v1 <> 0 by FINSEQ_1:25;
      then len v1 > 0 by NAT_1:19;
      then 0+1 <= len v1 by NAT_1:38;
      then A189: len v1 in dom v1 & len Upper_Seq(C,n) in dom Upper_Seq(C,n)
                                                     by A7,FINSEQ_3:27;
      A190: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:47;
        E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:50;
      then A191: E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                           by A190,FINSEQ_6:96;
      A192: Upper_Seq(C,n).len Upper_Seq(C,n) =
            Upper_Seq(C,n)/.len Upper_Seq(C,n) by A189,FINSEQ_4:def 4
         .= (Rotate(Cage(C,n),W-min L~Cage(C,n))-:E-max L~Cage(C,n))/.
            ((E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)))
                                                       by A6,A191,FINSEQ_5:45
         .= E-max L~Cage(C,n) by A191,FINSEQ_5:48;
      A193: rng Lower_Seq(C,n) c= L~Lower_Seq(C,n) by A7,SPPOL_2:18;
      A194: E-max L~Cage(C,n) in rng Lower_Seq(C,n) by A5,FINSEQ_6:66;
        Gij in LSeg(Gauge(C,n)*(i,1),Gij) by TOPREAL1:6;
      hence LSeg(Gauge(C,n)*(i,1),Gij) meets L~Lower_Seq(C,n)
                                         by A188,A192,A193,A194,XBOOLE_0:3;
    end;
    hence thesis;
   end;

  theorem Th55:
   for C be compact connected non vertical non horizontal Subset of TOP-REAL 2
   for n be Nat st n > 0 holds
    First_Point(L~Upper_Seq(C,n),W-min L~Cage(C,n),E-max L~Cage(C,n),
     Vertical_Line ((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))
      in rng Upper_Seq(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let n be Nat;
    assume A1: n > 0;
    set Wmin = W-min L~Cage(C,n);
    set Emax = E-max L~Cage(C,n);
    set Wbo = W-bound L~Cage(C,n);
    set Ebo = E-bound L~Cage(C,n);
    set sr = (W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2;
    set FiP = First_Point(L~Upper_Seq(C,n),Wmin,Emax,Vertical_Line sr);
    A2: Upper_Seq(C,n)/.1 = W-min L~Cage(C,n) by JORDAN1F:5;
    A3: Upper_Seq(C,n)/.len Upper_Seq(C,n) = E-max L~Cage(C,n) by JORDAN1F:7;
    then A4: L~Upper_Seq(C,n) is_an_arc_of Wmin,Emax by A2,TOPREAL1:31;
      Wbo < Ebo by SPRECT_1:33;
    then Wbo < sr & sr < Ebo by TOPREAL3:3;
    then Wmin`1 <= sr & sr <= Emax`1 by PSCOMP_1:84,104;
    then A5: L~Upper_Seq(C,n) meets Vertical_Line(sr) &
         L~Upper_Seq(C,n) /\ Vertical_Line(sr) is closed by A4,JORDAN6:64;
    then FiP in L~Upper_Seq(C,n) /\ Vertical_Line sr by A4,JORDAN5C:def 1;
    then A6: FiP in L~Upper_Seq(C,n) & FiP in Vertical_Line sr
                                                             by XBOOLE_0:def 3;
    then consider t be Nat such that
     A7: 1 <= t & t+1 <= len Upper_Seq(C,n) and
     A8: FiP in LSeg(Upper_Seq(C,n),t) by SPPOL_2:13;
      Vertical_Line sr is closed by JORDAN6:33;
    then A9: FiP = First_Point (LSeg (Upper_Seq(C,n),t),Upper_Seq(C,n)/.t,
     Upper_Seq(C,n)/.(t+1),Vertical_Line sr)
                                          by A2,A3,A5,A7,A8,JORDAN5C:19;
    A10: LSeg(Upper_Seq(C,n),t) = LSeg(Upper_Seq(C,n)/.t,Upper_Seq(C,n)/.(t+1))
                                                      by A7,TOPREAL1:def 5;
    A11: FiP`1 = sr by A6,JORDAN6:34;
      t < len Upper_Seq(C,n) & 1 <= t+1 by A7,NAT_1:38;
    then A12: t in dom Upper_Seq(C,n) & t+1 in dom Upper_Seq(C,n)
                                                          by A7,FINSEQ_3:27;
    A13: 1 <= Center Gauge(C,n) & Center Gauge(C,n) <= len Gauge(C,n)
                                                           by JORDAN1B:12,14;
      now per cases by SPPOL_1:41;
     suppose LSeg(Upper_Seq(C,n),t) is vertical;
      then (Upper_Seq(C,n)/.t)`1 = sr & (Upper_Seq(C,n)/.(t+1))`1 = sr
                                              by A8,A10,A11,SPRECT_3:20;
      then Upper_Seq(C,n)/.t in {p where p is Point of TOP-REAL 2: p`1 = sr} &
       Upper_Seq(C,n)/.(t+1) in {p where p is Point of TOP-REAL 2: p`1 = sr};
      then Upper_Seq(C,n)/.t in Vertical_Line sr &
       Upper_Seq(C,n)/.(t+1) in Vertical_Line sr by JORDAN6:def 6;
      then A14: LSeg(Upper_Seq(C,n),t) c= Vertical_Line sr by A10,JORDAN1A:23;
      A15: LSeg(Upper_Seq(C,n),t) is closed by SPPOL_1:40;
        Upper_Seq(C,n)/.t <> Upper_Seq(C,n)/.(t+1) by A12,GOBOARD7:31;
      then LSeg(Upper_Seq(C,n),t) is_an_arc_of Upper_Seq(C,n)/.t,
                Upper_Seq(C,n)/.(t+1) by A10,TOPREAL1:15;
      then First_Point (LSeg(Upper_Seq(C,n),t),Upper_Seq(C,n)/.t,
      Upper_Seq(C,n)/.(t+1),Vertical_Line sr) = Upper_Seq(C,n)/.t
                                                   by A14,A15,JORDAN5C:7;
      hence FiP in rng Upper_Seq(C,n) by A9,A12,PARTFUN2:4;
     suppose LSeg(Upper_Seq(C,n),t) is horizontal;
      then A16: (Upper_Seq(C,n)/.t)`2 = (Upper_Seq(C,n)/.(t+1))`2
                                                           by A10,SPPOL_1:36;
      then A17: FiP`2 = (Upper_Seq(C,n)/.t)`2 by A8,A10,GOBOARD7:6;
        Upper_Seq(C,n) is_sequence_on Gauge(C,n) by Th4;
      then consider i1,j1,i2,j2 be Nat such that
       A18: [i1,j1] in Indices Gauge(C,n) and
       A19: Upper_Seq(C,n)/.t = Gauge(C,n)*(i1,j1) and
       A20: [i2,j2] in Indices Gauge(C,n) and
       A21: Upper_Seq(C,n)/.(t+1) = Gauge(C,n)*(i2,j2) and
       A22: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or
            i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A7,JORDAN8:6;
      A23: 1 <= i1 & i1 <= len Gauge(C,n) & 1 <= j1 & j1 <= width Gauge(C,n)
                                                           by A18,GOBOARD5:1;
      A24: 1 <= i2 & i2 <= len Gauge(C,n) & 1 <= j2 & j2 <= width Gauge(C,n)
                                                           by A20,GOBOARD5:1;
      A25: j1 = j2 by A16,A18,A19,A20,A21,Th6;
      A26: Gauge(C,n)*(Center Gauge(C,n),j1)`1 = (W-bound C + E-bound C)/2
                                                            by A1,A23,Th43
         .= FiP`1 by A11,Th41;
        Gauge(C,n)*(Center Gauge(C,n),j1)`2 = Gauge(C,n)*(1,j1)`2
                                                        by A13,A23,GOBOARD5:2
         .= FiP`2 by A17,A19,A23,GOBOARD5:2;
      then A27: FiP = Gauge(C,n)*(Center Gauge(C,n),j1) by A26,TOPREAL3:11;
        now per cases by A22,A25,NAT_1:38;
       suppose A28: i1+1 = i2;
          i1 < i1+1 by NAT_1:38;
        then Gauge(C,n)*(i1,j1)`1 <= Gauge(C,n)*(i1+1,j1)`1
                                                  by A23,A24,A28,SPRECT_3:25;
        then Gauge(C,n)*(i1,j1)`1 <= FiP`1 & FiP`1 <= Gauge(C,n)*(i1+1,j1)`1
                                       by A8,A10,A19,A21,A25,A28,TOPREAL1:9;
        then A29: i1 <= Center Gauge(C,n) & Center Gauge(C,n) <= i1+1
                                     by A13,A23,A24,A26,A28,GOBOARD5:4;
        then i1 = Center Gauge(C,n) or i1 < Center Gauge(C,n) by REAL_1:def 5;
        then i1 = Center Gauge(C,n) or i1+1 <= Center Gauge(C,n) by NAT_1:38;
        then i1 = Center Gauge(C,n) or i1+1 = Center Gauge(C,n)
                                                            by A29,AXIOMS:21;
        hence FiP in rng Upper_Seq(C,n) by A12,A19,A21,A25,A27,A28,PARTFUN2:4;
       suppose A30: i1 = i2+1;
          i2 < i2+1 by NAT_1:38;
        then Gauge(C,n)*(i2,j1)`1 <= Gauge(C,n)*(i2+1,j1)`1
                                                  by A23,A24,A30,SPRECT_3:25;
        then Gauge(C,n)*(i2,j1)`1 <= FiP`1 & FiP`1 <= Gauge(C,n)*(i2+1,j1)`1
                                       by A8,A10,A19,A21,A25,A30,TOPREAL1:9;
        then A31: i2 <= Center Gauge(C,n) & Center Gauge(C,n) <= i2+1
                                     by A13,A23,A24,A26,A30,GOBOARD5:4;
        then i2 = Center Gauge(C,n) or i2 < Center Gauge(C,n) by REAL_1:def 5;
        then i2 = Center Gauge(C,n) or i2+1 <= Center Gauge(C,n) by NAT_1:38;
        then i2 = Center Gauge(C,n) or i2+1 = Center Gauge(C,n)
                                                            by A31,AXIOMS:21;
        hence FiP in rng Upper_Seq(C,n) by A12,A19,A21,A25,A27,A30,PARTFUN2:4;
      end;
      hence FiP in rng Upper_Seq(C,n);
    end;
    hence thesis;
   end;

  theorem Th56:
   for C be compact connected non vertical non horizontal Subset of TOP-REAL 2
   for n be Nat st n > 0 holds
    Last_Point(L~Lower_Seq(C,n),E-max L~Cage(C,n),W-min L~Cage(C,n),
     Vertical_Line ((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))
      in rng Lower_Seq(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let n be Nat;
    assume A1: n > 0;
    set Wmin = W-min L~Cage(C,n);
    set Emax = E-max L~Cage(C,n);
    set Wbo = W-bound L~Cage(C,n);
    set Ebo = E-bound L~Cage(C,n);
    set sr = (W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2;
    set LaP = Last_Point(L~Lower_Seq(C,n),Emax,Wmin,Vertical_Line sr);
    A2: Lower_Seq(C,n)/.1 = E-max L~Cage(C,n) by JORDAN1F:6;
    A3: Lower_Seq(C,n)/.len Lower_Seq(C,n) = W-min L~Cage(C,n) by JORDAN1F:8;
    then A4: L~Lower_Seq(C,n) is_an_arc_of Emax,Wmin by A2,TOPREAL1:31;
    then A5: L~Lower_Seq(C,n) is_an_arc_of Wmin,Emax by JORDAN5B:14;
      Wbo <= Ebo by SPRECT_1:23;
    then Wbo <= sr & sr <= Ebo by JORDAN6:2;
    then Wmin`1 <= sr & sr <= Emax`1 by PSCOMP_1:84,104;
    then A6: L~Lower_Seq(C,n) meets Vertical_Line(sr) &
         L~Lower_Seq(C,n) /\ Vertical_Line(sr) is closed by A5,JORDAN6:64;
    then LaP in L~Lower_Seq(C,n) /\ Vertical_Line sr by A4,JORDAN5C:def 2;
    then A7: LaP in L~Lower_Seq(C,n) & LaP in Vertical_Line sr
                                                             by XBOOLE_0:def 3;
    then consider t be Nat such that
     A8: 1 <= t & t+1 <= len Lower_Seq(C,n) and
     A9: LaP in LSeg(Lower_Seq(C,n),t) by SPPOL_2:13;
      Vertical_Line sr is closed by JORDAN6:33;
    then A10: LaP = Last_Point (LSeg (Lower_Seq(C,n),t),Lower_Seq(C,n)/.t,
     Lower_Seq(C,n)/.(t+1),Vertical_Line sr)
                                          by A2,A3,A6,A8,A9,JORDAN5C:20;
    A11: LSeg(Lower_Seq(C,n),t) = LSeg(Lower_Seq(C,n)/.t,Lower_Seq(C,n)/.(t+1))
                                                      by A8,TOPREAL1:def 5;
    A12: LaP`1 = sr by A7,JORDAN6:34;
      t < len Lower_Seq(C,n) & 1 <= t+1 by A8,NAT_1:38;
    then A13: t in dom Lower_Seq(C,n) & t+1 in dom Lower_Seq(C,n)
                                                          by A8,FINSEQ_3:27;
      now per cases by SPPOL_1:41;
     suppose LSeg(Lower_Seq(C,n),t) is vertical;
      then (Lower_Seq(C,n)/.t)`1 = sr & (Lower_Seq(C,n)/.(t+1))`1 = sr
                                              by A9,A11,A12,SPRECT_3:20;
      then Lower_Seq(C,n)/.t in {p where p is Point of TOP-REAL 2: p`1 = sr} &
       Lower_Seq(C,n)/.(t+1) in {p where p is Point of TOP-REAL 2: p`1 = sr};
      then Lower_Seq(C,n)/.t in Vertical_Line sr &
       Lower_Seq(C,n)/.(t+1) in Vertical_Line sr by JORDAN6:def 6;
      then A14: LSeg(Lower_Seq(C,n),t) c= Vertical_Line sr by A11,JORDAN1A:23;
      A15: LSeg(Lower_Seq(C,n),t) is closed by SPPOL_1:40;
        Lower_Seq(C,n)/.t <> Lower_Seq(C,n)/.(t+1) by A13,GOBOARD7:31;
      then LSeg(Lower_Seq(C,n),t) is_an_arc_of Lower_Seq(C,n)/.t,
                Lower_Seq(C,n)/.(t+1) by A11,TOPREAL1:15;
      then Last_Point (LSeg(Lower_Seq(C,n),t),Lower_Seq(C,n)/.t,
      Lower_Seq(C,n)/.(t+1),Vertical_Line sr) = Lower_Seq(C,n)/.(t+1)
                                                   by A14,A15,JORDAN5C:7;
      hence LaP in rng Lower_Seq(C,n) by A10,A13,PARTFUN2:4;
     suppose LSeg(Lower_Seq(C,n),t) is horizontal;
      then A16: (Lower_Seq(C,n)/.t)`2 = (Lower_Seq(C,n)/.(t+1))`2
                                                           by A11,SPPOL_1:36;
      then A17: LaP`2 = (Lower_Seq(C,n)/.t)`2 by A9,A11,GOBOARD7:6;
        Lower_Seq(C,n) is_sequence_on Gauge(C,n) by Th5;
      then consider i1,j1,i2,j2 be Nat such that
       A18: [i1,j1] in Indices Gauge(C,n) and
       A19: Lower_Seq(C,n)/.t = Gauge(C,n)*(i1,j1) and
       A20: [i2,j2] in Indices Gauge(C,n) and
       A21: Lower_Seq(C,n)/.(t+1) = Gauge(C,n)*(i2,j2) and
       A22: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or
            i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A8,JORDAN8:6;
      A23: 1 <= i1 & i1 <= len Gauge(C,n) & 1 <= j1 & j1 <= width Gauge(C,n)
                                                           by A18,GOBOARD5:1;
      A24: 1 <= i2 & i2 <= len Gauge(C,n) & 1 <= j2 & j2 <= width Gauge(C,n)
                                                           by A20,GOBOARD5:1;
      A25: j1 = j2 by A16,A18,A19,A20,A21,Th6;
      A26: 1 <= Center Gauge(C,n) & Center Gauge(C,n) <= len Gauge(C,n)
                                                           by JORDAN1B:12,14;
      A27: Gauge(C,n)*(Center Gauge(C,n),j1)`1 = (W-bound C + E-bound C)/2
                                                            by A1,A23,Th43
         .= LaP`1 by A12,Th41;
        Gauge(C,n)*(Center Gauge(C,n),j1)`2 = Gauge(C,n)*(1,j1)`2
                                                        by A23,A26,GOBOARD5:2
         .= LaP`2 by A17,A19,A23,GOBOARD5:2;
      then A28: LaP = Gauge(C,n)*(Center Gauge(C,n),j1) by A27,TOPREAL3:11;
        now per cases by A22,A25,NAT_1:38;
       suppose A29: i1+1 = i2;
          i1 < i1+1 by NAT_1:38;
        then Gauge(C,n)*(i1,j1)`1 <= Gauge(C,n)*(i1+1,j1)`1
                                                  by A23,A24,A29,SPRECT_3:25;
        then Gauge(C,n)*(i1,j1)`1 <= LaP`1 & LaP`1 <= Gauge(C,n)*(i1+1,j1)`1
                                       by A9,A11,A19,A21,A25,A29,TOPREAL1:9;
        then A30: i1 <= Center Gauge(C,n) & Center Gauge(C,n) <= i1+1
                                     by A23,A24,A26,A27,A29,GOBOARD5:4;
        then i1 = Center Gauge(C,n) or i1 < Center Gauge(C,n) by REAL_1:def 5;
        then i1 = Center Gauge(C,n) or i1+1 <= Center Gauge(C,n) by NAT_1:38;
        then i1 = Center Gauge(C,n) or i1+1 = Center Gauge(C,n)
                                                            by A30,AXIOMS:21;
        hence LaP in rng Lower_Seq(C,n) by A13,A19,A21,A25,A28,A29,PARTFUN2:4;
       suppose A31: i1 = i2+1;
          i2 < i2+1 by NAT_1:38;
        then Gauge(C,n)*(i2,j1)`1 <= Gauge(C,n)*(i2+1,j1)`1
                                                  by A23,A24,A31,SPRECT_3:25;
        then Gauge(C,n)*(i2,j1)`1 <= LaP`1 & LaP`1 <= Gauge(C,n)*(i2+1,j1)`1
                                       by A9,A11,A19,A21,A25,A31,TOPREAL1:9;
        then A32: i2 <= Center Gauge(C,n) & Center Gauge(C,n) <= i2+1
                                     by A23,A24,A26,A27,A31,GOBOARD5:4;
        then i2 = Center Gauge(C,n) or i2 < Center Gauge(C,n) by REAL_1:def 5;
        then i2 = Center Gauge(C,n) or i2+1 <= Center Gauge(C,n) by NAT_1:38;
        then i2 = Center Gauge(C,n) or i2+1 = Center Gauge(C,n)
                                                            by A32,AXIOMS:21;
        hence LaP in rng Lower_Seq(C,n) by A13,A19,A21,A25,A28,A31,PARTFUN2:4;
      end;
      hence LaP in rng Lower_Seq(C,n);
    end;
    hence thesis;
   end;

  theorem Th57:
   for f be S-Sequence_in_R2
   for p be Point of TOP-REAL 2 st p in rng f holds
    R_Cut(f,p) = mid(f,1,p..f)
   proof
    let f be S-Sequence_in_R2;
    let p be Point of TOP-REAL 2;
    assume A1: p in rng f;
    then consider i be Nat such that
     A2: i in dom f and
     A3: f.i = p by FINSEQ_2:11;
    A4: 0+1 <= i & i <= len f by A2,FINSEQ_3:27;
    then A5: i-1 >= 0 by REAL_1:84;
      len f >= 2 by TOPREAL1:def 10;
    then rng f c= L~f by SPPOL_2:18;
    then A6: 1 <= Index(p,f) & Index(p,f) < len f by A1,JORDAN3:41;
    per cases by A4,REAL_1:def 5;
     suppose A7: 1 < i;
      then A8: Index(p,f) + 1 = i by A3,A4,JORDAN3:45;
        1 <= len f by A4,AXIOMS:22;
      then 1 in dom f by FINSEQ_3:27;
      then p <> f.1 by A2,A3,A7,FUNCT_1:def 8;
      then A9: R_Cut(f,p) = mid(f,1,Index(p,f))^<*p*> by JORDAN3:def 5;
      A10: len mid(f,1,i) = i-'1+1 by A4,JORDAN4:20
         .= i by A4,AMI_5:4;
      A11: len mid(f,1,Index(p,f)) = Index(p,f)-'1+1 by A6,JORDAN4:20
         .= i-'1 by A6,A8,JORDAN4:3;
      then A12: len (mid(f,1,Index(p,f))^<*p*>) = i-'1+1 by FINSEQ_2:19
         .= i by A4,AMI_5:4;
        now let j be Nat;
       assume j in Seg i;
       then A13: 1 <= j & j <= i by FINSEQ_1:3;
         now per cases by A13,REAL_1:def 5;
        suppose j < i;
         then A14: j <= Index(p,f) by A8,NAT_1:38;
         then j <= i-1 by A8,XCMPLX_1:26;
         then j <= i-'1 by A5,BINARITH:def 3;
         then A15: j in dom mid(f,1,Index(p,f)) by A11,A13,FINSEQ_3:27;
         thus mid(f,1,i).j = f.j by A4,A13,JORDAN3:32
            .= mid(f,1,Index(p,f)).j by A6,A13,A14,JORDAN3:32
            .= (mid(f,1,Index(p,f))^<*p*>).j by A15,FINSEQ_1:def 7;
        suppose A16: j = i;
         A17: i-'1+1 = i by A4,AMI_5:4;
         thus mid(f,1,i).j = f.j by A4,A13,JORDAN3:32
            .= (mid(f,1,Index(p,f))^<*p*>).j by A3,A11,A16,A17,FINSEQ_1:59;
       end;
       hence mid(f,1,i).j = (mid(f,1,Index(p,f))^<*p*>).j;
      end;
      then mid(f,1,i) = R_Cut(f,p) by A9,A10,A12,FINSEQ_2:10;
      hence R_Cut(f,p) = mid(f,1,p..f) by A2,A3,FINSEQ_5:12;
     suppose A18: 1 = i;
      then A19: R_Cut(f,p) = <*p*> by A3,JORDAN3:def 5;
      A20: p = f/.1 by A2,A3,A18,FINSEQ_4:def 4;
      then p..f = 1 by FINSEQ_6:47;
      hence R_Cut(f,p) = mid(f,1,p..f) by A4,A18,A19,A20,JORDAN4:27;
   end;

  theorem Th58:
   for f be S-Sequence_in_R2
   for Q be closed Subset of TOP-REAL 2 st L~f meets Q & not f/.1 in Q &
    First_Point(L~f,f/.1,f/.len f,Q) in rng f holds
     L~mid(f,1,First_Point(L~f,f/.1,f/.len f,Q)..f) /\ Q =
      {First_Point(L~f,f/.1,f/.len f,Q)}
   proof
    let f be S-Sequence_in_R2;
    let Q be closed Subset of TOP-REAL 2;
    assume that
     A1: L~f meets Q and
     A2: not f/.1 in Q and
     A3: First_Point(L~f,f/.1,f/.len f,Q) in rng f;
      L~R_Cut(f,First_Point(L~f,f/.1,f/.len f,Q)) /\ Q =
     {First_Point(L~f,f/.1,f/.len f,Q)} by A1,A2,SPRECT_4:1;
    hence thesis by A3,Th57;
   end;

  theorem Th59:
   for C be compact connected non vertical non horizontal Subset of TOP-REAL 2
   for n be Nat st n > 0
   for k be Nat st
    1 <= k &
    k < First_Point(L~Upper_Seq(C,n),W-min L~Cage(C,n),E-max L~Cage(C,n),
     Vertical_Line ((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))..
      Upper_Seq(C,n) holds
     (Upper_Seq(C,n)/.k)`1 < (W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let n be Nat;
    assume A1: n > 0;
    let k be Nat such that
     A2: 1 <= k and
     A3: k < First_Point(L~Upper_Seq(C,n),W-min L~Cage(C,n),E-max L~Cage(C,n),
      Vertical_Line ((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))..
       Upper_Seq(C,n);
    set Wmin = W-min L~Cage(C,n);
    set Emax = E-max L~Cage(C,n);
    set Wbo = W-bound L~Cage(C,n);
    set Ebo = E-bound L~Cage(C,n);
    set sr = (W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2;
    set US = Upper_Seq(C,n);
    set FiP = First_Point(L~US,Wmin,Emax,Vertical_Line sr);
    A4: k > 0 by A2,AXIOMS:22;
      Wbo < Ebo by SPRECT_1:33;
    then A5: Wbo < sr & sr < Ebo by TOPREAL3:3;
      defpred P[Nat] means 1 <= $1 & $1 < FiP..US implies (US/.$1)`1 < sr;
      for k be non empty Nat holds P[k]
    proof
     A6: P[1]
     proof
      assume 1 <= 1 & 1 < FiP..US;
        US/.1 = Wmin by JORDAN1F:5;
      hence (US/.1)`1 < sr by A5,PSCOMP_1:84;
     end;
     A7: for k be non empty Nat st P[k] holds P[k+1]
     proof
      let k be non empty Nat;
      assume A8: 1 <= k & k < FiP..US implies (US/.k)`1 < sr;
      assume that
       A9: 1 <= k+1 and
       A10: k+1 < FiP..US;
      A11: k >= 1 by RLVECT_1:99;
      A12: FiP in rng US by A1,Th55;
      then A13: FiP..Upper_Seq(C,n) in dom Upper_Seq(C,n) by FINSEQ_4:30;
      then A14: 1 <= FiP..US & FiP..US <= len US by FINSEQ_3:27;
      then A15: k+1 <= len US by A10,AXIOMS:22;
        US is_sequence_on Gauge(C,n) by Th4;
      then consider i1,j1,i2,j2 be Nat such that
       A16: [i1,j1] in Indices Gauge(C,n) and
       A17: US/.k = Gauge(C,n)*(i1,j1) and
       A18: [i2,j2] in Indices Gauge(C,n) and
       A19: US/.(k+1) = Gauge(C,n)*(i2,j2) and
       A20: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or
            i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A11,A15,JORDAN8:6;
      A21: 1 <= i1 & i1 <= len Gauge(C,n) & 1 <= j1 & j1 <= width Gauge(C,n)
                                                           by A16,GOBOARD5:1;
      A22: 1 <= i2 & i2 <= len Gauge(C,n) & 1 <= j2 & j2 <= width Gauge(C,n)
                                                           by A18,GOBOARD5:1;
      A23: 1 <= Center Gauge(C,n) & Center Gauge(C,n) <= len Gauge(C,n)
                                                           by JORDAN1B:12,14;
      set GC1 = Gauge(C,n)*(Center Gauge(C,n),1);
        4 <= len Gauge(C,n) by JORDAN8:13;
      then 1 <= len Gauge(C,n) by AXIOMS:22;
      then A24: 1 <= width Gauge(C,n) by JORDAN8:def 1;
      then A25: GC1`1 = (W-bound C + E-bound C)/2 by A1,Th43
         .= sr by Th41;
      A26: i1+1 >= 1 by NAT_1:29;
      A27: US/.1 = Wmin by JORDAN1F:5;
      A28: US/.len US = Emax by JORDAN1F:7;
        now per cases by A20;
       suppose i1 = i2 & j1+1 = j2;
        then (US/.k)`1 = Gauge(C,n)*(i2,1)`1 by A17,A21,GOBOARD5:3
           .= (US/.(k+1))`1 by A19,A22,GOBOARD5:3;
        hence (US/.(k+1))`1 < sr by A8,A10,NAT_1:38,RLVECT_1:99;
       suppose A29: i1+1 = i2 & j1 = j2;
          i1 < Center Gauge(C,n) by A8,A10,A17,A21,A23,A24,A25,JORDAN1A:39,
NAT_1:38,RLVECT_1:99;
        then i1+1 <= Center Gauge(C,n) by NAT_1:38;
        then A30: (US/.(k+1))`1 <= sr
                                  by A19,A21,A23,A24,A25,A26,A29,JORDAN1A:39;
          now assume (US/.(k+1))`1 = sr;
         then US/.(k+1) in {p where p is Point of TOP-REAL 2 : p`1 = sr};
         then A31: US/.(k+1) in Vertical_Line sr by JORDAN6:def 6;
         A32: k+1 >= 1+1 by A11,REAL_1:55;
           len mid(US,1,FiP..US) = FiP..US-'1+1 by A14,JORDAN4:20
            .= FiP..US by A14,AMI_5:4;
         then len mid(US,1,FiP..US) >= 2 by A10,A32,AXIOMS:22;
         then A33: rng mid(US,1,FiP..US) c= L~mid(US,1,FiP..US) by SPPOL_2:18;
         A34: mid(US,1,FiP..US) = US|(FiP..US) by A14,JORDAN3:25;
         A35: US|Seg(FiP..US) = US|(FiP..US) by TOPREAL1:def 1;
         A36: k+1 in dom US by A9,A15,FINSEQ_3:27;
           k+1 in Seg (FiP..US) by A9,A10,FINSEQ_1:3;
         then A37: US/.(k+1) in rng mid(US,1,FiP..US)
                                                  by A34,A35,A36,PARTFUN2:36;
         A38: Vertical_Line sr is closed by JORDAN6:33;
         A39: L~US is_an_arc_of Wmin,Emax by A27,A28,TOPREAL1:31;
           Wmin`1 <= sr & sr <= Emax`1 by A5,PSCOMP_1:84,104;
         then A40: L~US meets Vertical_Line sr by A39,JORDAN6:64;
         A41: now assume US/.1 in Vertical_Line sr;
          then Wmin`1 = sr by A27,JORDAN6:34;
          hence contradiction by A5,PSCOMP_1:84;
         end;
           First_Point(L~US,US/.1,US/.len US,Vertical_Line sr) in rng US
                                                      by A1,A27,A28,Th55;
         then L~mid(US,1,FiP..US) /\ Vertical_Line sr = {FiP}
                                              by A27,A28,A38,A40,A41,Th58;
         then US/.(k+1) in {FiP} by A31,A33,A37,XBOOLE_0:def 3;
         then A42: US/.(k+1) = FiP by TARSKI:def 1;
           US/.(FiP..US) = FiP by A12,FINSEQ_5:41;
         hence contradiction by A10,A13,A36,A42,PARTFUN2:17;
        end;
        hence (US/.(k+1))`1 < sr by A30,REAL_1:def 5;
       suppose i1 = i2+1 & j1 = j2;
        then i2 < i1 by NAT_1:38;
        then (US/.(k+1))`1 <= (US/.k)`1 by A17,A19,A21,A22,JORDAN1A:39;
        hence (US/.(k+1))`1 < sr by A8,A10,AXIOMS:22,NAT_1:38,RLVECT_1:99;
       suppose i1 = i2 & j1 = j2+1;
        then (US/.k)`1 = Gauge(C,n)*(i2,1)`1 by A17,A21,GOBOARD5:3
           .= (US/.(k+1))`1 by A19,A22,GOBOARD5:3;
        hence (US/.(k+1))`1 < sr by A8,A10,NAT_1:38,RLVECT_1:99;
      end;
      hence (US/.(k+1))`1 < sr;
     end;
     thus for k being non empty Nat holds P[k] from Ind_from_1(A6,A7);
    end;
    hence thesis by A2,A3,A4;
   end;

  theorem Th60:
   for C be compact connected non vertical non horizontal Subset of TOP-REAL 2
   for n be Nat st n > 0
   for k be Nat st
    1 <= k &
    k < First_Point(L~Rev Lower_Seq(C,n),W-min L~Cage(C,n),E-max L~Cage(C,n),
     Vertical_Line ((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))..
      Rev Lower_Seq(C,n) holds
     (Rev Lower_Seq(C,n)/.k)`1 < (W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let n be Nat;
    assume A1: n > 0;
    let k be Nat such that
     A2: 1 <= k and
     A3: k < First_Point(L~Rev Lower_Seq(C,n),W-min L~Cage(C,n),E-max
      L~Cage(C,n),Vertical_Line ((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))
       ..Rev Lower_Seq(C,n);
    set Wmin = W-min L~Cage(C,n);
    set Emax = E-max L~Cage(C,n);
    set Wbo = W-bound L~Cage(C,n);
    set Ebo = E-bound L~Cage(C,n);
    set sr = (W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2;
    set LS = Lower_Seq(C,n);
    set RLS = Rev LS;
    set FiP = First_Point(L~RLS,Wmin,Emax,Vertical_Line sr);
    set LaP = Last_Point(L~LS,Emax,Wmin,Vertical_Line sr);
    A4: L~RLS = L~LS by SPPOL_2:22;
    A5: rng RLS = rng LS by FINSEQ_5:60;
    A6: len RLS = len LS by FINSEQ_5:def 3;
    A7: k > 0 by A2,AXIOMS:22;
      Wbo < Ebo by SPRECT_1:33;
    then A8: Wbo < sr & sr < Ebo by TOPREAL3:3;
    defpred P[Nat] means 1 <= $1 & $1 < FiP..RLS implies (RLS/.$1)`1 < sr;
      for k be non empty Nat holds P[k]
    proof
     A9: P[1]
     proof
      assume 1 <= 1 & 1 < FiP..RLS;
        RLS/.1 = LS/.len LS by FINSEQ_5:68
         .= Wmin by JORDAN1F:8;
      hence (RLS/.1)`1 < sr by A8,PSCOMP_1:84;
     end;
     A10: for k be non empty Nat st P[k] holds P[k+1]
     proof
      let k be non empty Nat;
      assume A11: 1 <= k & k < FiP..RLS implies (RLS/.k)`1 < sr;
      assume that
       A12: 1 <= k+1 and
       A13: k+1 < FiP..RLS;
      A14: k >= 1 by RLVECT_1:99;
      A15: LS/.1 = Emax by JORDAN1F:6;
        LS/.len LS = Wmin by JORDAN1F:8;
      then A16: L~LS is_an_arc_of Emax,Wmin by A15,TOPREAL1:31;
      then A17: L~LS is_an_arc_of Wmin,Emax by JORDAN5B:14;
        Wbo <= Ebo by SPRECT_1:23;
      then Wbo <= sr & sr <= Ebo by JORDAN6:2;
      then Wmin`1 <= sr & sr <= Emax`1 by PSCOMP_1:84,104;
      then L~LS meets Vertical_Line(sr) &
           L~LS /\ Vertical_Line(sr) is closed by A17,JORDAN6:64;
      then A18: FiP = LaP by A4,A16,JORDAN5C:18;
      then A19: FiP in rng RLS by A1,A5,Th56;
      then A20: FiP..RLS in dom RLS by FINSEQ_4:30;
      then A21: 1 <= FiP..RLS & FiP..RLS <= len RLS by FINSEQ_3:27;
      then A22: k+1 <= len RLS by A13,AXIOMS:22;
        LS is_sequence_on Gauge(C,n) by Th5;
      then RLS is_sequence_on Gauge(C,n) by JORDAN9:7;
      then consider i1,j1,i2,j2 be Nat such that
       A23: [i1,j1] in Indices Gauge(C,n) and
       A24: RLS/.k = Gauge(C,n)*(i1,j1) and
       A25: [i2,j2] in Indices Gauge(C,n) and
       A26: RLS/.(k+1) = Gauge(C,n)*(i2,j2) and
       A27: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or
            i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A14,A22,JORDAN8:6;
      A28: 1 <= i1 & i1 <= len Gauge(C,n) & 1 <= j1 & j1 <= width Gauge(C,n)
                                                           by A23,GOBOARD5:1;
      A29: 1 <= i2 & i2 <= len Gauge(C,n) & 1 <= j2 & j2 <= width Gauge(C,n)
                                                           by A25,GOBOARD5:1;
      A30: 1 <= Center Gauge(C,n) & Center Gauge(C,n) <= len Gauge(C,n)
                                                           by JORDAN1B:12,14;
      set GC1 = Gauge(C,n)*(Center Gauge(C,n),1);
        4 <= len Gauge(C,n) by JORDAN8:13;
      then 1 <= len Gauge(C,n) by AXIOMS:22;
      then A31: 1 <= width Gauge(C,n) by JORDAN8:def 1;
      then A32: GC1`1 = (W-bound C + E-bound C)/2 by A1,Th43
         .= sr by Th41;
      A33: i1+1 >= 1 by NAT_1:29;
      A34: RLS/.1 = LS/.len LS by FINSEQ_5:68
         .= Wmin by JORDAN1F:8;
      A35: RLS/.len RLS = LS/.1 by A6,FINSEQ_5:68
         .= Emax by JORDAN1F:6;
        now per cases by A27;
       suppose i1 = i2 & j1+1 = j2;
        then (RLS/.k)`1 = Gauge(C,n)*(i2,1)`1 by A24,A28,GOBOARD5:3
           .= (RLS/.(k+1))`1 by A26,A29,GOBOARD5:3;
        hence (RLS/.(k+1))`1 < sr by A11,A13,NAT_1:38,RLVECT_1:99;
       suppose A36: i1+1 = i2 & j1 = j2;
          i1 < Center Gauge(C,n) by A11,A13,A24,A28,A30,A31,A32,JORDAN1A:39,
NAT_1:38,RLVECT_1:99;
        then i1+1 <= Center Gauge(C,n) by NAT_1:38;
        then A37: (RLS/.(k+1))`1 <= sr
                                  by A26,A28,A30,A31,A32,A33,A36,JORDAN1A:39;
          now assume (RLS/.(k+1))`1 = sr;
         then RLS/.(k+1) in {p where p is Point of TOP-REAL 2 : p`1 = sr};
         then A38: RLS/.(k+1) in Vertical_Line sr by JORDAN6:def 6;
         A39: k+1 >= 1+1 by A14,REAL_1:55;
           len mid(RLS,1,FiP..RLS) = FiP..RLS-'1+1 by A21,JORDAN4:20
            .= FiP..RLS by A21,AMI_5:4;
         then len mid(RLS,1,FiP..RLS) >= 2 by A13,A39,AXIOMS:22;
         then A40: rng mid(RLS,1,FiP..RLS) c= L~mid(RLS,1,FiP..RLS)
                                                               by SPPOL_2:18;
         A41: mid(RLS,1,FiP..RLS) = RLS|(FiP..RLS) by A21,JORDAN3:25;
         A42: RLS|Seg(FiP..RLS) = RLS|(FiP..RLS) by TOPREAL1:def 1;
         A43: k+1 in dom RLS by A12,A22,FINSEQ_3:27;
           k+1 in Seg (FiP..RLS) by A12,A13,FINSEQ_1:3;
         then A44: RLS/.(k+1) in rng mid(RLS,1,FiP..RLS)
                                                  by A41,A42,A43,PARTFUN2:36;
         A45: Vertical_Line sr is closed by JORDAN6:33;
         A46: RLS is_S-Seq by SPPOL_2:47;
         then A47: L~RLS is_an_arc_of Wmin,Emax by A34,A35,TOPREAL1:31;
           Wmin`1 <= sr & sr <= Emax`1 by A8,PSCOMP_1:84,104;
         then A48: L~RLS meets Vertical_Line sr by A47,JORDAN6:64;
         A49: now assume RLS/.1 in Vertical_Line sr;
          then Wmin`1 = sr by A34,JORDAN6:34;
          hence contradiction by A8,PSCOMP_1:84;
         end;
           First_Point(L~RLS,RLS/.1,RLS/.len RLS,Vertical_Line sr) in rng RLS
                                               by A1,A5,A18,A34,A35,Th56;
         then L~mid(RLS,1,FiP..RLS) /\ Vertical_Line sr = {FiP}
                                           by A34,A35,A45,A46,A48,A49,Th58;
         then RLS/.(k+1) in {FiP} by A38,A40,A44,XBOOLE_0:def 3;
         then A50: RLS/.(k+1) = FiP by TARSKI:def 1;
           RLS/.(FiP..RLS) = FiP by A19,FINSEQ_5:41;
         hence contradiction by A13,A20,A43,A50,PARTFUN2:17;
        end;
        hence (RLS/.(k+1))`1 < sr by A37,REAL_1:def 5;
       suppose i1 = i2+1 & j1 = j2;
        then i2 < i1 by NAT_1:38;
        then (RLS/.(k+1))`1 <= (RLS/.k)`1 by A24,A26,A28,A29,JORDAN1A:39;
        hence (RLS/.(k+1))`1 < sr by A11,A13,AXIOMS:22,NAT_1:38,RLVECT_1:99;
       suppose i1 = i2 & j1 = j2+1;
        then (RLS/.k)`1 = Gauge(C,n)*(i2,1)`1 by A24,A28,GOBOARD5:3
           .= (RLS/.(k+1))`1 by A26,A29,GOBOARD5:3;
        hence (RLS/.(k+1))`1 < sr by A11,A13,NAT_1:38,RLVECT_1:99;
      end;
      hence (RLS/.(k+1))`1 < sr;
     end;
     thus for k being non empty Nat holds P[k] from Ind_from_1(A9,A10);
    end;
    hence thesis by A2,A3,A7;
   end;

  theorem Th61:
   for C be compact connected non vertical non horizontal Subset of TOP-REAL 2
   for n be Nat st n > 0
   for q be Point of TOP-REAL 2 holds
    q in rng mid(Upper_Seq(C,n),2,First_Point(L~Upper_Seq(C,n),
     W-min L~Cage(C,n),E-max L~Cage(C,n),Vertical_Line
      ((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))..Upper_Seq(C,n)) implies
     q`1 <= (W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let n be Nat;
    assume A1: n > 0;
    let q be Point of TOP-REAL 2;
    set Wmin = W-min L~Cage(C,n);
    set Emax = E-max L~Cage(C,n);
    set Wbo = W-bound L~Cage(C,n);
    set Ebo = E-bound L~Cage(C,n);
    set sr = (W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2;
    set US = Upper_Seq(C,n);
    set FiP = First_Point(L~US,Wmin,Emax,Vertical_Line sr);
    assume q in rng mid(US,2,FiP..US);
    then consider k be Nat such that
     A2: k in dom mid(US,2,FiP..US) and
     A3: q = mid(US,2,FiP..US)/.k by PARTFUN2:4;
    A4: FiP in rng US by A1,Th55;
    then A5: FiP..US in dom US by FINSEQ_4:30;
    then A6: 1 <= FiP..US & FiP..US <= len US by FINSEQ_3:27;
    A7: US/.1 = Wmin by JORDAN1F:5;
    then A8: Wmin in rng US by FINSEQ_6:46;
      US/.len US = Emax by JORDAN1F:7;
    then A9: L~US is_an_arc_of Wmin,Emax by A7,TOPREAL1:31;
      Wbo < Ebo by SPRECT_1:33;
    then A10: Wbo < sr & sr < Ebo by TOPREAL3:3;
    then Wmin`1 <= sr & sr <= Emax`1 by PSCOMP_1:84,104;
    then L~US meets Vertical_Line sr & L~US /\ Vertical_Line sr is closed
                                                            by A9,JORDAN6:64;
    then FiP in L~US /\ Vertical_Line sr by A9,JORDAN5C:def 1;
    then FiP in L~US & FiP in Vertical_Line sr by XBOOLE_0:def 3;
    then A11: FiP`1 = sr by JORDAN6:34;
      now assume FiP..US = 1;
     then FiP..US = (US/.1)..US by FINSEQ_6:47
        .= Wmin..US by JORDAN1F:5;
     then FiP = Wmin by A4,A8,FINSEQ_5:10;
     hence contradiction by A10,A11,PSCOMP_1:84;
    end;
    then FiP..US > 1 by A6,REAL_1:def 5;
    then A12: 1+1+0 <= FiP..US by NAT_1:38;
      k+2 >= 1+1 by NAT_1:29;
    then k+2-1 >= 1+1-1 by REAL_1:49;
    then A13: k+2-1 >= 0 by AXIOMS:22;
      len US >= 3 by JORDAN1E:19;
    then len US >= 2 by AXIOMS:22;
    then 2 in dom US by FINSEQ_3:27;
    then A14: mid(US,2,FiP..US)/.k = US/.(k+2-'1) by A2,A5,A12,SPRECT_2:7
       .= US/.(k+2-1) by A13,BINARITH:def 3
       .= US/.(k+(2-1)) by XCMPLX_1:29;
    A15: 1 <= k+1 by NAT_1:29;
    A16: k <= len mid(US,2,FiP..US) by A2,FINSEQ_3:27;
      FiP..US-2 >= 0 by A12,REAL_1:84;
    then FiP..US-'2 = FiP..US-2 by BINARITH:def 3;
    then len mid(US,2,FiP..US) = FiP..US-2+1 by A6,A12,JORDAN4:20;
    then k < FiP..US-2+1+1 by A16,NAT_1:38;
    then k < FiP..US-2+(1+1) by XCMPLX_1:1;
    then k < FiP..US by XCMPLX_1:27;
    then A17: k+1 <= FiP..US by NAT_1:38;
    per cases by A17,REAL_1:def 5;
     suppose k+1 < FiP..US;
      hence q`1 <= sr by A1,A3,A14,A15,Th59;
     suppose k+1 = FiP..US;
      hence q`1 <= sr by A3,A4,A11,A14,FINSEQ_5:41;
   end;

  theorem Th62:
   for C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2
   for n be Nat st n > 0 holds
    First_Point(L~Upper_Seq(C,n),W-min L~Cage(C,n),E-max L~Cage(C,n),
     Vertical_Line((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))`2 >
    Last_Point(L~Lower_Seq(C,n),E-max L~Cage(C,n),W-min L~Cage(C,n),
     Vertical_Line((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))`2
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let n be Nat;
    set sr = (W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2;
    set Wmin = W-min L~Cage(C,n);
    set Emax = E-max L~Cage(C,n);
    set Nbo = N-bound L~Cage(C,n);
    set Ebo = E-bound L~Cage(C,n);
    set Sbo = S-bound L~Cage(C,n);
    set Wbo = W-bound L~Cage(C,n);
    set SW = SW-corner L~Cage(C,n);
    set FiP = First_Point(L~Upper_Seq(C,n),Wmin,Emax,Vertical_Line sr);
    set LaP = Last_Point(L~Lower_Seq(C,n),Emax,Wmin,Vertical_Line sr);
    set g = mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n))^<*|[Ebo,FiP`2]|*>;
    set h = <*SW*>^((Rev Lower_Seq(C,n))-:LaP)^<*|[sr,Nbo]|*>;
    assume A1: n > 0;
    A2: Upper_Seq(C,n) is_in_the_area_of Cage(C,n) by JORDAN1E:21;
      len Upper_Seq(C,n) >= 3 by JORDAN1E:19;
    then A3: len Upper_Seq(C,n) > 2 by AXIOMS:22;
    then A4: 2 in dom Upper_Seq(C,n) by FINSEQ_3:27;
    A5: Upper_Seq(C,n)/.1 = W-min L~Cage(C,n) by JORDAN1F:5;
    A6: Upper_Seq(C,n)/.len Upper_Seq(C,n) = E-max L~Cage(C,n) by JORDAN1F:7;
    then A7: L~Upper_Seq(C,n) is_an_arc_of Wmin,Emax by A5,TOPREAL1:31;
      Wbo < Ebo by SPRECT_1:33;
    then A8: Wbo < sr & sr < Ebo by TOPREAL3:3;
    then Wmin`1 <= sr & sr <= Emax`1 by PSCOMP_1:84,104;
    then L~Upper_Seq(C,n) meets Vertical_Line(sr) &
         L~Upper_Seq(C,n) /\ Vertical_Line(sr) is closed by A7,JORDAN6:64;
    then FiP in L~Upper_Seq(C,n) /\ Vertical_Line sr by A7,JORDAN5C:def 1;
    then A9: FiP in L~Upper_Seq(C,n) & FiP in Vertical_Line sr
                                                             by XBOOLE_0:def 3;
    then FiP in L~Upper_Seq(C,n) \/ L~Lower_Seq(C,n) by XBOOLE_0:def 2;
    then A10: FiP in L~Cage(C,n) by JORDAN1E:17;
    A11: FiP`1 = sr by A9,JORDAN6:34;
    A12: 1 <= Center Gauge(C,n) & Center Gauge(C,n) <= len Gauge(C,n)
                                                           by JORDAN1B:12,14;
    A13: FiP in rng Upper_Seq(C,n) by A1,Th55;
    then A14: FiP..Upper_Seq(C,n) in dom Upper_Seq(C,n) by FINSEQ_4:30;
    A15: Lower_Seq(C,n)/.1 = E-max L~Cage(C,n) by JORDAN1F:6;
      Lower_Seq(C,n)/.len Lower_Seq(C,n) = W-min L~Cage(C,n) by JORDAN1F:8;
    then A16: L~Lower_Seq(C,n) is_an_arc_of Emax,Wmin by A15,TOPREAL1:31;
    then A17: L~Lower_Seq(C,n) is_an_arc_of Wmin,Emax by JORDAN5B:14;
      Wbo <= Ebo by SPRECT_1:23;
    then Wbo <= sr & sr <= Ebo by JORDAN6:2;
    then Wmin`1 <= sr & sr <= Emax`1 by PSCOMP_1:84,104;
    then A18: L~Lower_Seq(C,n) meets Vertical_Line(sr) &
         L~Lower_Seq(C,n) /\ Vertical_Line(sr) is closed by A17,JORDAN6:64;
    then LaP in L~Lower_Seq(C,n) /\ Vertical_Line sr by A16,JORDAN5C:def 2;
    then A19: LaP in L~Lower_Seq(C,n) & LaP in Vertical_Line sr
                                                             by XBOOLE_0:def 3;
    then LaP in L~Upper_Seq(C,n) \/ L~Lower_Seq(C,n) by XBOOLE_0:def 2;
    then A20: LaP in L~Cage(C,n) by JORDAN1E:17;
    A21: LaP`1 = sr by A19,JORDAN6:34;
    assume A22: FiP`2 <= LaP`2;
    A23: now assume FiP`2 = LaP`2;
     then FiP = LaP by A11,A21,TOPREAL3:11;
     then FiP in L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n) by A9,A19,XBOOLE_0:def 3
;
     then FiP in {Wmin,Emax} by JORDAN1E:20;
     then FiP = Wmin or FiP = Emax by TARSKI:def 2;
     hence contradiction by A8,A11,PSCOMP_1:84,104;
    end;
    then A24: FiP`2 < LaP`2 by A22,REAL_1:def 5;
      LaP in rng Lower_Seq(C,n) by A1,Th56;
    then A25: LaP in rng Rev Lower_Seq(C,n) by FINSEQ_5:60;
      now assume rng mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) /\
      {|[Ebo,FiP`2]|} <> {};
     then consider x be set such that
      A26: x in rng mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) /\
                                             {|[Ebo,FiP`2]|} by XBOOLE_0:def 1;
       x in rng mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) & x in {|[Ebo,FiP`2]|
}
                                                         by A26,XBOOLE_0:def 3;
     then |[Ebo,FiP`2]| in rng mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n))
                                                            by TARSKI:def 1;
     then |[Ebo,FiP`2]|`1 <= sr by A1,Th61;
     hence contradiction by A8,EUCLID:56;
    end;
    then rng mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) misses {|[Ebo,FiP`2]|}
                                                                by XBOOLE_0:def
7;
    then A27: rng mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) misses
     rng <*|[Ebo,FiP`2]|*> by FINSEQ_1:55;
    A28: 1<=FiP..Upper_Seq(C,n) & FiP..Upper_Seq(C,n)<=len Upper_Seq(C,n)
                                                            by A14,FINSEQ_3:27;
      2 <> FiP..Upper_Seq(C,n)
    proof
     assume 2 = FiP..Upper_Seq(C,n);
     then Upper_Seq(C,n)/.2 = FiP by A13,FINSEQ_5:41;
     then FiP`1 = Wbo by Th39;
     then Wbo = sr by A9,JORDAN6:34;
     then Wbo = Ebo by XCMPLX_1:67;
     hence contradiction by SPRECT_1:33;
    end;
    then mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) is_S-Seq
                                                       by A3,A28,JORDAN3:39;
    then A29: mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) is one-to-one special
                                                         by TOPREAL1:def 10;
    A30: <*|[Ebo,FiP`2]|*> is one-to-one by FINSEQ_3:102;
    A31: (mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n))/.len mid(Upper_Seq(C,n),2,
          FiP..Upper_Seq(C,n)))`2 = (Upper_Seq(C,n)/.(FiP..Upper_Seq(C,n)))`2
                                                        by A4,A14,SPRECT_2:13
       .= FiP`2 by A13,FINSEQ_5:41
       .= |[Ebo,FiP`2]|`2 by EUCLID:56
       .= (<*|[Ebo,FiP`2]|*>/.1)`2 by FINSEQ_4:25;
      <*|[Ebo,FiP`2]|*> is special by SPPOL_2:39;
    then reconsider g as one-to-one special FinSequence of TOP-REAL 2
                                  by A27,A29,A30,A31,FINSEQ_3:98,GOBOARD2:13;
    A32: rng ((Rev Lower_Seq(C,n))-:LaP) c= rng (Rev Lower_Seq(C,n))
                                                              by FINSEQ_5:51;
    set GCw = Gauge(C,n)*(Center Gauge(C,n),width Gauge(C,n));
    A33: 4 <= len Gauge(C,n) by JORDAN8:13;
    then 1 <= len Gauge(C,n) by AXIOMS:22;
    then 1 <= width Gauge(C,n) by JORDAN8:def 1;
    then A34: GCw`1 = (W-bound C + E-bound C)/2 by A1,Th43
       .= sr by Th41;
      len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
    then GCw`2 = Nbo by A12,JORDAN1A:91;
    then A35: GCw = |[sr,Nbo]| by A34,EUCLID:57;
      len Gauge(C,n) >= 3 by A33,AXIOMS:22;
    then Center Gauge(C,n) < len Gauge(C,n) by JORDAN1B:16;
    then not |[sr,Nbo]| in rng Lower_Seq(C,n) by A12,A35,Th51;
    then not |[sr,Nbo]| in rng (Rev Lower_Seq(C,n)) by FINSEQ_5:60;
    then A36: not |[sr,Nbo]| in rng ((Rev Lower_Seq(C,n))-:LaP) by A32;
    A37: |[sr,Nbo]|`2 = Nbo & SW`2 = Sbo by EUCLID:56,PSCOMP_1:73;
    then |[sr,Nbo]| <> SW by SPRECT_1:34;
    then not |[sr,Nbo]| in {SW} by TARSKI:def 1;
    then not |[sr,Nbo]| in rng <*SW*> by FINSEQ_1:55;
    then not |[sr,Nbo]| in rng <*SW*> \/ rng ((Rev Lower_Seq(C,n))-:LaP)
                                                         by A36,XBOOLE_0:def 2;
    then not |[sr,Nbo]| in rng (<*SW*>^((Rev Lower_Seq(C,n))-:LaP))
                                                              by FINSEQ_1:44;
    then rng (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)) misses {|[sr,Nbo]|}
                                                              by ZFMISC_1:56;
    then rng (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)) /\ {|[sr,Nbo]|} = {}
                                                         by XBOOLE_0:def 7;
    then rng (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)) /\ rng <*|[sr,Nbo]|*> = {}
                                                              by FINSEQ_1:55;
    then A38: rng (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)) misses rng <*|[sr,Nbo]|
*>
                                                          by XBOOLE_0:def 7;
    A39: SW`2 <= Wmin`2 by PSCOMP_1:87;
    A40: rng Lower_Seq(C,n) c= rng Cage(C,n) by Th47;
    A41: mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n))
                        is_in_the_area_of Cage(C,n) by A2,A4,A14,SPRECT_2:26;
      now let m be Nat;
     assume m in dom <*|[Ebo,FiP`2]|*>;
     then m in Seg 1 by FINSEQ_1:55;
     then m = 1 by FINSEQ_1:4,TARSKI:def 1;
     then <*|[Ebo,FiP`2]|*>/.m = |[Ebo,FiP`2]| by FINSEQ_4:25;
     then A42: (<*|[Ebo,FiP`2]|*>/.m)`1 = Ebo &
      (<*|[Ebo,FiP`2]|*>/.m)`2 = FiP`2 by EUCLID:56;
     hence W-bound L~Cage(C,n) <= (<*|[Ebo,FiP`2]|*>/.m)`1 &
          (<*|[Ebo,FiP`2]|*>/.m)`1 <= E-bound L~Cage(C,n) by SPRECT_1:23;
     thus S-bound L~Cage(C,n) <= (<*|[Ebo,FiP`2]|*>/.m)`2 &
      (<*|[Ebo,FiP`2]|*>/.m)`2 <= N-bound L~Cage(C,n) by A10,A42,PSCOMP_1:71;
    end;
    then <*|[Ebo,FiP`2]|*> is_in_the_area_of Cage(C,n) by SPRECT_2:def 1;
    then A43: g is_in_the_area_of Cage(C,n) by A41,SPRECT_2:28;
    A44: 1 <= len mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n))
                                                        by A4,A14,SPRECT_2:9;
    then 1 in dom mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) by FINSEQ_3:27;
    then A45: (g/.1)`1 = (mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n))/.1)`1
                                                                by GROUP_5:95
       .= (Upper_Seq(C,n)/.2)`1 by A4,A14,SPRECT_2:12
       .= W-bound L~Cage(C,n) by Th39;
      (g/.len g)`1 = (<*|[Ebo,FiP`2]|*>/.len <*|[Ebo,FiP`2]|*>)`1 by SPRECT_3:
11
       .= (<*|[Ebo,FiP`2]|*>/.1)`1 by FINSEQ_1:56
       .= (|[Ebo,FiP`2]|)`1 by FINSEQ_4:25
       .= E-bound L~Cage(C,n) by EUCLID:56;
    then A46: g is_a_h.c._for Cage(C,n) by A43,A45,SPRECT_2:def 2;
    set RevL = (Rev Lower_Seq(C,n))-:LaP;
    A47: RevL is non empty by A25,FINSEQ_5:50;
    A48: len RevL = LaP..Rev Lower_Seq(C,n) by A25,FINSEQ_5:45;
      now per cases;
     suppose A49: SW <> Wmin;
        not SW in rng Lower_Seq(C,n)
      proof
       assume SW in rng Lower_Seq(C,n);
       then A50: SW in rng Cage(C,n) by A40;
         len Cage(C,n) > 4 by GOBOARD7:36;
       then len Cage(C,n) >= 2 by AXIOMS:22;
       then A51: rng Cage(C,n) c= L~Cage(C,n) by SPPOL_2:18;
         SW`1 = W-bound L~Cage(C,n) by PSCOMP_1:72;
       then SW in W-most L~Cage(C,n) by A50,A51,SPRECT_2:16;
       then A52: Wmin`2 <= SW`2 by PSCOMP_1:88;
         SW`1 = Wmin`1 by PSCOMP_1:85;
       then SW`2 <> Wmin`2 by A49,TOPREAL3:11;
       hence contradiction by A39,A52,REAL_1:def 5;
      end;
      then not SW in rng (Rev Lower_Seq(C,n)) by FINSEQ_5:60;
      then not SW in rng ((Rev Lower_Seq(C,n))-:LaP) by A32;
      then {SW} misses rng ((Rev Lower_Seq(C,n))-:LaP) by ZFMISC_1:56;
      then {SW} /\ rng ((Rev Lower_Seq(C,n))-:LaP) = {} by XBOOLE_0:def 7;
      then rng <*SW*> /\ rng ((Rev Lower_Seq(C,n))-:LaP) = {} by FINSEQ_1:55;
      then A53: rng <*SW*> misses rng ((Rev Lower_Seq(C,n))-:LaP) by XBOOLE_0:
def 7;
        <*SW*> is one-to-one by FINSEQ_3:102;
      then A54: <*SW*>^((Rev Lower_Seq(C,n))-:LaP) is one-to-one
                                                           by A53,FINSEQ_3:98;
      A55: <*|[sr,Nbo]|*> is one-to-one by FINSEQ_3:102;
        (Rev Lower_Seq(C,n))-:LaP is non empty by A25,FINSEQ_5:50;
      then A56: ((<*SW*>^((Rev Lower_Seq(C,n))-:LaP))/.
            len (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)))`1
          = (((Rev Lower_Seq(C,n))-:LaP)/.len ((Rev Lower_Seq(C,n))-:LaP))`1
                                                            by SPRECT_3:11
         .= (((Rev Lower_Seq(C,n))-:LaP)/.(LaP..Rev Lower_Seq(C,n)))`1
                                                           by A25,FINSEQ_5:45
         .= LaP`1 by A25,FINSEQ_5:48
         .= |[sr,Nbo]|`1 by A21,EUCLID:56
         .= (<*|[sr,Nbo]|*>/.1)`1 by FINSEQ_4:25;
      A57: (<*SW*>/.len <*SW*>)`1 = (<*SW*>/.1)`1 by FINSEQ_1:56
         .= SW`1 by FINSEQ_4:25
         .= W-bound L~Cage(C,n) by PSCOMP_1:72
         .= (W-min L~Cage(C,n))`1 by PSCOMP_1:84
         .= (Lower_Seq(C,n)/.len Lower_Seq(C,n))`1 by JORDAN1F:8
         .= ((Rev Lower_Seq(C,n))/.1)`1 by FINSEQ_5:68
         .= (((Rev Lower_Seq(C,n))-:LaP)/.1)`1 by A25,FINSEQ_5:47;
      reconsider RevLS = Rev Lower_Seq(C,n) as
                             special FinSequence of TOP-REAL 2 by SPPOL_2:42;
        <*SW*> is special by SPPOL_2:39;
      then A58: <*SW*>^(RevLS-:LaP) is special by A57,GOBOARD2:13;
        <*|[sr,Nbo]|*> is special by SPPOL_2:39;
      then reconsider h as one-to-one special FinSequence of TOP-REAL 2
                        by A38,A54,A55,A56,A58,FINSEQ_3:98,GOBOARD2:13;
        now let m be Nat;
       assume m in dom <*SW*>;
       then m in Seg 1 by FINSEQ_1:55;
       then m = 1 by FINSEQ_1:4,TARSKI:def 1;
       then <*SW*>/.m = SW by FINSEQ_4:25;
       then A59: (<*SW*>/.m)`1 = Wbo & (<*SW*>/.m)`2 = Sbo
         by PSCOMP_1:72,73;
       hence W-bound L~Cage(C,n) <= (<*SW*>/.m)`1 &
            (<*SW*>/.m)`1 <= E-bound L~Cage(C,n) by SPRECT_1:23;
       thus S-bound L~Cage(C,n) <= (<*SW*>/.m)`2 &
            (<*SW*>/.m)`2 <= N-bound L~Cage(C,n) by A59,SPRECT_1:24;
      end;
      then A60: <*SW*> is_in_the_area_of Cage(C,n) by SPRECT_2:def 1;
        Lower_Seq(C,n) is_in_the_area_of Cage(C,n) by JORDAN1E:22;
      then Rev Lower_Seq(C,n) is_in_the_area_of Cage(C,n) by SPRECT_3:68;
      then ((Rev Lower_Seq(C,n))-:LaP) is_in_the_area_of Cage(C,n)
                                                           by A25,JORDAN1E:5;
      then A61: <*SW*>^((Rev Lower_Seq(C,n))-:LaP) is_in_the_area_of Cage(C,n)
                                                          by A60,SPRECT_2:28;
        now let m be Nat;
       assume m in dom <*|[sr,Nbo]|*>;
       then m in Seg 1 by FINSEQ_1:55;
       then m = 1 by FINSEQ_1:4,TARSKI:def 1;
       then <*|[sr,Nbo]|*>/.m = |[sr,Nbo]| by FINSEQ_4:25;
       then A62: (<*|[sr,Nbo]|*>/.m)`1 = sr & (<*|[sr,Nbo]|*>/.m)`2 = Nbo
                                                                by EUCLID:56;
         W-bound L~Cage(C,n) <= E-bound L~Cage(C,n) by SPRECT_1:23;
       hence W-bound L~Cage(C,n) <= (<*|[sr,Nbo]|*>/.m)`1 &
            (<*|[sr,Nbo]|*>/.m)`1 <= E-bound L~Cage(C,n) by A62,JORDAN6:2;
       thus S-bound L~Cage(C,n) <= (<*|[sr,Nbo]|*>/.m)`2 &
            (<*|[sr,Nbo]|*>/.m)`2 <= N-bound L~Cage(C,n) by A62,SPRECT_1:24;
      end;
      then <*|[sr,Nbo]|*> is_in_the_area_of Cage(C,n) by SPRECT_2:def 1;
      then A63: h is_in_the_area_of Cage(C,n) by A61,SPRECT_2:28;
        len (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)) =
       1 + len ((Rev Lower_Seq(C,n))-:LaP) by FINSEQ_5:8;
      then A64: len (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)) >= 1 by NAT_1:29;
      A65: len h = len (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)) + 1 by FINSEQ_2:19;
      then A66: 1+1 <= len h by A64,REAL_1:55;
        1 in dom h by FINSEQ_5:6;
      then h/.1 = h.1 by FINSEQ_4:def 4;
      then A67: (h/.1)`2 = ((<*SW*>^((Rev Lower_Seq(C,n))-:LaP))/.1)`2
                                                            by A64,JORDAN3:17
         .= SW`2 by FINSEQ_5:16
         .= S-bound L~Cage(C,n) by PSCOMP_1:73;
        (h/.len h)`2 = |[sr,Nbo]|`2 by A65,TOPREAL4:1
         .= N-bound L~Cage(C,n) by EUCLID:56;
      then A68: h is_a_v.c._for Cage(C,n) by A63,A67,SPRECT_2:def 3;
        len g = len mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n))+1 by FINSEQ_2:19;
      then 1+1 <= len g by A44,REAL_1:55;
      then L~g meets L~h by A46,A66,A68,SPRECT_2:33;
      then consider x be set such that
       A69: x in L~g and
       A70: x in L~h by XBOOLE_0:3;
      reconsider x as Point of TOP-REAL 2 by A69;
      set midU = mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n));
        midU is non empty by A4,A14,SPRECT_2:11;
      then A71: L~g = L~midU \/ LSeg(midU/.len midU,|[Ebo,FiP`2]|)
                                                               by SPPOL_2:19;
        L~h = L~(<*SW*>^(((Rev Lower_Seq(C,n))-:LaP)^<*|[sr,Nbo]|*>))
                                                               by FINSEQ_1:45
         .= LSeg(SW,(RevL^<*|[sr,Nbo]|*>)/.1) \/ L~(RevL^<*|[sr,Nbo]|*>)
                                                                by SPPOL_2:20
         .= LSeg(SW,(RevL^<*|[sr,Nbo]|*>)/.1) \/
            (L~RevL \/ LSeg(RevL/.len RevL,|[sr,Nbo]|)) by A47,SPPOL_2:19;
      then A72: x in LSeg(SW,(RevL^<*|[sr,Nbo]|*>)/.1) or
       x in L~RevL \/ LSeg(RevL/.len RevL,|[sr,Nbo]|) by A70,XBOOLE_0:def 2;
        1 in dom RevL by A47,FINSEQ_5:6;
      then A73: (RevL^<*|[sr,Nbo]|*>)/.1 = RevL/.1 by GROUP_5:95
         .= (Rev Lower_Seq(C,n))/.1 by A25,FINSEQ_5:47
         .= Lower_Seq(C,n)/.len Lower_Seq(C,n) by FINSEQ_5:68
         .= Wmin by JORDAN1F:8;
      A74: L~midU c= L~Upper_Seq(C,n) by A4,A14,SPRECT_3:34;
        L~Rev Lower_Seq(C,n) = L~RevL \/ L~((Rev Lower_Seq(C,n)):-LaP)
                                                           by A25,SPPOL_2:25;
      then L~RevL c= L~Rev Lower_Seq(C,n) by XBOOLE_1:7;
      then A75: L~RevL c= L~Lower_Seq(C,n) by SPPOL_2:22;
      A76: midU/.len midU = Upper_Seq(C,n)/.(FiP..Upper_Seq(C,n))
                                                        by A4,A14,SPRECT_2:13
         .= FiP by A13,FINSEQ_5:41;
      A77: RevL/.len RevL = RevL/.(LaP..Rev Lower_Seq(C,n)) by A25,FINSEQ_5:45
         .= LaP by A25,FINSEQ_5:48;
      A78: |[Ebo,FiP`2]|`1 = Ebo by EUCLID:56;
      A79: SW`1 = Wmin`1 by PSCOMP_1:85;
      then A80: LSeg(SW,Wmin) is vertical by SPPOL_1:37;
      A81: SW`2 <= Wmin`2 by PSCOMP_1:87;
        L~Cage(C,n) = L~Upper_Seq(C,n) \/ L~Lower_Seq(C,n) by JORDAN1E:17;
      then A82: L~Upper_Seq(C,n) c= L~Cage(C,n) by XBOOLE_1:7;
        LaP`1 = |[sr,Nbo]|`1 by A21,EUCLID:56;
      then A83: LSeg(LaP,|[sr,Nbo]|) is vertical by SPPOL_1:37;
      A84: |[Ebo,FiP`2]|`2 = FiP`2 by EUCLID:56;
      then A85: LSeg(FiP,|[Ebo,FiP`2]|) is horizontal by SPPOL_1:36;
      A86: LaP`2 <= Nbo by A20,PSCOMP_1:71;
      A87: Wmin in rng Upper_Seq(C,n) by A5,FINSEQ_6:46;
        now assume FiP..Upper_Seq(C,n) = 1;
       then FiP..Upper_Seq(C,n) = (Upper_Seq(C,n)/.1)..Upper_Seq(C,n)
                                                               by FINSEQ_6:47
          .= Wmin..Upper_Seq(C,n) by JORDAN1F:5;
       then FiP = Wmin by A13,A87,FINSEQ_5:10;
       hence contradiction by A8,A11,PSCOMP_1:84;
      end;
      then FiP..Upper_Seq(C,n) > 1 by A28,REAL_1:def 5;
      then A88: 1+1+0 <= FiP..Upper_Seq(C,n) by NAT_1:38;
      then FiP..Upper_Seq(C,n)-2 >= 0 by REAL_1:84;
      then FiP..Upper_Seq(C,n)-'2 = FiP..Upper_Seq(C,n)-2 by BINARITH:def 3;
      then A89: len midU = FiP..Upper_Seq(C,n)-2+1 by A28,A88,JORDAN4:20
         .= FiP..Upper_Seq(C,n)-(2-1) by XCMPLX_1:37;
      set FiP2 = First_Point(L~Lower_Seq(C,n),Wmin,Emax,Vertical_Line sr);
      A90: L~Rev Lower_Seq(C,n) = L~Lower_Seq(C,n) by SPPOL_2:22;
      A91: FiP2 = LaP by A16,A18,JORDAN5C:18;
        now per cases by A72,A73,A77,XBOOLE_0:def 2;
       suppose A92: x in LSeg(SW,Wmin);
        then A93: x`1 = SW`1 by A80,SPRECT_3:20;
        then A94: x`1 = Wbo by PSCOMP_1:72;
        A95: x`2 <= Wmin`2 by A81,A92,TOPREAL1:10;
          now per cases by A69,A71,A76,XBOOLE_0:def 2;
         suppose A96: x in L~midU;
          then x in L~Upper_Seq(C,n) by A74;
          then x in W-most L~Cage(C,n) by A82,A94,SPRECT_2:16;
          then x`2 >= Wmin`2 by PSCOMP_1:88;
          then x`2 = Wmin`2 by A95,AXIOMS:21;
          then x = Wmin by A79,A93,TOPREAL3:11;
          then FiP..Upper_Seq(C,n) = 1 by A3,A5,A28,A96,Th45;
          then Wmin = FiP by A5,A13,FINSEQ_5:41;
          hence contradiction by A8,A11,PSCOMP_1:84;
         suppose x in LSeg(FiP,|[Ebo,FiP`2]|);
          hence contradiction by A8,A11,A78,A94,TOPREAL1:9;
        end;
        hence contradiction;
       suppose A97: x in L~RevL;
          now per cases by A69,A71,A76,XBOOLE_0:def 2;
         suppose A98: x in L~midU;
          then x in L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n)
                                                 by A74,A75,A97,XBOOLE_0:def 3;
          then A99: x in {Wmin,Emax} by JORDAN1E:20;
            now per cases by A99,TARSKI:def 2;
           suppose x = Wmin;
            then FiP..Upper_Seq(C,n) = 1 by A3,A5,A28,A98,Th45;
            then Wmin = FiP by A5,A13,FINSEQ_5:41;
            hence contradiction by A8,A11,PSCOMP_1:84;
           suppose x = Emax;
            then FiP..Upper_Seq(C,n) = len Upper_Seq(C,n)
                                                   by A3,A6,A28,A98,Th46;
            then Emax = FiP by A6,A13,FINSEQ_5:41;
            hence contradiction by A8,A11,PSCOMP_1:104;
          end;
          hence contradiction;
         suppose A100: x in LSeg(FiP,|[Ebo,FiP`2]|);
            LSeg(FiP,|[Ebo,FiP`2]|) is horizontal by A84,SPPOL_1:36;
          then A101: x`2 = FiP`2 by A100,SPRECT_3:19;
          A102: FiP`1 <= x`1 & x`1 <= |[Ebo,FiP`2]|`1
                                               by A8,A11,A78,A100,TOPREAL1:9;
          consider i be Nat such that
           A103: 1 <= i & i+1 <= len RevL and
           A104: x in LSeg(RevL/.i,RevL/.(i+1)) by A97,SPPOL_2:14;
          A105: i < len RevL by A103,NAT_1:38;
          then i in Seg (LaP..Rev Lower_Seq(C,n)) by A48,A103,FINSEQ_1:3;
          then A106: RevL/.i = (Rev Lower_Seq(C,n))/.i by A25,FINSEQ_5:46;
          A107: i+1 >= 1 by NAT_1:29;
          then i+1 in Seg (LaP..Rev Lower_Seq(C,n)) by A48,A103,FINSEQ_1:3;
          then A108: RevL/.(i+1) = (Rev Lower_Seq(C,n))/.(i+1)
                                                          by A25,FINSEQ_5:46;
          A109: (Rev Lower_Seq(C,n)/.i)`1 < sr
                                           by A1,A48,A90,A91,A103,A105,Th60;
            now per cases by A103,REAL_1:def 5;
           suppose i+1 < len RevL;
            then A110: (Rev Lower_Seq(C,n)/.(i+1))`1 < sr
                                               by A1,A48,A90,A91,A107,Th60;
              (RevL/.i)`1 <= (RevL/.(i+1))`1 or (RevL/.(i+1))`1 <= (RevL/.i)`1;
            then x`1 <= (RevL/.(i+1))`1 or x`1 <= (RevL/.i)`1
                                                           by A104,TOPREAL1:9;
            hence contradiction by A11,A102,A106,A108,A109,A110,AXIOMS:22;
           suppose A111: i+1 = len RevL;
            A112: Rev Lower_Seq(C,n) is special by SPPOL_2:42;
              i+1 <= len Rev Lower_Seq(C,n) by A25,A48,A111,FINSEQ_4:31;
            then LSeg(Rev Lower_Seq(C,n)/.i,Rev Lower_Seq(C,n)/.(i+1)) =
             LSeg(Rev Lower_Seq(C,n),i) by A103,TOPREAL1:def 5;
            then LSeg(RevL/.i,RevL/.(i+1)) is vertical or
             LSeg(RevL/.i,RevL/.(i+1)) is horizontal by A106,A108,A112,SPPOL_1:
41;
            hence contradiction by A21,A23,A77,A101,A104,A106,A109,A111,SPPOL_1
:37,SPRECT_3:19;
          end;
          hence contradiction;
        end;
        hence contradiction;
       suppose A113: x in LSeg(LaP,|[sr,Nbo]|);
        then A114: x`1 = LaP`1 by A83,SPRECT_3:20;
        A115: LaP`2 <= x`2 by A37,A86,A113,TOPREAL1:10;
          now per cases by A69,A71,A76,XBOOLE_0:def 2;
         suppose x in L~midU;
          then consider i be Nat such that
           A116: 1 <= i & i+1 <= len midU and
           A117: x in LSeg(midU/.i,midU/.(i+1)) by SPPOL_2:14;
            i < len midU by A116,NAT_1:38;
          then A118: i in dom midU by A116,FINSEQ_3:27;
            i+2 >= 1+1 by NAT_1:29;
          then i+2-1 >= 1+1-1 by REAL_1:49;
          then A119: i+2-1 >= 0 by AXIOMS:22;
          A120: midU/.i = Upper_Seq(C,n)/.(i+2-'1) by A4,A14,A88,A118,SPRECT_2:
7
             .= Upper_Seq(C,n)/.(i+2-1) by A119,BINARITH:def 3
             .= Upper_Seq(C,n)/.(i+(2-1)) by XCMPLX_1:29;
          A121: 1 <= i+1 by NAT_1:29;
          then A122: i+1 in dom midU by A116,FINSEQ_3:27;
            i+1+2 >= 1+1 by NAT_1:29;
          then i+1+2-1 >= 1+1-1 by REAL_1:49;
          then A123: i+1+2-1 >= 0 by AXIOMS:22;
          A124: midU/.(i+1) = Upper_Seq(C,n)/.(i+1+2-'1)
                                                 by A4,A14,A88,A122,SPRECT_2:7
             .= Upper_Seq(C,n)/.(i+1+2-1) by A123,BINARITH:def 3
             .= Upper_Seq(C,n)/.(i+1+(2-1)) by XCMPLX_1:29;
            i+1+1 <= FiP..Upper_Seq(C,n)-1+1 by A89,A116,REAL_1:55;
          then A125: i+1+1 <= FiP..Upper_Seq(C,n) by XCMPLX_1:27;
          then i+1 < FiP..Upper_Seq(C,n) by NAT_1:38;
          then A126: (midU/.i)`1 < sr by A1,A120,A121,Th59;
          A127: 1 <= i+1+1 by NAT_1:29;
            i+1+1 <= len Upper_Seq(C,n) by A28,A125,AXIOMS:22;
          then LSeg(midU/.i,midU/.(i+1)) = LSeg(Upper_Seq(C,n),i+1)
                                              by A120,A121,A124,TOPREAL1:def 5;
          then A128: LSeg(midU/.i,midU/.(i+1)) is vertical or
           LSeg(midU/.i,midU/.(i+1)) is horizontal by SPPOL_1:41;
            now per cases by A116,REAL_1:def 5;
           suppose i+1 < len midU;
            then i+1+1 <= len midU by NAT_1:38;
            then i+1+1+1 <= FiP..Upper_Seq(C,n)-1+1 by A89,REAL_1:55;
            then i+1+1+1 <= FiP..Upper_Seq(C,n) by XCMPLX_1:27;
            then i+1+1 < FiP..Upper_Seq(C,n) by NAT_1:38;
            then A129: (midU/.(i+1))`1 < sr by A1,A124,A127,Th59;
              (midU/.i)`1 <= (midU/.(i+1))`1 or (midU/.(i+1))`1 <= (midU/.i)`1;
            hence contradiction by A21,A114,A117,A126,A129,TOPREAL1:9;
           suppose A130: i+1 = len midU;
            then (midU/.i)`2 = (midU/.(i+1))`2
                                            by A11,A76,A126,A128,SPPOL_1:36,37;
            hence contradiction by A24,A76,A115,A117,A130,GOBOARD7:6;
          end;
          hence contradiction;
         suppose x in LSeg(FiP,|[Ebo,FiP`2]|);
          hence contradiction by A24,A85,A115,SPRECT_3:19;
        end;
        hence contradiction;
      end;
      hence contradiction;
     suppose A131: SW = Wmin;
      set h = ((Rev Lower_Seq(C,n))-:LaP)^<*|[sr,Nbo]|*>;
        rng ((Rev Lower_Seq(C,n))-:LaP) misses {|[sr,Nbo]|} by A36,ZFMISC_1:56;
      then rng ((Rev Lower_Seq(C,n))-:LaP) /\ {|[sr,Nbo]|} = {} by XBOOLE_0:def
7;
      then rng ((Rev Lower_Seq(C,n))-:LaP) /\ rng <*|[sr,Nbo]|*> = {}
                                                              by FINSEQ_1:55;
      then A132: rng ((Rev Lower_Seq(C,n))-:LaP) misses rng <*|[sr,Nbo]|*>
                                                                by XBOOLE_0:def
7;
      A133: <*|[sr,Nbo]|*> is one-to-one by FINSEQ_3:102;
      A134: (((Rev Lower_Seq(C,n))-:LaP)/.len ((Rev Lower_Seq(C,n))-:LaP))`1
          = (((Rev Lower_Seq(C,n))-:LaP)/.(LaP..Rev Lower_Seq(C,n)))`1
                                                           by A25,FINSEQ_5:45
         .= LaP`1 by A25,FINSEQ_5:48
         .= |[sr,Nbo]|`1 by A21,EUCLID:56
         .= (<*|[sr,Nbo]|*>/.1)`1 by FINSEQ_4:25;
      reconsider RevLS = Rev Lower_Seq(C,n) as
                             special FinSequence of TOP-REAL 2 by SPPOL_2:42;
      A135: RevLS-:LaP is special;
        <*|[sr,Nbo]|*> is special by SPPOL_2:39;
      then reconsider h as one-to-one special FinSequence of TOP-REAL 2
                         by A132,A133,A134,A135,FINSEQ_3:98,GOBOARD2:13;
        Lower_Seq(C,n) is_in_the_area_of Cage(C,n) by JORDAN1E:22;
      then Rev Lower_Seq(C,n) is_in_the_area_of Cage(C,n) by SPRECT_3:68;
      then A136: ((Rev Lower_Seq(C,n))-:LaP) is_in_the_area_of Cage(C,n)
                                                           by A25,JORDAN1E:5;
        now let m be Nat;
       assume m in dom <*|[sr,Nbo]|*>;
       then m in Seg 1 by FINSEQ_1:55;
       then m = 1 by FINSEQ_1:4,TARSKI:def 1;
       then <*|[sr,Nbo]|*>/.m = |[sr,Nbo]| by FINSEQ_4:25;
       then A137: (<*|[sr,Nbo]|*>/.m)`1 = sr & (<*|[sr,Nbo]|*>/.m)`2 = Nbo
                                                                by EUCLID:56;
         W-bound L~Cage(C,n) <= E-bound L~Cage(C,n) by SPRECT_1:23;
       hence W-bound L~Cage(C,n) <= (<*|[sr,Nbo]|*>/.m)`1 &
            (<*|[sr,Nbo]|*>/.m)`1 <= E-bound L~Cage(C,n) by A137,JORDAN6:2;
       thus S-bound L~Cage(C,n) <= (<*|[sr,Nbo]|*>/.m)`2 &
            (<*|[sr,Nbo]|*>/.m)`2 <= N-bound L~Cage(C,n) by A137,SPRECT_1:24;
      end;
      then <*|[sr,Nbo]|*> is_in_the_area_of Cage(C,n) by SPRECT_2:def 1;
      then A138: h is_in_the_area_of Cage(C,n) by A136,SPRECT_2:28;
        LaP..(Rev Lower_Seq(C,n)) <> 0 by A47,A48,FINSEQ_1:25;
      then LaP..(Rev Lower_Seq(C,n)) > 0 by NAT_1:19;
      then LaP..(Rev Lower_Seq(C,n)) >= 0+1 by NAT_1:38;
      then A139: len ((Rev Lower_Seq(C,n))-:LaP) >= 1 by A25,FINSEQ_5:45;
      A140: len h = len ((Rev Lower_Seq(C,n))-:LaP) + 1 by FINSEQ_2:19;
      then A141: 1+1 <= len h by A139,REAL_1:55;
        1 in dom h by FINSEQ_5:6;
      then h/.1 = h.1 by FINSEQ_4:def 4;
      then A142: (h/.1)`2 = (((Rev Lower_Seq(C,n))-:LaP)/.1)`2
                                                            by A139,JORDAN3:17
         .= ((Rev Lower_Seq(C,n))/.1)`2 by A25,FINSEQ_5:47
         .= (Lower_Seq(C,n)/.len Lower_Seq(C,n))`2 by FINSEQ_5:68
         .= Wmin`2 by JORDAN1F:8
         .= S-bound L~Cage(C,n) by A131,PSCOMP_1:73;
        (h/.len h)`2 = |[sr,Nbo]|`2 by A140,TOPREAL4:1
         .= N-bound L~Cage(C,n) by EUCLID:56;
      then A143: h is_a_v.c._for Cage(C,n) by A138,A142,SPRECT_2:def 3;
        len g = len mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n))+1 by FINSEQ_2:19;
      then 1+1 <= len g by A44,REAL_1:55;
      then L~g meets L~h by A46,A141,A143,SPRECT_2:33;
      then consider x be set such that
       A144: x in L~g and
       A145: x in L~h by XBOOLE_0:3;
      reconsider x as Point of TOP-REAL 2 by A144;
      set midU = mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n));
        midU is non empty by A4,A14,SPRECT_2:11;
      then A146: L~g = L~midU \/ LSeg(midU/.len midU,|[Ebo,FiP`2]|)
                                                               by SPPOL_2:19;
      A147: L~h = L~RevL \/ LSeg(RevL/.len RevL,|[sr,Nbo]|) by A47,SPPOL_2:19;
      A148: L~midU c= L~Upper_Seq(C,n) by A4,A14,SPRECT_3:34;
        L~Rev Lower_Seq(C,n) = L~RevL \/ L~((Rev Lower_Seq(C,n)):-LaP)
                                                           by A25,SPPOL_2:25;
      then L~RevL c= L~Rev Lower_Seq(C,n) by XBOOLE_1:7;
      then A149: L~RevL c= L~Lower_Seq(C,n) by SPPOL_2:22;
      A150: midU/.len midU = Upper_Seq(C,n)/.(FiP..Upper_Seq(C,n))
                                                        by A4,A14,SPRECT_2:13
         .= FiP by A13,FINSEQ_5:41;
      A151: RevL/.len RevL = RevL/.(LaP..Rev Lower_Seq(C,n)) by A25,FINSEQ_5:45
         .= LaP by A25,FINSEQ_5:48;
      A152: |[Ebo,FiP`2]|`1 = Ebo by EUCLID:56;
        LaP`1 = |[sr,Nbo]|`1 by A21,EUCLID:56;
      then A153: LSeg(LaP,|[sr,Nbo]|) is vertical by SPPOL_1:37;
      A154: |[Ebo,FiP`2]|`2 = FiP`2 by EUCLID:56;
      then A155: LSeg(FiP,|[Ebo,FiP`2]|) is horizontal by SPPOL_1:36;
      A156: LaP`2 <= Nbo by A20,PSCOMP_1:71;
      A157: Wmin in rng Upper_Seq(C,n) by A5,FINSEQ_6:46;
        now assume FiP..Upper_Seq(C,n) = 1;
       then FiP..Upper_Seq(C,n) = (Upper_Seq(C,n)/.1)..Upper_Seq(C,n)
                                                               by FINSEQ_6:47
          .= Wmin..Upper_Seq(C,n) by JORDAN1F:5;
       then FiP = Wmin by A13,A157,FINSEQ_5:10;
       hence contradiction by A8,A11,PSCOMP_1:84;
      end;
      then FiP..Upper_Seq(C,n) > 1 by A28,REAL_1:def 5;
      then A158: 1+1+0 <= FiP..Upper_Seq(C,n) by NAT_1:38;
      then FiP..Upper_Seq(C,n)-2 >= 0 by REAL_1:84;
      then FiP..Upper_Seq(C,n)-'2 = FiP..Upper_Seq(C,n)-2 by BINARITH:def 3;
      then A159: len midU = FiP..Upper_Seq(C,n)-2+1 by A28,A158,JORDAN4:20
         .= FiP..Upper_Seq(C,n)-(2-1) by XCMPLX_1:37;
      set FiP2 = First_Point(L~Lower_Seq(C,n),Wmin,Emax,Vertical_Line sr);
      A160: L~Rev Lower_Seq(C,n) = L~Lower_Seq(C,n) by SPPOL_2:22;
      A161: FiP2 = LaP by A16,A18,JORDAN5C:18;
        now per cases by A145,A147,A151,XBOOLE_0:def 2;
       suppose A162: x in L~RevL;
          now per cases by A144,A146,A150,XBOOLE_0:def 2;
         suppose A163: x in L~midU;
          then x in L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n)
                                        by A148,A149,A162,XBOOLE_0:def 3;
          then A164: x in {Wmin,Emax} by JORDAN1E:20;
            now per cases by A164,TARSKI:def 2;
           suppose x = Wmin;
            then FiP..Upper_Seq(C,n) = 1 by A3,A5,A28,A163,Th45;
            then Wmin = FiP by A5,A13,FINSEQ_5:41;
            hence contradiction by A8,A11,PSCOMP_1:84;
           suppose x = Emax;
            then FiP..Upper_Seq(C,n) = len Upper_Seq(C,n)
                                                   by A3,A6,A28,A163,Th46;
            then Emax = FiP by A6,A13,FINSEQ_5:41;
            hence contradiction by A8,A11,PSCOMP_1:104;
          end;
          hence contradiction;
         suppose A165: x in LSeg(FiP,|[Ebo,FiP`2]|);
            LSeg(FiP,|[Ebo,FiP`2]|) is horizontal by A154,SPPOL_1:36;
          then A166: x`2 = FiP`2 by A165,SPRECT_3:19;
          A167: FiP`1 <= x`1 & x`1 <= |[Ebo,FiP`2]|`1
                                               by A8,A11,A152,A165,TOPREAL1:9;
          consider i be Nat such that
           A168: 1 <= i & i+1 <= len RevL and
           A169: x in LSeg(RevL/.i,RevL/.(i+1)) by A162,SPPOL_2:14;
          A170: i < len RevL by A168,NAT_1:38;
          then i in Seg (LaP..Rev Lower_Seq(C,n)) by A48,A168,FINSEQ_1:3;
          then A171: RevL/.i = (Rev Lower_Seq(C,n))/.i by A25,FINSEQ_5:46;
          A172: i+1 >= 1 by NAT_1:29;
          then i+1 in Seg (LaP..Rev Lower_Seq(C,n)) by A48,A168,FINSEQ_1:3;
          then A173: RevL/.(i+1) = (Rev Lower_Seq(C,n))/.(i+1)
                                                          by A25,FINSEQ_5:46;
          A174: (Rev Lower_Seq(C,n)/.i)`1 < sr
                                           by A1,A48,A160,A161,A168,A170,Th60;
            now per cases by A168,REAL_1:def 5;
           suppose i+1 < len RevL;
            then A175: (Rev Lower_Seq(C,n)/.(i+1))`1 < sr
                                               by A1,A48,A160,A161,A172,Th60;
              (RevL/.i)`1 <= (RevL/.(i+1))`1 or (RevL/.(i+1))`1 <= (RevL/.i)`1;
            then x`1 <= (RevL/.(i+1))`1 or x`1 <= (RevL/.i)`1
                                                           by A169,TOPREAL1:9;
            hence contradiction by A11,A167,A171,A173,A174,A175,AXIOMS:22;
           suppose A176: i+1 = len RevL;
            A177: Rev Lower_Seq(C,n) is special by SPPOL_2:42;
              i+1 <= len Rev Lower_Seq(C,n) by A25,A48,A176,FINSEQ_4:31;
            then LSeg(Rev Lower_Seq(C,n)/.i,Rev Lower_Seq(C,n)/.(i+1)) =
             LSeg(Rev Lower_Seq(C,n),i) by A168,TOPREAL1:def 5;
            then LSeg(RevL/.i,RevL/.(i+1)) is vertical or
             LSeg(RevL/.i,RevL/.(i+1)) is horizontal by A171,A173,A177,SPPOL_1:
41;
            hence contradiction by A21,A23,A151,A166,A169,A171,A174,A176,
SPPOL_1:37,SPRECT_3:19;
          end;
          hence contradiction;
        end;
        hence contradiction;
       suppose A178: x in LSeg(LaP,|[sr,Nbo]|);
        then A179: x`1 = LaP`1 by A153,SPRECT_3:20;
        A180: LaP`2 <= x`2 by A37,A156,A178,TOPREAL1:10;
          now per cases by A144,A146,A150,XBOOLE_0:def 2;
         suppose x in L~midU;
          then consider i be Nat such that
           A181: 1 <= i & i+1 <= len midU and
           A182: x in LSeg(midU/.i,midU/.(i+1)) by SPPOL_2:14;
            i < len midU by A181,NAT_1:38;
          then A183: i in dom midU by A181,FINSEQ_3:27;
            i+2 >= 1+1 by NAT_1:29;
          then i+2-1 >= 1+1-1 by REAL_1:49;
          then A184: i+2-1 >= 0 by AXIOMS:22;
          A185: midU/.i = Upper_Seq(C,n)/.(i+2-'1) by A4,A14,A158,A183,SPRECT_2
:7
             .= Upper_Seq(C,n)/.(i+2-1) by A184,BINARITH:def 3
             .= Upper_Seq(C,n)/.(i+(2-1)) by XCMPLX_1:29;
          A186: 1 <= i+1 by NAT_1:29;
          then A187: i+1 in dom midU by A181,FINSEQ_3:27;
            i+1+2 >= 1+1 by NAT_1:29;
          then i+1+2-1 >= 1+1-1 by REAL_1:49;
          then A188: i+1+2-1 >= 0 by AXIOMS:22;
          A189: midU/.(i+1) = Upper_Seq(C,n)/.(i+1+2-'1)
                                                 by A4,A14,A158,A187,SPRECT_2:7
             .= Upper_Seq(C,n)/.(i+1+2-1) by A188,BINARITH:def 3
             .= Upper_Seq(C,n)/.(i+1+(2-1)) by XCMPLX_1:29;
            i+1+1 <= FiP..Upper_Seq(C,n)-1+1 by A159,A181,REAL_1:55;
          then A190: i+1+1 <= FiP..Upper_Seq(C,n) by XCMPLX_1:27;
          then i+1 < FiP..Upper_Seq(C,n) by NAT_1:38;
          then A191: (midU/.i)`1 < sr by A1,A185,A186,Th59;
          A192: 1 <= i+1+1 by NAT_1:29;
            i+1+1 <= len Upper_Seq(C,n) by A28,A190,AXIOMS:22;
          then LSeg(midU/.i,midU/.(i+1)) = LSeg(Upper_Seq(C,n),i+1)
                                              by A185,A186,A189,TOPREAL1:def 5;
          then A193: LSeg(midU/.i,midU/.(i+1)) is vertical or
           LSeg(midU/.i,midU/.(i+1)) is horizontal by SPPOL_1:41;
            now per cases by A181,REAL_1:def 5;
           suppose i+1 < len midU;
            then i+1+1 <= len midU by NAT_1:38;
            then i+1+1+1 <= FiP..Upper_Seq(C,n)-1+1 by A159,REAL_1:55;
            then i+1+1+1 <= FiP..Upper_Seq(C,n) by XCMPLX_1:27;
            then i+1+1 < FiP..Upper_Seq(C,n) by NAT_1:38;
            then A194: (midU/.(i+1))`1 < sr by A1,A189,A192,Th59;
              (midU/.i)`1 <= (midU/.(i+1))`1 or (midU/.(i+1))`1 <= (midU/.i)`1;
            hence contradiction by A21,A179,A182,A191,A194,TOPREAL1:9;
           suppose A195: i+1 = len midU;
            then (midU/.i)`2 = (midU/.(i+1))`2
                                  by A11,A150,A191,A193,SPPOL_1:36,37;
            hence contradiction by A24,A150,A180,A182,A195,GOBOARD7:6;
          end;
          hence contradiction;
         suppose x in LSeg(FiP,|[Ebo,FiP`2]|);
          hence contradiction by A24,A155,A180,SPRECT_3:19;
        end;
        hence contradiction;
      end;
      hence contradiction;
    end;
    hence contradiction;
   end;

  theorem Th63:
   for C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2
   for n be Nat st n > 0 holds
    L~Upper_Seq(C,n) = Upper_Arc L~Cage(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let n be Nat;
    assume A1: n > 0;
    A2: Upper_Seq(C,n)=Rotate(Cage(C,n),W-min L~Cage(C,n))-:E-max L~Cage(C,n)
                                                          by JORDAN1E:def 1;
    A3: Lower_Seq(C,n)=Rotate(Cage(C,n),W-min L~Cage(C,n)):-E-max L~Cage(C,n)
                                                          by JORDAN1E:def 2;
    A4: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:47;
      E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:50;
    then A5: E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                          by A4,FINSEQ_6:96;
    then Upper_Seq(C,n)/.1 = Rotate(Cage(C,n),W-min L~Cage(C,n))/.1
                                                       by A2,FINSEQ_5:47;
    then A6: Upper_Seq(C,n)/.1 = W-min L~Cage(C,n) by A4,FINSEQ_6:98;
    A7: Upper_Seq(C,n)/.len Upper_Seq(C,n) =
          (Rotate(Cage(C,n),W-min L~Cage(C,n))-:E-max L~Cage(C,n))/.
           ((E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)))
                                                        by A2,A5,FINSEQ_5:45
       .= E-max L~Cage(C,n) by A5,FINSEQ_5:48;
    A8: Lower_Seq(C,n)/.1 = (Rotate(Cage(C,n),W-min L~Cage(C,n)):-
                                     E-max L~Cage(C,n))/.1 by JORDAN1E:def 2
       .= E-max L~Cage(C,n) by FINSEQ_5:56;
    A9: Lower_Seq(C,n)/.len Lower_Seq(C,n) =
          Rotate(Cage(C,n),W-min L~Cage(C,n))/.
           (len Rotate(Cage(C,n),W-min L~Cage(C,n))) by A3,A5,FINSEQ_5:57
       .= Rotate(Cage(C,n),W-min L~Cage(C,n))/.1 by FINSEQ_6:def 1
       .= W-min L~Cage(C,n) by A4,FINSEQ_6:98;
    A10: L~Upper_Seq(C,n) is_an_arc_of W-min L~Cage(C,n),E-max L~Cage(C,n)
                                                        by A6,A7,TOPREAL1:31;
    A11: L~Lower_Seq(C,n) is_an_arc_of E-max L~Cage(C,n),W-min L~Cage(C,n)
                                                        by A8,A9,TOPREAL1:31;
    A12: L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n) =
     {W-min L~Cage(C,n),E-max L~Cage(C,n)} by JORDAN1E:20;
    A13: L~Upper_Seq(C,n) \/ L~Lower_Seq(C,n) = L~Cage(C,n) by JORDAN1E:17;
      First_Point(L~Upper_Seq(C,n),W-min L~Cage(C,n),E-max L~Cage(C,n),
      Vertical_Line((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))`2 >
     Last_Point(L~Lower_Seq(C,n),E-max L~Cage(C,n),W-min L~Cage(C,n),
      Vertical_Line((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))`2
                                                               by A1,Th62;
    hence thesis by A10,A11,A12,A13,JORDAN6:def 8;
   end;

  theorem Th64:
   for C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2
   for n be Nat st n > 0 holds
    L~Lower_Seq(C,n) = Lower_Arc L~Cage(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let n be Nat;
    assume A1: n > 0;
    A2: Lower_Seq(C,n)=Rotate(Cage(C,n),W-min L~Cage(C,n)):-E-max L~Cage(C,n)
                                                          by JORDAN1E:def 2;
    A3: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:47;
      E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:50;
    then A4: E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n))
                                                          by A3,FINSEQ_6:96;
    A5: Lower_Seq(C,n)/.1 = (Rotate(Cage(C,n),W-min L~Cage(C,n)):-
                                     E-max L~Cage(C,n))/.1 by JORDAN1E:def 2
       .= E-max L~Cage(C,n) by FINSEQ_5:56;
    A6: Lower_Seq(C,n)/.len Lower_Seq(C,n) =
          Rotate(Cage(C,n),W-min L~Cage(C,n))/.
           (len Rotate(Cage(C,n),W-min L~Cage(C,n))) by A2,A4,FINSEQ_5:57
       .= Rotate(Cage(C,n),W-min L~Cage(C,n))/.1 by FINSEQ_6:def 1
       .= W-min L~Cage(C,n) by A3,FINSEQ_6:98;
    A7: L~Upper_Seq(C,n) = Upper_Arc L~Cage(C,n) by A1,Th63;
    A8: L~Lower_Seq(C,n) is_an_arc_of E-max L~Cage(C,n),W-min L~Cage(C,n)
                                                        by A5,A6,TOPREAL1:31;
    A9: L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n) =
     {W-min L~Cage(C,n),E-max L~Cage(C,n)} by JORDAN1E:20;
    A10: L~Upper_Seq(C,n) \/ L~Lower_Seq(C,n) = L~Cage(C,n) by JORDAN1E:17;
      First_Point(L~Upper_Seq(C,n),W-min L~Cage(C,n),E-max L~Cage(C,n),
      Vertical_Line((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))`2 >
     Last_Point(L~Lower_Seq(C,n),E-max L~Cage(C,n),W-min L~Cage(C,n),
      Vertical_Line((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))`2
                                                               by A1,Th62;
    hence thesis by A7,A8,A9,A10,JORDAN6:def 9;
   end;

  theorem
     for C be compact connected non vertical non horizontal Subset of TOP-REAL
2
   for n be Nat st n > 0
   for i,j be Nat st 1 <= i & i <= len Gauge(C,n) &
    1 <= j & j <= width Gauge(C,n) & Gauge(C,n)*(i,j) in L~Cage(C,n) holds
     LSeg(Gauge(C,n)*(i,1),Gauge(C,n)*(i,j)) meets Lower_Arc L~Cage(C,n)
   proof
    let C be compact connected non vertical non horizontal
     Subset of TOP-REAL 2;
    let n be Nat;
    assume A1: n > 0;
    let i,j be Nat;
    assume A2: 1 <= i & i <= len Gauge(C,n) & 1 <= j & j <= width Gauge(C,n) &
     Gauge(C,n)*(i,j) in L~Cage(C,n);
      L~Lower_Seq(C,n) = Lower_Arc L~Cage(C,n) by A1,Th64;
    hence thesis by A2,Th54;
   end;


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