let i, n be Element of NAT ; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in S-most C & p in south_halfline x & 1 <= i & i < len (Cage C,n) & p in LSeg (Cage C,n),i holds
LSeg (Cage C,n),i is horizontal
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for x, p being Point of (TOP-REAL 2) st x in S-most C & p in south_halfline x & 1 <= i & i < len (Cage C,n) & p in LSeg (Cage C,n),i holds
LSeg (Cage C,n),i is horizontal
let x, p be Point of (TOP-REAL 2); :: thesis: ( x in S-most C & p in south_halfline x & 1 <= i & i < len (Cage C,n) & p in LSeg (Cage C,n),i implies LSeg (Cage C,n),i is horizontal )
set G = Gauge C,n;
set f = Cage C,n;
assume that
A1: x in S-most C and
A2: p in south_halfline x and
A3: 1 <= i and
A4: i < len (Cage C,n) and
A5: p in LSeg (Cage C,n),i ; :: thesis: LSeg (Cage C,n),i is horizontal
A6: i + 1 <= len (Cage C,n) by A4, NAT_1:13;
then A7: LSeg (Cage C,n),i = LSeg ((Cage C,n) /. i),((Cage C,n) /. (i + 1)) by A3, TOPREAL1:def 5;
assume A8: not LSeg (Cage C,n),i is horizontal ; :: thesis: contradiction
A9: x in C by A1, XBOOLE_0:def 4;
p in L~ (Cage C,n) by A5, SPPOL_2:17;
then A10: p in (south_halfline x) /\ (L~ (Cage C,n)) by A2, XBOOLE_0:def 4;
A11: x `1 = p `1 by A2, TOPREAL1:def 14
.= ((Cage C,n) /. (i + 1)) `1 by A5, A7, A8, SPPOL_1:41, SPPOL_1:64 ;
A12: x `1 = p `1 by A2, TOPREAL1:def 14
.= ((Cage C,n) /. i) `1 by A5, A7, A8, SPPOL_1:41, SPPOL_1:64 ;
A13: Cage C,n is_sequence_on Gauge C,n by JORDAN9:def 1;
i in Seg (len (Cage C,n)) by A3, A4, FINSEQ_1:3;
then A14: i in dom (Cage C,n) by FINSEQ_1:def 3;
1 <= i + 1 by A3, NAT_1:13;
then i + 1 in Seg (len (Cage C,n)) by A6, FINSEQ_1:3;
then A15: i + 1 in dom (Cage C,n) by FINSEQ_1:def 3;

per cases ( ((Cage C,n) /. i) `2 <= ((Cage C,n) /. (i + 1)) `2 or ((Cage C,n) /. i) `2 >= ((Cage C,n) /. (i + 1)) `2 ) ;

suppose A16: ((Cage C,n) /. i) `2 <= ((Cage C,n) /. (i + 1)) `2 ; :: thesis: contradiction

then ((Cage C,n) /. i) `2 <= p `2 by A5, A7, TOPREAL1:10;
then A17: ((Cage C,n) /. i) `2 < x `2 by A9, A10, Th97, XXREAL_0:2;
consider i1, i2 being Element of NAT such that
A18: [i1,i2] in Indices (Gauge C,n) and
A19: (Cage C,n) /. i = (Gauge C,n) * i1,i2 by A13, A14, GOBOARD1:def 11;
A20: ( 1 <= i2 & i2 <= width (Gauge C,n) & 1 <= i1 & i1 <= len (Gauge C,n) ) by A18, MATRIX_1:39;
A21: x `2 = (S-min C) `2 by A1, PSCOMP_1:118
.= S-bound C by EUCLID:56
.= ((Gauge C,n) * i1,2) `2 by A20, JORDAN8:16 ;
then i2 < 1 + 1 by A17, A19, A20, SPRECT_3:24;
then A22: i2 <= 1 by NAT_1:13;
consider j1, j2 being Element of NAT such that
A23: [j1,j2] in Indices (Gauge C,n) and
A24: (Cage C,n) /. (i + 1) = (Gauge C,n) * j1,j2 by A13, A15, GOBOARD1:def 11;
A25: ( 1 <= j2 & j2 <= width (Gauge C,n) & 1 <= j1 & j1 <= len (Gauge C,n) ) by A23, MATRIX_1:39;

now

assume ((Cage C,n) /. i) `2 = ((Cage C,n) /. (i + 1)) `2 ; :: thesis: contradiction
then (Cage C,n) /. i = (Cage C,n) /. (i + 1) by A11, A12, TOPREAL3:11;
then A26: ( i1 = j1 & i2 = j2 ) by A18, A19, A23, A24, GOBOARD1:21;
(abs (i1 - j1)) + (abs (i2 - j2)) = 1 by A13, A14, A15, A18, A19, A23, A24, GOBOARD1:def 11;
then 1 = 0 + (abs (i2 - j2)) by A26, GOBOARD7:2
.= 0 + 0 by A26, GOBOARD7:2 ;
hence contradiction ; :: thesis: verum

end;

then ((Cage C,n) /. i) `2 < ((Cage C,n) /. (i + 1)) `2 by A16, XXREAL_0:1;
then i2 < j2 by A19, A20, A24, A25, Th40;
then 1 < j2 by A20, A22, XXREAL_0:1;
then 1 + 1 <= j2 by NAT_1:13;
then x `2 <= ((Cage C,n) /. (i + 1)) `2 by A20, A21, A24, A25, Th40;
then x in L~ (Cage C,n) by A7, A11, A12, A17, GOBOARD7:8, SPPOL_2:17;
then x in (L~ (Cage C,n)) /\ C by A9, XBOOLE_0:def 4;
then L~ (Cage C,n) meets C by XBOOLE_0:4;
hence contradiction by JORDAN10:5; :: thesis: verum

end;

suppose A27: ((Cage C,n) /. i) `2 >= ((Cage C,n) /. (i + 1)) `2 ; :: thesis: contradiction

then ((Cage C,n) /. (i + 1)) `2 <= p `2 by A5, A7, TOPREAL1:10;
then A28: ((Cage C,n) /. (i + 1)) `2 < x `2 by A9, A10, Th97, XXREAL_0:2;
consider i1, i2 being Element of NAT such that
A29: [i1,i2] in Indices (Gauge C,n) and
A30: (Cage C,n) /. (i + 1) = (Gauge C,n) * i1,i2 by A13, A15, GOBOARD1:def 11;
A31: ( 1 <= i2 & i2 <= width (Gauge C,n) & 1 <= i1 & i1 <= len (Gauge C,n) ) by A29, MATRIX_1:39;
A32: x `2 = (S-min C) `2 by A1, PSCOMP_1:118
.= S-bound C by EUCLID:56
.= ((Gauge C,n) * i1,2) `2 by A31, JORDAN8:16 ;
then i2 < 1 + 1 by A28, A30, A31, SPRECT_3:24;
then A33: i2 <= 1 by NAT_1:13;
consider j1, j2 being Element of NAT such that
A34: [j1,j2] in Indices (Gauge C,n) and
A35: (Cage C,n) /. i = (Gauge C,n) * j1,j2 by A13, A14, GOBOARD1:def 11;
A36: ( 1 <= j2 & j2 <= width (Gauge C,n) & 1 <= j1 & j1 <= len (Gauge C,n) ) by A34, MATRIX_1:39;

now

assume ((Cage C,n) /. i) `2 = ((Cage C,n) /. (i + 1)) `2 ; :: thesis: contradiction
then (Cage C,n) /. i = (Cage C,n) /. (i + 1) by A11, A12, TOPREAL3:11;
then A37: ( i1 = j1 & i2 = j2 ) by A29, A30, A34, A35, GOBOARD1:21;
(abs (j1 - i1)) + (abs (j2 - i2)) = 1 by A13, A14, A15, A29, A30, A34, A35, GOBOARD1:def 11;
then 1 = 0 + (abs (i2 - j2)) by A37, GOBOARD7:2
.= 0 + 0 by A37, GOBOARD7:2 ;
hence contradiction ; :: thesis: verum

end;

then ((Cage C,n) /. (i + 1)) `2 < ((Cage C,n) /. i) `2 by A27, XXREAL_0:1;
then i2 < j2 by A30, A31, A35, A36, Th40;
then 1 < j2 by A31, A33, XXREAL_0:1;
then 1 + 1 <= j2 by NAT_1:13;
then x `2 <= ((Cage C,n) /. i) `2 by A31, A32, A35, A36, Th40;
then x in L~ (Cage C,n) by A7, A11, A12, A28, GOBOARD7:8, SPPOL_2:17;
then x in (L~ (Cage C,n)) /\ C by A9, XBOOLE_0:def 4;
then L~ (Cage C,n) meets C by XBOOLE_0:4;
hence contradiction by JORDAN10:5; :: thesis: verum

end;

end;