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
assume A6: not LSeg (Cage C,n),i is horizontal ; :: thesis: contradiction
A7: i + 1 <= len (Cage C,n) by A4, NAT_1:13;
then A8: LSeg (Cage C,n),i = LSeg ((Cage C,n) /. i),((Cage C,n) /. (i + 1)) by A3, TOPREAL1:def 5;
1 <= i + 1 by A3, NAT_1:13;
then i + 1 in Seg (len (Cage C,n)) by A7, FINSEQ_1:3;
then A9: i + 1 in dom (Cage C,n) by FINSEQ_1:def 3;
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: Cage C,n is_sequence_on Gauge C,n by JORDAN9:def 1;
A12: x `1 = p `1 by A2, TOPREAL1:def 14
.= ((Cage C,n) /. i) `1 by A5, A8, A6, SPPOL_1:41, SPPOL_1:64 ;
i in Seg (len (Cage C,n)) by A3, A4, FINSEQ_1:3;
then A13: i in dom (Cage C,n) by FINSEQ_1:def 3;
A14: x `1 = p `1 by A2, TOPREAL1:def 14
.= ((Cage C,n) /. (i + 1)) `1 by A5, A8, A6, SPPOL_1:41, SPPOL_1:64 ;
A15: x in C by A1, XBOOLE_0:def 4;

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, A8, TOPREAL1:10;
then A17: ((Cage C,n) /. i) `2 < x `2 by A15, 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 A11, A13, GOBOARD1:def 11;
A20: i2 <= width (Gauge C,n) by A18, MATRIX_1:39;
A21: 1 <= i2 by A18, MATRIX_1:39;
A22: ( 1 <= i1 & i1 <= len (Gauge C,n) ) by A18, MATRIX_1:39;
A23: x `2 = (S-min C) `2 by A1, PSCOMP_1:118
.= S-bound C by EUCLID:56
.= ((Gauge C,n) * i1,2) `2 by A22, JORDAN8:16 ;
then i2 < 1 + 1 by A17, A19, A20, A22, SPRECT_3:24;
then A24: i2 <= 1 by NAT_1:13;
consider j1, j2 being Element of NAT such that
A25: [j1,j2] in Indices (Gauge C,n) and
A26: (Cage C,n) /. (i + 1) = (Gauge C,n) * j1,j2 by A11, A9, GOBOARD1:def 11;
A27: j2 <= width (Gauge C,n) by A25, MATRIX_1:39;

now

assume ((Cage C,n) /. i) `2 = ((Cage C,n) /. (i + 1)) `2 ; :: thesis: contradiction
then A28: (Cage C,n) /. i = (Cage C,n) /. (i + 1) by A14, A12, TOPREAL3:11;
then A29: i1 = j1 by A18, A19, A25, A26, GOBOARD1:21;
A30: i2 = j2 by A18, A19, A25, A26, A28, GOBOARD1:21;
(abs (i1 - j1)) + (abs (i2 - j2)) = 1 by A11, A13, A9, A18, A19, A25, A26, GOBOARD1:def 11;
then 1 = 0 + (abs (i2 - j2)) by A29, GOBOARD7:2
.= 0 + 0 by A30, GOBOARD7:2 ;
hence contradiction ; :: thesis: verum

end;

then A31: ((Cage C,n) /. i) `2 < ((Cage C,n) /. (i + 1)) `2 by A16, XXREAL_0:1;
A32: ( 1 <= j1 & j1 <= len (Gauge C,n) ) by A25, MATRIX_1:39;
1 <= j2 by A25, MATRIX_1:39;
then i2 < j2 by A19, A20, A22, A26, A32, A31, Th40;
then 1 < j2 by A21, A24, XXREAL_0:1;
then 1 + 1 <= j2 by NAT_1:13;
then x `2 <= ((Cage C,n) /. (i + 1)) `2 by A22, A23, A26, A27, A32, Th40;
then x in L~ (Cage C,n) by A8, A14, A12, A17, GOBOARD7:8, SPPOL_2:17;
then x in (L~ (Cage C,n)) /\ C by A15, XBOOLE_0:def 4;
then L~ (Cage C,n) meets C by XBOOLE_0:4;
hence contradiction by JORDAN10:5; :: thesis: verum

end;

suppose A33: ((Cage C,n) /. i) `2 >= ((Cage C,n) /. (i + 1)) `2 ; :: thesis: contradiction

then ((Cage C,n) /. (i + 1)) `2 <= p `2 by A5, A8, TOPREAL1:10;
then A34: ((Cage C,n) /. (i + 1)) `2 < x `2 by A15, A10, Th97, XXREAL_0:2;
consider i1, i2 being Element of NAT such that
A35: [i1,i2] in Indices (Gauge C,n) and
A36: (Cage C,n) /. (i + 1) = (Gauge C,n) * i1,i2 by A11, A9, GOBOARD1:def 11;
A37: i2 <= width (Gauge C,n) by A35, MATRIX_1:39;
A38: 1 <= i2 by A35, MATRIX_1:39;
A39: ( 1 <= i1 & i1 <= len (Gauge C,n) ) by A35, MATRIX_1:39;
A40: x `2 = (S-min C) `2 by A1, PSCOMP_1:118
.= S-bound C by EUCLID:56
.= ((Gauge C,n) * i1,2) `2 by A39, JORDAN8:16 ;
then i2 < 1 + 1 by A34, A36, A37, A39, SPRECT_3:24;
then A41: i2 <= 1 by NAT_1:13;
consider j1, j2 being Element of NAT such that
A42: [j1,j2] in Indices (Gauge C,n) and
A43: (Cage C,n) /. i = (Gauge C,n) * j1,j2 by A11, A13, GOBOARD1:def 11;
A44: j2 <= width (Gauge C,n) by A42, MATRIX_1:39;

now

assume ((Cage C,n) /. i) `2 = ((Cage C,n) /. (i + 1)) `2 ; :: thesis: contradiction
then A45: (Cage C,n) /. i = (Cage C,n) /. (i + 1) by A14, A12, TOPREAL3:11;
then A46: i1 = j1 by A35, A36, A42, A43, GOBOARD1:21;
A47: i2 = j2 by A35, A36, A42, A43, A45, GOBOARD1:21;
(abs (j1 - i1)) + (abs (j2 - i2)) = 1 by A11, A13, A9, A35, A36, A42, A43, GOBOARD1:def 11;
then 1 = 0 + (abs (i2 - j2)) by A46, A47, GOBOARD7:2
.= 0 + 0 by A47, GOBOARD7:2 ;
hence contradiction ; :: thesis: verum

end;

then A48: ((Cage C,n) /. (i + 1)) `2 < ((Cage C,n) /. i) `2 by A33, XXREAL_0:1;
A49: ( 1 <= j1 & j1 <= len (Gauge C,n) ) by A42, MATRIX_1:39;
1 <= j2 by A42, MATRIX_1:39;
then i2 < j2 by A36, A37, A39, A43, A49, A48, Th40;
then 1 < j2 by A38, A41, XXREAL_0:1;
then 1 + 1 <= j2 by NAT_1:13;
then x `2 <= ((Cage C,n) /. i) `2 by A39, A40, A43, A44, A49, Th40;
then x in L~ (Cage C,n) by A8, A14, A12, A34, GOBOARD7:8, SPPOL_2:17;
then x in (L~ (Cage C,n)) /\ C by A15, XBOOLE_0:def 4;
then L~ (Cage C,n) meets C by XBOOLE_0:4;
hence contradiction by JORDAN10:5; :: thesis: verum

end;

end;