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 N-most C & p in north_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 N-most C & p in north_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 N-most C & p in north_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 N-most C and
A2: p in north_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;
i in Seg (len (Cage C,n)) by A3, A4, FINSEQ_1:3;
then A10: i in dom (Cage C,n) by FINSEQ_1:def 3;
A11: len (Gauge C,n) = width (Gauge C,n) by JORDAN8:def 1;
then A12: (len (Gauge C,n)) -' 1 <= width (Gauge C,n) by NAT_D:35;
A13: x in C by A1, XBOOLE_0:def 4;
p in L~ (Cage C,n) by A5, SPPOL_2:17;
then A14: p in (north_halfline x) /\ (L~ (Cage C,n)) by A2, XBOOLE_0:def 4;
A15: Cage C,n is_sequence_on Gauge C,n by JORDAN9:def 1;
A16: x `1 = p `1 by A2, TOPREAL1:def 12
.= ((Cage C,n) /. i) `1 by A5, A8, A6, SPPOL_1:41, SPPOL_1:64 ;
A17: x `1 = p `1 by A2, TOPREAL1:def 12
.= ((Cage C,n) /. (i + 1)) `1 by A5, A8, A6, SPPOL_1:41, SPPOL_1:64 ;

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 A18: ((Cage C,n) /. i) `2 <= ((Cage C,n) /. (i + 1)) `2 ; :: thesis: contradiction

then p `2 <= ((Cage C,n) /. (i + 1)) `2 by A5, A8, TOPREAL1:10;
then A19: ((Cage C,n) /. (i + 1)) `2 > x `2 by A13, A14, Th95, XXREAL_0:2;
consider i1, i2 being Element of NAT such that
A20: [i1,i2] in Indices (Gauge C,n) and
A21: (Cage C,n) /. (i + 1) = (Gauge C,n) * i1,i2 by A15, A9, GOBOARD1:def 11;
A22: 1 <= i2 by A20, MATRIX_1:39;
i2 <= width (Gauge C,n) by A20, MATRIX_1:39;
then A23: i2 <= len (Gauge C,n) by JORDAN8:def 1;
A24: ( 1 <= i1 & i1 <= len (Gauge C,n) ) by A20, MATRIX_1:39;
consider j1, j2 being Element of NAT such that
A25: [j1,j2] in Indices (Gauge C,n) and
A26: (Cage C,n) /. i = (Gauge C,n) * j1,j2 by A15, A10, GOBOARD1:def 11;
A27: ( 1 <= j1 & j1 <= len (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 A17, A16, TOPREAL3:11;
then A29: i2 = j2 by A20, A21, A25, A26, GOBOARD1:21;
( i1 = j1 & (abs (i1 - j1)) + (abs (i2 - j2)) = 1 ) by A15, A10, A9, A20, A21, A25, A26, A28, GOBOARD1:21, GOBOARD1:def 11;
then 1 = 0 + (abs (i2 - j2)) by GOBOARD7:2
.= 0 + 0 by A29, GOBOARD7:2 ;
hence contradiction ; :: thesis: verum

end;

then A30: ((Cage C,n) /. i) `2 < ((Cage C,n) /. (i + 1)) `2 by A18, XXREAL_0:1;
A31: 1 <= j2 by A25, MATRIX_1:39;
j2 <= width (Gauge C,n) by A25, MATRIX_1:39;
then i2 > j2 by A21, A22, A24, A26, A27, A30, Th40;
then len (Gauge C,n) > j2 by A23, XXREAL_0:2;
then A32: (len (Gauge C,n)) -' 1 >= j2 by NAT_D:49;
x `2 = (N-min C) `2 by A1, PSCOMP_1:98
.= N-bound C by EUCLID:56
.= ((Gauge C,n) * i1,((len (Gauge C,n)) -' 1)) `2 by A24, JORDAN8:17 ;
then x `2 >= ((Cage C,n) /. i) `2 by A12, A24, A26, A31, A27, A32, Th40;
then x in L~ (Cage C,n) by A8, A17, A16, A19, GOBOARD7:8, SPPOL_2:17;
then L~ (Cage C,n) meets C by A13, XBOOLE_0:3;
hence contradiction by JORDAN10:5; :: thesis: verum

end;

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

then p `2 <= ((Cage C,n) /. i) `2 by A5, A8, TOPREAL1:10;
then A34: ((Cage C,n) /. i) `2 > x `2 by A13, A14, Th95, 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 = (Gauge C,n) * i1,i2 by A15, A10, GOBOARD1:def 11;
A37: 1 <= i2 by A35, MATRIX_1:39;
consider j1, j2 being Element of NAT such that
A38: [j1,j2] in Indices (Gauge C,n) and
A39: (Cage C,n) /. (i + 1) = (Gauge C,n) * j1,j2 by A15, A9, GOBOARD1:def 11;
A40: ( 1 <= j1 & j1 <= len (Gauge C,n) ) by A38, MATRIX_1:39;

now

assume ((Cage C,n) /. i) `2 = ((Cage C,n) /. (i + 1)) `2 ; :: thesis: contradiction
then A41: (Cage C,n) /. i = (Cage C,n) /. (i + 1) by A17, A16, TOPREAL3:11;
then A42: i2 = j2 by A35, A36, A38, A39, GOBOARD1:21;
( i1 = j1 & (abs (j1 - i1)) + (abs (j2 - i2)) = 1 ) by A15, A10, A9, A35, A36, A38, A39, A41, GOBOARD1:21, GOBOARD1:def 11;
then 1 = 0 + (abs (i2 - j2)) by A42, GOBOARD7:2
.= 0 + 0 by A42, GOBOARD7:2 ;
hence contradiction ; :: thesis: verum

end;

then A43: ((Cage C,n) /. (i + 1)) `2 < ((Cage C,n) /. i) `2 by A33, XXREAL_0:1;
A44: i2 <= width (Gauge C,n) by A35, MATRIX_1:39;
A45: ( 1 <= i1 & i1 <= len (Gauge C,n) ) by A35, MATRIX_1:39;
A46: 1 <= j2 by A38, MATRIX_1:39;
j2 <= width (Gauge C,n) by A38, MATRIX_1:39;
then i2 > j2 by A36, A37, A45, A39, A40, A43, Th40;
then len (Gauge C,n) > j2 by A11, A44, XXREAL_0:2;
then A47: (len (Gauge C,n)) -' 1 >= j2 by NAT_D:49;
x `2 = (N-min C) `2 by A1, PSCOMP_1:98
.= N-bound C by EUCLID:56
.= ((Gauge C,n) * i1,((len (Gauge C,n)) -' 1)) `2 by A45, JORDAN8:17 ;
then x `2 >= ((Cage C,n) /. (i + 1)) `2 by A12, A45, A39, A46, A40, A47, Th40;
then x in L~ (Cage C,n) by A8, A17, A16, A34, GOBOARD7:8, SPPOL_2:17;
then x in (L~ (Cage C,n)) /\ C by A13, XBOOLE_0:def 4;
then L~ (Cage C,n) meets C by XBOOLE_0:4;
hence contradiction by JORDAN10:5; :: thesis: verum

end;

end;