reserve i, i1, i2, j, j1, j2, k, m, n, t for Nat,
  D for non empty Subset of TOP-REAL 2,
  E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p, q, x for Point of TOP-REAL 2,
  r, s for Real;

theorem Th81:
  x in W-most C & p in west_halfline x & 1 <= i & i < len Cage(C,
  n) & p in LSeg(Cage(C,n),i) implies LSeg(Cage(C,n),i) is vertical
proof
  set G = Gauge(C,n), f = Cage(C,n);
  assume that
A1: x in W-most C and
A2: p in west_halfline x and
A3: 1 <= i and
A4: i < len f and
A5: p in LSeg(f,i);
  assume
A6: not thesis;
A7: i+1 <= len f by A4,NAT_1:13;
  then
A8: LSeg(f,i) = LSeg(f/.i,f/.(i+1)) by A3,TOPREAL1:def 3;
  1 <= i+1 by A3,NAT_1:13;
  then i+1 in Seg len f by A7,FINSEQ_1:1;
  then
A9: i+1 in dom f by FINSEQ_1:def 3;
  p in L~f by A5,SPPOL_2:17;
  then
A10: p in west_halfline x /\ L~f by A2,XBOOLE_0:def 4;
A11: f is_sequence_on G by JORDAN9:def 1;
A12: x`2 = p`2 by A2,TOPREAL1:def 13
    .= (f/.i)`2 by A5,A8,A6,SPPOL_1:19,40;
  i in Seg len f by A3,A4,FINSEQ_1:1;
  then
A13: i in dom f by FINSEQ_1:def 3;
A14: x`2 = p`2 by A2,TOPREAL1:def 13
    .= (f/.(i+1))`2 by A5,A8,A6,SPPOL_1:19,40;
A15: x in C by A1,XBOOLE_0:def 4;
  per cases;
  suppose
A16: (f/.i)`1 <= (f/.(i+1))`1;
    consider i1,i2 being Nat such that
A17: [i1,i2] in Indices G and
A18: f/.i = G*(i1,i2) by A11,A13,GOBOARD1:def 9;
A19: 1 <= i2 by A17,MATRIX_0:32;
A20: i2 <= width G by A17,MATRIX_0:32;
    then
A21: i2 <= len G by JORDAN8:def 1;
    consider j1,j2 being Nat such that
A22: [j1,j2] in Indices G and
A23: f/.(i+1) = G*(j1,j2) by A11,A9,GOBOARD1:def 9;
A24: 1 <= j2 & j2 <= width G by A22,MATRIX_0:32;
    now
      assume (f/.i)`1 = (f/.(i+1))`1;
      then
A25:  f/.i = f/.(i+1) by A14,A12,TOPREAL3:6;
      then
A26:  i1 = j1 by A17,A18,A22,A23,GOBOARD1:5;
A27:  i2=j2 by A17,A18,A22,A23,A25,GOBOARD1:5;
      |.i1-j1.|+|.i2-j2.| = 1 by A11,A13,A9,A17,A18,A22,A23,GOBOARD1:def 9;
      then 1 = 0 + |.i2-j2.| by A26,GOBOARD7:2
        .= 0 + 0 by A27,GOBOARD7:2;
      hence contradiction;
    end;
    then
A28: (f/.i)`1 < (f/.(i+1))`1 by A16,XXREAL_0:1;
    (f/.i)`1 <= p`1 by A5,A8,A16,TOPREAL1:3;
    then
A29: (f/.i)`1 < x`1 by A15,A10,Th77,XXREAL_0:2;
A30: 1 <= i1 by A17,MATRIX_0:32;
A31: i1 <= len G by A17,MATRIX_0:32;
A32: x`1 = (W-min C)`1 by A1,PSCOMP_1:31
      .= W-bound C by EUCLID:52
      .= G*(2,i2)`1 by A19,A21,JORDAN8:11;
    then i1 < 1+1 by A29,A18,A19,A20,A31,SPRECT_3:13;
    then
A33: i1 <= 1 by NAT_1:13;
A34: j1 <= len G by A22,MATRIX_0:32;
    1 <= j1 by A22,MATRIX_0:32;
    then i1 < j1 by A18,A19,A20,A31,A23,A24,A28,Th18;
    then 1 < j1 by A30,A33,XXREAL_0:1;
    then 1+1 <= j1 by NAT_1:13;
    then x`1 <= (f/.(i+1))`1 by A19,A20,A32,A23,A24,A34,Th18;
    then x in L~f by A8,A14,A12,A29,GOBOARD7:8,SPPOL_2:17;
    then x in L~f /\ C by A15,XBOOLE_0:def 4;
    then L~f meets C;
    hence contradiction by JORDAN10:5;
  end;
  suppose
A35: (f/.i)`1 >= (f/.(i+1))`1;
    consider i1,i2 being Nat such that
A36: [i1,i2] in Indices G and
A37: f/.(i+1) = G*(i1,i2) by A11,A9,GOBOARD1:def 9;
A38: 1 <= i2 by A36,MATRIX_0:32;
A39: i2 <= width G by A36,MATRIX_0:32;
    then
A40: i2 <= len G by JORDAN8:def 1;
    consider j1,j2 being Nat such that
A41: [j1,j2] in Indices G and
A42: f/.i = G*(j1,j2) by A11,A13,GOBOARD1:def 9;
A43: 1 <= j2 & j2 <= width G by A41,MATRIX_0:32;
    now
      assume (f/.i)`1 = (f/.(i+1))`1;
      then
A44:  f/.i = f/.(i+1) by A14,A12,TOPREAL3:6;
      then
A45:  i1 = j1 by A36,A37,A41,A42,GOBOARD1:5;
A46:  i2=j2 by A36,A37,A41,A42,A44,GOBOARD1:5;
      |.j1-i1.|+|.j2-i2.| = 1 by A11,A13,A9,A36,A37,A41,A42,GOBOARD1:def 9;
      then 1 = 0 + |.i2-j2.| by A45,A46,GOBOARD7:2
        .= 0 + 0 by A46,GOBOARD7:2;
      hence contradiction;
    end;
    then
A47: (f/.(i+1))`1 < (f/.i)`1 by A35,XXREAL_0:1;
    (f/.(i+1))`1 <= p`1 by A5,A8,A35,TOPREAL1:3;
    then
A48: (f/.(i+1))`1 < x`1 by A15,A10,Th77,XXREAL_0:2;
A49: 1 <= i1 by A36,MATRIX_0:32;
A50: i1 <= len G by A36,MATRIX_0:32;
A51: x`1 = (W-min C)`1 by A1,PSCOMP_1:31
      .= W-bound C by EUCLID:52
      .= G*(2,i2)`1 by A38,A40,JORDAN8:11;
    then i1 < 1+1 by A48,A37,A38,A39,A50,SPRECT_3:13;
    then
A52: i1 <= 1 by NAT_1:13;
A53: j1 <= len G by A41,MATRIX_0:32;
    1 <= j1 by A41,MATRIX_0:32;
    then i1 < j1 by A37,A38,A39,A50,A42,A43,A47,Th18;
    then 1 < j1 by A49,A52,XXREAL_0:1;
    then 1+1 <= j1 by NAT_1:13;
    then x`1 <= (f/.i)`1 by A38,A39,A51,A42,A43,A53,Th18;
    then x in L~f by A8,A14,A12,A48,GOBOARD7:8,SPPOL_2:17;
    then x in L~f /\ C by A15,XBOOLE_0:def 4;
    then L~f meets C;
    hence contradiction by JORDAN10:5;
  end;
end;
