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 Th78:
  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
proof
  set G = Gauge(C,n), 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 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;
  i in Seg len f by A3,A4,FINSEQ_1:1;
  then
A10: i in dom f by FINSEQ_1:def 3;
A11: len G = width G by JORDAN8:def 1;
  then
A12: len G-'1 <= width G by NAT_D:35;
A13: x in C by A1,XBOOLE_0:def 4;
  p in L~f by A5,SPPOL_2:17;
  then
A14: p in north_halfline x /\ L~f by A2,XBOOLE_0:def 4;
A15: f is_sequence_on G by JORDAN9:def 1;
A16: x`1 = p`1 by A2,TOPREAL1:def 10
    .= (f/.i)`1 by A5,A8,A6,SPPOL_1:19,41;
A17: x`1 = p`1 by A2,TOPREAL1:def 10
    .= (f/.(i+1))`1 by A5,A8,A6,SPPOL_1:19,41;
  per cases;
  suppose
A18: (f/.i)`2 <= (f/.(i+1))`2;
    then p`2 <= (f/.(i+1))`2 by A5,A8,TOPREAL1:4;
    then
A19: (f/.(i+1))`2 > x`2 by A13,A14,Th74,XXREAL_0:2;
    consider i1,i2 being Nat such that
A20: [i1,i2] in Indices G and
A21: f/.(i+1) = G*(i1,i2) by A15,A9,GOBOARD1:def 9;
A22: 1 <= i2 by A20,MATRIX_0:32;
    i2 <= width G by A20,MATRIX_0:32;
    then
A23: i2 <= len G by JORDAN8:def 1;
A24: 1 <= i1 & i1 <= len G by A20,MATRIX_0:32;
    consider j1,j2 being Nat such that
A25: [j1,j2] in Indices G and
A26: f/.i = G*(j1,j2) by A15,A10,GOBOARD1:def 9;
A27: 1 <= j1 & j1 <= len G by A25,MATRIX_0:32;
    now
      assume (f/.i)`2 = (f/.(i+1))`2;
      then
A28:  f/.i = f/.(i+1) by A17,A16,TOPREAL3:6;
      then
A29:  i2=j2 by A20,A21,A25,A26,GOBOARD1:5;
      i1 = j1 & |.i1-j1.|+|.i2-j2.| = 1 by A15,A10,A9,A20,A21,A25,A26,A28,
GOBOARD1:5,def 9;
      then 1 = 0 + |.i2-j2.| by GOBOARD7:2
        .= 0 + 0 by A29,GOBOARD7:2;
      hence contradiction;
    end;
    then
A30: (f/.i)`2 < (f/.(i+1))`2 by A18,XXREAL_0:1;
A31: 1 <= j2 by A25,MATRIX_0:32;
    j2 <= width G by A25,MATRIX_0:32;
    then i2 > j2 by A21,A22,A24,A26,A27,A30,Th19;
    then len G > j2 by A23,XXREAL_0:2;
    then
A32: len G-'1 >= j2 by NAT_D:49;
    x`2 = (N-min C)`2 by A1,PSCOMP_1:39
      .= N-bound C by EUCLID:52
      .= G*(i1,len G-'1)`2 by A24,JORDAN8:14;
    then x`2 >= (f/.i)`2 by A12,A24,A26,A31,A27,A32,Th19;
    then x in L~f by A8,A17,A16,A19,GOBOARD7:7,SPPOL_2:17;
    then L~f meets C by A13,XBOOLE_0:3;
    hence contradiction by JORDAN10:5;
  end;
  suppose
A33: (f/.i)`2 >= (f/.(i+1))`2;
    then p`2 <= (f/.i)`2 by A5,A8,TOPREAL1:4;
    then
A34: (f/.i)`2 > x`2 by A13,A14,Th74,XXREAL_0:2;
    consider i1,i2 being Nat such that
A35: [i1,i2] in Indices G and
A36: f/.i = G*(i1,i2) by A15,A10,GOBOARD1:def 9;
A37: 1 <= i2 by A35,MATRIX_0:32;
    consider j1,j2 being Nat such that
A38: [j1,j2] in Indices G and
A39: f/.(i+1) = G*(j1,j2) by A15,A9,GOBOARD1:def 9;
A40: 1 <= j1 & j1 <= len G by A38,MATRIX_0:32;
    now
      assume (f/.i)`2 = (f/.(i+1))`2;
      then
A41:  f/.i = f/.(i+1) by A17,A16,TOPREAL3:6;
      then
A42:  i2=j2 by A35,A36,A38,A39,GOBOARD1:5;
      i1 = j1 & |.j1-i1.|+|.j2-i2.| = 1 by A15,A10,A9,A35,A36,A38,A39,A41,
GOBOARD1:5,def 9;
      then 1 = 0 + |.i2-j2.| by A42,GOBOARD7:2
        .= 0 + 0 by A42,GOBOARD7:2;
      hence contradiction;
    end;
    then
A43: (f/.(i+1))`2 < (f/.i)`2 by A33,XXREAL_0:1;
A44: i2 <= width G by A35,MATRIX_0:32;
A45: 1 <= i1 & i1 <= len G by A35,MATRIX_0:32;
A46: 1 <= j2 by A38,MATRIX_0:32;
    j2 <= width G by A38,MATRIX_0:32;
    then i2 > j2 by A36,A37,A45,A39,A40,A43,Th19;
    then len G > j2 by A11,A44,XXREAL_0:2;
    then
A47: len G-'1 >= j2 by NAT_D:49;
    x`2 = (N-min C)`2 by A1,PSCOMP_1:39
      .= N-bound C by EUCLID:52
      .= G*(i1,len G-'1)`2 by A45,JORDAN8:14;
    then x`2 >= (f/.(i+1))`2 by A12,A45,A39,A46,A40,A47,Th19;
    then x in L~f by A8,A17,A16,A34,GOBOARD7:7,SPPOL_2:17;
    then x in L~f /\ C by A13,XBOOLE_0:def 4;
    then L~f meets C;
    hence contradiction by JORDAN10:5;
  end;
end;
