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 Th80:
  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
proof
  set G = Gauge(C,n), 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 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 south_halfline x /\ L~f by A2,XBOOLE_0:def 4;
A11: f is_sequence_on G by JORDAN9:def 1;
A12: x`1 = p`1 by A2,TOPREAL1:def 12
    .= (f/.i)`1 by A5,A8,A6,SPPOL_1:19,41;
  i in Seg len f by A3,A4,FINSEQ_1:1;
  then
A13: i in dom f by FINSEQ_1:def 3;
A14: x`1 = p`1 by A2,TOPREAL1:def 12
    .= (f/.(i+1))`1 by A5,A8,A6,SPPOL_1:19,41;
A15: x in C by A1,XBOOLE_0:def 4;
  per cases;
  suppose
A16: (f/.i)`2 <= (f/.(i+1))`2;
    then (f/.i)`2 <= p`2 by A5,A8,TOPREAL1:4;
    then
A17: (f/.i)`2 < x`2 by A15,A10,Th76,XXREAL_0:2;
    consider i1,i2 being Nat such that
A18: [i1,i2] in Indices G and
A19: f/.i = G*(i1,i2) by A11,A13,GOBOARD1:def 9;
A20: i2 <= width G by A18,MATRIX_0:32;
A21: 1 <= i2 by A18,MATRIX_0:32;
A22: 1 <= i1 & i1 <= len G by A18,MATRIX_0:32;
A23: x`2 = (S-min C)`2 by A1,PSCOMP_1:55
      .= S-bound C by EUCLID:52
      .= G*(i1,2)`2 by A22,JORDAN8:13;
    then i2 < 1+1 by A17,A19,A20,A22,SPRECT_3:12;
    then
A24: i2 <= 1 by NAT_1:13;
    consider j1,j2 being Nat such that
A25: [j1,j2] in Indices G and
A26: f/.(i+1) = G*(j1,j2) by A11,A9,GOBOARD1:def 9;
A27: j2 <= width G by A25,MATRIX_0:32;
    now
      assume (f/.i)`2 = (f/.(i+1))`2;
      then
A28:  f/.i = f/.(i+1) by A14,A12,TOPREAL3:6;
      then
A29:  i1 = j1 by A18,A19,A25,A26,GOBOARD1:5;
A30:  i2=j2 by A18,A19,A25,A26,A28,GOBOARD1:5;
      |.i1-j1.|+|.i2-j2.| = 1 by A11,A13,A9,A18,A19,A25,A26,GOBOARD1:def 9;
      then 1 = 0 + |.i2-j2.| by A29,GOBOARD7:2
        .= 0 + 0 by A30,GOBOARD7:2;
      hence contradiction;
    end;
    then
A31: (f/.i)`2 < (f/.(i+1))`2 by A16,XXREAL_0:1;
A32: 1 <= j1 & j1 <= len G by A25,MATRIX_0:32;
    1 <= j2 by A25,MATRIX_0:32;
    then i2 < j2 by A19,A20,A22,A26,A32,A31,Th19;
    then 1 < j2 by A21,A24,XXREAL_0:1;
    then 1+1 <= j2 by NAT_1:13;
    then x`2 <= (f/.(i+1))`2 by A22,A23,A26,A27,A32,Th19;
    then x in L~f by A8,A14,A12,A17,GOBOARD7:7,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
A33: (f/.i)`2 >= (f/.(i+1))`2;
    then (f/.(i+1))`2 <= p`2 by A5,A8,TOPREAL1:4;
    then
A34: (f/.(i+1))`2 < x`2 by A15,A10,Th76,XXREAL_0:2;
    consider i1,i2 being Nat such that
A35: [i1,i2] in Indices G and
A36: f/.(i+1) = G*(i1,i2) by A11,A9,GOBOARD1:def 9;
A37: i2 <= width G by A35,MATRIX_0:32;
A38: 1 <= i2 by A35,MATRIX_0:32;
A39: 1 <= i1 & i1 <= len G by A35,MATRIX_0:32;
A40: x`2 = (S-min C)`2 by A1,PSCOMP_1:55
      .= S-bound C by EUCLID:52
      .= G*(i1,2)`2 by A39,JORDAN8:13;
    then i2 < 1+1 by A34,A36,A37,A39,SPRECT_3:12;
    then
A41: i2 <= 1 by NAT_1:13;
    consider j1,j2 being Nat such that
A42: [j1,j2] in Indices G and
A43: f/.i = G*(j1,j2) by A11,A13,GOBOARD1:def 9;
A44: j2 <= width G by A42,MATRIX_0:32;
    now
      assume (f/.i)`2 = (f/.(i+1))`2;
      then
A45:  f/.i = f/.(i+1) by A14,A12,TOPREAL3:6;
      then
A46:  i1 = j1 by A35,A36,A42,A43,GOBOARD1:5;
A47:  i2=j2 by A35,A36,A42,A43,A45,GOBOARD1:5;
      |.j1-i1.|+|.j2-i2.| = 1 by A11,A13,A9,A35,A36,A42,A43,GOBOARD1:def 9;
      then 1 = 0 + |.i2-j2.| by A46,A47,GOBOARD7:2
        .= 0 + 0 by A47,GOBOARD7:2;
      hence contradiction;
    end;
    then
A48: (f/.(i+1))`2 < (f/.i)`2 by A33,XXREAL_0:1;
A49: 1 <= j1 & j1 <= len G by A42,MATRIX_0:32;
    1 <= j2 by A42,MATRIX_0:32;
    then i2 < j2 by A36,A37,A39,A43,A49,A48,Th19;
    then 1 < j2 by A38,A41,XXREAL_0:1;
    then 1+1 <= j2 by NAT_1:13;
    then x`2 <= (f/.i)`2 by A39,A40,A43,A44,A49,Th19;
    then x in L~f by A8,A14,A12,A34,GOBOARD7:7,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;
