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 Th85:
  x in W-most C & p in west_halfline x /\ L~Cage(C,n) implies p`1
  = W-bound L~Cage(C,n)
proof
  set G = Gauge(C,n), f = Cage(C,n);
A1: f is_sequence_on G by JORDAN9:def 1;
  assume
A2: x in W-most C;
  then
A3: x in C by XBOOLE_0:def 4;
A4: len G = width G by JORDAN8:def 1;
  assume
A5: p in west_halfline x /\ L~f;
  then p in L~f by XBOOLE_0:def 4;
  then consider i such that
A6: 1 <= i and
A7: i+1 <= len f and
A8: p in LSeg(f,i) by SPPOL_2:13;
A9: LSeg(f,i) = LSeg(f/.i,f/.(i+1)) by A6,A7,TOPREAL1:def 3;
A10: i < len f by A7,NAT_1:13;
  then i in Seg len f by A6,FINSEQ_1:1;
  then i in dom f by FINSEQ_1:def 3;
  then consider i1, i2 being Nat such that
A11: [i1,i2] in Indices G and
A12: f/.i = G*(i1,i2) by A1,GOBOARD1:def 9;
A13: 1 <= i2 & i2 <= width G by A11,MATRIX_0:32;
  p in west_halfline x by A5,XBOOLE_0:def 4;
  then LSeg(f,i) is vertical by A2,A6,A8,A10,Th81;
  then (f/.i)`1 = (f/.(i+1))`1 by A9,SPPOL_1:16;
  then
A14: p`1 = (f/.i)`1 by A8,A9,GOBOARD7:5;
A15: i1 <= len G by A11,MATRIX_0:32;
A16: 1 <= i1 by A11,MATRIX_0:32;
  x`1 = (W-min C)`1 by A2,PSCOMP_1:31
    .= W-bound C by EUCLID:52
    .= G*(2,i2)`1 by A4,A13,JORDAN8:11;
  then i1 < 1+1 by A3,A5,A12,A13,A15,A14,Th77,SPRECT_3:13;
  then i1 <= 1 by NAT_1:13;
  then i1 = 1 by A16,XXREAL_0:1;
  then f/.i in W-most L~f by A6,A10,A12,A13,Th59;
  hence thesis by A14,Th6;
end;
