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 Th83:
  x in E-most C & p in east_halfline x /\ L~Cage(C,n) implies p`1
  = E-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 E-most C;
  then
A3: x in C by XBOOLE_0:def 4;
A4: len G-'1 <= len G by NAT_D:35;
A5: len G = width G by JORDAN8:def 1;
  assume
A6: p in east_halfline x /\ L~f;
  then p in L~f by XBOOLE_0:def 4;
  then consider i such that
A7: 1 <= i and
A8: i+1 <= len f and
A9: p in LSeg(f,i) by SPPOL_2:13;
A10: LSeg(f,i) = LSeg(f/.i,f/.(i+1)) by A7,A8,TOPREAL1:def 3;
A11: i < len f by A8,NAT_1:13;
  then i in Seg len f by A7,FINSEQ_1:1;
  then i in dom f by FINSEQ_1:def 3;
  then consider i1, i2 being Nat such that
A12: [i1,i2] in Indices G and
A13: f/.i = G*(i1,i2) by A1,GOBOARD1:def 9;
A14: 1 <= i2 & i2 <= width G by A12,MATRIX_0:32;
  p in east_halfline x by A6,XBOOLE_0:def 4;
  then LSeg(f,i) is vertical by A2,A7,A9,A11,Th79;
  then (f/.i)`1 = (f/.(i+1))`1 by A10,SPPOL_1:16;
  then
A15: p`1 = (f/.i)`1 by A9,A10,GOBOARD7:5;
A16: i1 <= len G by A12,MATRIX_0:32;
A17: 1 <= i1 by A12,MATRIX_0:32;
  x`1 = (E-min C)`1 by A2,PSCOMP_1:47
    .= E-bound C by EUCLID:52
    .= G*(len G-'1,i2)`1 by A5,A14,JORDAN8:12;
  then i1 > len G-'1 by A3,A6,A13,A14,A17,A15,A4,Th75,SPRECT_3:13;
  then i1 >= len G-'1+1 by NAT_1:13;
  then i1 >= len G by A17,XREAL_1:235,XXREAL_0:2;
  then i1 = len G by A16,XXREAL_0:1;
  then f/.i in E-most L~f by A7,A11,A13,A14,Th61;
  hence thesis by A15,Th4;
end;
