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 Th82:
  x in N-most C & p in north_halfline x /\ L~Cage(C,n) implies p
  `2 = N-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 N-most C;
  then
A3: x in C by XBOOLE_0:def 4;
  assume
A4: p in north_halfline x /\ L~f;
  then p in L~f by XBOOLE_0:def 4;
  then consider i such that
A5: 1 <= i and
A6: i+1 <= len f and
A7: p in LSeg(f,i) by SPPOL_2:13;
A8: LSeg(f,i) = LSeg(f/.i,f/.(i+1)) by A5,A6,TOPREAL1:def 3;
A9: i < len f by A6,NAT_1:13;
  then i in Seg len f by A5,FINSEQ_1:1;
  then i in dom f by FINSEQ_1:def 3;
  then consider i1, i2 being Nat such that
A10: [i1,i2] in Indices G and
A11: f/.i = G*(i1,i2) by A1,GOBOARD1:def 9;
A12: 1 <= i2 by A10,MATRIX_0:32;
  p in north_halfline x by A4,XBOOLE_0:def 4;
  then LSeg(f,i) is horizontal by A2,A5,A7,A9,Th78;
  then (f/.i)`2 = (f/.(i+1))`2 by A8,SPPOL_1:15;
  then
A13: p`2 = (f/.i)`2 by A7,A8,GOBOARD7:6;
A14: i2 <= width G by A10,MATRIX_0:32;
A15: 1 <= i1 & i1 <= len G by A10,MATRIX_0:32;
A16: len G-'1 <= len G by NAT_D:35;
A17: len G = width G by JORDAN8:def 1;
  x`2 = (N-min C)`2 by A2,PSCOMP_1:39
    .= N-bound C by EUCLID:52
    .= G*(i1,len G-'1)`2 by A15,JORDAN8:14;
  then i2 > len G-'1 by A3,A4,A11,A17,A12,A15,A13,A16,Th74,SPRECT_3:12;
  then i2 >= len G-'1+1 by NAT_1:13;
  then i2 >= len G by A12,XREAL_1:235,XXREAL_0:2;
  then i2 = len G by A17,A14,XXREAL_0:1;
  then f/.i in N-most L~f by A5,A9,A11,A17,A15,Th58;
  hence thesis by A13,Th3;
end;
