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 Th19:
  1 <= i1 & i1 <= len G & 1 <= i2 & i2 <= len G & 1 <= j1 & j1 <=
  j2 & j2 <= width G implies G*(i1,j1)`2 <= G*(i2,j2)`2
proof
  assume that
A1: 1 <= i1 & i1 <= len G and
A2: 1 <= i2 & i2 <= len G and
A3: 1 <= j1 & j1 <= j2 and
A4: j2 <= width G;
A5: 1 <= j2 by A3,XXREAL_0:2;
  then G*(i1,j2)`2 = G*(1,j2)`2 by A1,A4,GOBOARD5:1
    .= G*(i2,j2)`2 by A2,A4,A5,GOBOARD5:1;
  hence thesis by A1,A3,A4,SPRECT_3:12;
end;
