reserve m,k,j,j1,i,i1,i2,n for Nat,
  r,s for Real,
  C for compact non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p for Point of TOP-REAL 2;

theorem Th33:
  for f being non empty FinSequence of TOP-REAL 2, G being
  Go-board st proj1.:rng f = proj1.:Values G & proj2.:rng f = proj2.:Values G
  holds G = GoB f
proof
  let f be non empty FinSequence of TOP-REAL 2, G being Go-board;
  X_axis f = proj1*f by Th10;
  then rng X_axis f = proj1.:rng f by RELAT_1:127;
  then
A1: Incr X_axis f = SgmX(RealOrd, proj1.:rng f) by Th8;
  Y_axis f = proj2*f by Th11;
  then rng Y_axis f = proj2.:rng f by RELAT_1:127;
  then
A2: Incr Y_axis f = SgmX(RealOrd, proj2.:rng f) by Th8;
  assume proj1.:rng f = proj1.:Values G & proj2.:rng f = proj2.:Values G;
  hence G = GoB(Incr X_axis f, Incr Y_axis f) by A1,A2,Th32
    .= GoB f by GOBOARD2:def 2;
end;
