reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve f for Function of X,Y;
reserve X,Y,Z for non empty TopSpace;
reserve f for Function of X,Y,
  g for Function of Y,Z;
reserve X, Y for non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve f for Function of X,Y;

theorem Th54:
  the TopStruct of X0 = the TopStruct of X implies f = f|X0
proof
  assume the TopStruct of X0 = the TopStruct of X;
  hence f|X0 = f*(id the carrier of X) by RELAT_1:65
    .= f by FUNCT_2:17;
end;
