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;
reserve f for Function of X,Y,
  X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace,
  X0, X1 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;

theorem Th70:
  for g being Function of X0,Y st g = f|X0 holds X1 is SubSpace of
  X0 implies g|X1 = f|X1
proof
  let g be Function of X0,Y;
  assume
A1: g = f|X0;
  assume
A2: X1 is SubSpace of X0;
  then
A3: the carrier of X1 c= the carrier of X0 by TSEP_1:4;
  thus g|X1 = g|(the carrier of X1) by A2,Def5
    .= f|X1 by A1,A3,FUNCT_1:51;
end;
