theorem
  X1 is SubSpace of X0 implies for B being Subset of Y st g"B c= the
  carrier of X1 holds g"B = (g|X1)"B
proof
  assume
A1: X1 is SubSpace of X0;
  let B be Subset of Y;
  assume g"B c= the carrier of X1;
  hence g"B = (g|(the carrier of X1))"B by FUNCT_2:98
    .= (g|X1)"B by A1,Def5;
end;
