
theorem Th2:
  for f being Function, Y1,Y2 being set st Y2 c= Y1 holds (Y1|`f)"Y2 = f"Y2
proof
  let f be Function;
  let Y1,Y2 be set;
  assume
A1: Y2 c= Y1;
  for x being object holds x in (Y1|`f)"Y2 iff x in f"Y2
  proof
    let x be object;
    hereby
      assume x in (Y1|`f)"Y2;
      then consider y be object such that
A2:   [x,y] in Y1|`f & y in Y2 by RELAT_1:def 14;
      [x,y] in f by A2,RELAT_1:def 12;
      hence x in f"Y2 by A2,RELAT_1:def 14;
    end;
    assume x in f"Y2;
    then consider y be object such that
A3: [x,y] in f & y in Y2 by RELAT_1:def 14;
    [x,y] in Y1|`f by A3,A1,RELAT_1:def 12;
    hence x in (Y1|`f)"Y2 by A3,RELAT_1:def 14;
  end;
  hence (Y1|`f)"Y2 = f"Y2 by TARSKI:2;
end;
