
theorem
  for f,g being Function st f tolerates g holds rng (f+*g) = (rng f)\/( rng g)
proof
  let f,g be Function such that
A1: f tolerates g;
  thus rng (f+*g) c= (rng f)\/(rng g) by FUNCT_4:17;
  let x be object;
  assume
A2: x in (rng f)\/(rng g);
A3: rng(f+*g) = f.:(dom f\dom g)\/rng g by Th12;
A4: rng g c= rng(f+*g) by FUNCT_4:18;
  per cases;
  suppose
    x in rng g;
    hence thesis by A4;
  end;
  suppose
A5: not x in rng g;
    then x in rng f by A2,XBOOLE_0:def 3;
    then consider a being object such that
A6: a in dom f and
A7: x = f.a by FUNCT_1:def 3;
    now
      assume
A8:   a in dom g;
      x = (f+*g).a by A1,A6,A7,FUNCT_4:15
        .= g.a by A8,FUNCT_4:13;
      hence contradiction by A5,A8,FUNCT_1:def 3;
    end;
    then a in dom f\dom g by A6,XBOOLE_0:def 5;
    then x in f.:(dom f\dom g) by A7,FUNCT_1:def 6;
    hence thesis by A3,XBOOLE_0:def 3;
  end;
end;
