reserve X,Y for set, x,y,z for object, i,j,n for natural number;

theorem Th1:
  for f,g being Function holds g is f-tolerating iff
  for x st x in dom f & x in dom g holds f.x = g.x
  proof
  let f,g be Function;
  thus g is f-tolerating implies
  for x st x in dom f & x in dom g holds f.x = g.x
  proof
    assume
A1: for x being object st x in dom f /\ dom g holds f.x = g.x;
    let x; assume x in dom f & x in dom g;
    then x in dom f /\ dom g by XBOOLE_0:def 4;
    hence thesis by A1;
  end;
  assume
A2: for x st x in dom f & x in dom g holds f.x = g.x;
    let x; assume x in dom f /\ dom g;
    then x in dom f & x in dom g by XBOOLE_0:def 4;
    hence thesis by A2;
end;
