
theorem Th3:
  for I being non empty set, a,b,c being set, i being Element of I
  holds c in (i-singleton a).b iff b = i & c = a
  proof
    let I be non empty set;
    let a,b,c be set;
    let i be Element of I;
A1: (i-singleton a).i = {a} & for b being set st b in I & b <> i holds
    (i-singleton a).b = {} by AOFA_A00:6;
    dom (i-singleton a) = I by PARTFUN1:def 2;
    then
A2: for b being set st b nin I holds (i-singleton a).b = {} by FUNCT_1:def 2;
    hereby
      assume A3: c in (i-singleton a).b;
      thus b = i by A3,A1,A2;
      hence c = a by A1,A3,TARSKI:def 1;
    end;
    thus thesis by A1,TARSKI:def 1;
  end;
