theorem
  for B being Element of Fin X holds FinUnion(B, singleton X) = B
proof
  let B be Element of Fin X;
  now
    let x be object;
    thus x in FinUnion(B, singleton X) implies x in B
    proof
      assume x in FinUnion(B, singleton X);
      then ex i being Element of X st i in B & x in singleton X.i by Th54;
      hence thesis by Th52;
    end;
    assume
A1: x in B;
    then reconsider x9 = x as Element of X by Th6;
    x in singleton X.x9 by Th52;
    hence x in FinUnion(B, singleton X) by A1,Th54;
  end;
  hence thesis by TARSKI:2;
end;
