reserve x,y,z,X,Y for set;
reserve X,Y for non empty set,
  f for Function of X,Y;
reserve X, Y for non empty set,
  F for (BinOp of Y),
  B for (Element of Fin X),
  f for Function of X,Y;
reserve A for set,
  x,y,z for Element of Fin A;
reserve X,Y for non empty set,
  A for set,
  f for (Function of X, Fin A),
  i,j,k for (Element of X);

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;
