
theorem Th5:
  for F being finite set, A being FinSequence of bool F,
      i being Element of NAT st i in dom A holds union (A, {i}) = A.i
proof
  let F be finite set, A be FinSequence of bool F, i be Element of NAT such
  that
A1: i in dom A;
  thus union (A, {i}) c= A.i
  proof
    let x be object;
    assume x in union (A, {i});
    then ex j be set st j in {i} & j in dom A & x in A.j by Def1;
    hence thesis by TARSKI:def 1;
  end;
  thus A.i c= union (A, {i})
  proof
    let x be object;
A2: i in {i} by TARSKI:def 1;
    assume x in A.i;
    hence thesis by A1,A2,Def1;
  end;
end;
