
theorem Th6:
  for F being finite set, A being FinSequence of bool F,
      i,j being Element of NAT st i in dom A & j in dom A holds
    union (A, {i,j}) = A.i \/ A.j
proof
  let F be finite set, A be FinSequence of bool F, i,j be Element of NAT such
  that
A1: i in dom A and
A2: j in dom A;
  thus union (A, { i,j }) c= A.i \/ A.j
  proof
    let x be object;
    assume x in union (A, { i,j });
    then consider k be set such that
A3: k in {i,j} & k in dom A & x in A.k by Def1;
    per cases by A3,TARSKI:def 2;
    suppose
      k = i & k in dom A & x in A.k;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
      k = j & k in dom A & x in A.k;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
  thus A.i \/ A.j c= union (A, { i,j })
  proof
    let x be object;
    assume
A4: x in A.i \/ A.j;
    per cases by A4,XBOOLE_0:def 3;
    suppose
A5:   x in A.i;
      i in {i,j} by TARSKI:def 2;
      hence thesis by A1,A5,Def1;
    end;
    suppose
A6:   x in A.j;
      j in {i,j} by TARSKI:def 2;
      hence thesis by A2,A6,Def1;
    end;
  end;
end;
