
theorem Th9:
  for J1, J2 being set, F being finite set, i being Element of NAT,
      A being FinSequence of bool F st
      i in dom A holds union (A,{i} \/ J1 \/ J2) = A.i \/ union (A,J1 \/ J2)
proof
  let J1, J2 be set;
  let F be finite set;
  let i be Element of NAT;
  let A be FinSequence of bool F;
  assume i in dom A;
  then
A1: union (A, {i}) = A.i by Th5;
  thus union (A,{i}\/J1\/J2) c= A.i \/ union (A,J1\/J2)
  proof
    let x be object;
    assume x in union (A,{i}\/J1\/J2);
    then consider j be set such that
A2: j in {i}\/J1\/J2 and
A3: j in dom A and
A4: x in A.j by Def1;
    per cases;
    suppose
      i = j;
      hence thesis by A4,XBOOLE_0:def 3;
    end;
    suppose
A5:   i <> j;
      j in {i}\/(J1\/J2) by A2,XBOOLE_1:4;
      then j in {i} or j in J1\/J2 by XBOOLE_0:def 3;
      then x in union (A, J1\/J2) by A3,A4,A5,Def1,TARSKI:def 1;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
  thus A.i \/ union (A,J1\/J2) c= union (A,{i}\/J1\/J2)
  proof
    let x be object;
    assume
A6: x in A.i \/ union (A,J1\/J2);
    per cases by A1,A6,XBOOLE_0:def 3;
    suppose
      x in union (A, {i});
      then consider j be set such that
A7:   j in {i} and
A8:   j in dom A & x in A.j by Def1;
      j in {i}\/(J1\/J2) by A7,XBOOLE_0:def 3;
      then j in {i}\/J1\/J2 by XBOOLE_1:4;
      hence thesis by A8,Def1;
    end;
    suppose
      x in union (A,J1\/J2);
      then consider j be set such that
A9:   j in J1 \/ J2 and
A10:  j in dom A & x in A.j by Def1;
      j in {i} \/ (J1\/J2) by A9,XBOOLE_0:def 3;
      then j in {i}\/J1\/J2 by XBOOLE_1:4;
      hence thesis by A10,Def1;
    end;
  end;
end;
