reserve n,m,k for Nat,
  x,X for set,
  A for Subset of X,
  A1,A2 for SetSequence of X;

theorem Th38:
  Union (A (/\) A1) = A /\ Union A1
proof
  thus Union (A (/\) A1) c= A /\ Union A1
  proof
    let x be object;
    assume x in Union (A (/\) A1);
    then consider n such that
A1: x in (A (/\) A1).n by PROB_1:12;
A2: x in A /\ A1.n by A1,Def5;
    then x in A1.n by XBOOLE_0:def 4;
    then
A3: x in Union A1 by PROB_1:12;
    x in A by A2,XBOOLE_0:def 4;
    hence thesis by A3,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A4: x in A /\ Union A1;
  then x in Union A1 by XBOOLE_0:def 4;
  then consider n such that
A5: x in A1.n by PROB_1:12;
  x in A by A4,XBOOLE_0:def 4;
  then x in A /\ A1.n by A5,XBOOLE_0:def 4;
  then x in (A (/\) A1).n by Def5;
  hence thesis by PROB_1:12;
end;
