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

theorem Th8:
  Union (A1 (/\) A2) c= Union A1 /\ Union A2
proof
  let x be object;
  assume x in Union (A1 (/\) A2);
  then consider n such that
A1: x in (A1 (/\) A2).n by PROB_1:12;
A2: x in A1.n /\ A2.n by A1,Def1;
  then x in A2.n by XBOOLE_0:def 4;
  then
A3: x in Union A2 by PROB_1:12;
  x in A1.n by A2,XBOOLE_0:def 4;
  then x in Union A1 by PROB_1:12;
  hence thesis by A3,XBOOLE_0:def 4;
end;
