reserve E, x, y, X for set;
reserve A, B, C for Subset of E^omega;
reserve a, b for Element of E^omega;
reserve i, k, l, kl, m, n, mn for Nat;

theorem
  <%x%> in A ^^ B iff <%>E in A & <%x%> in B or <%x%> in A & <%>E in B
proof
  thus <%x%> in A ^^ B implies <%>E in A & <%x%> in B or <%x%> in A & <%>E in B
  proof
    assume <%x%> in A ^^ B;
    then ex a, b st a in A & b in B & <%x%> = a ^ b by FLANG_1:def 1;
    hence thesis by FLANG_1:4;
  end;
A1: {} ^ <%x%> = <%x%> & <%x%> ^ {} = <%x%>;
  assume <%>E in A & <%x%> in B or <%x%> in A & <%>E in B;
  hence thesis by A1,FLANG_1:def 1;
end;
