reserve i, x, I for set,
  A, B, M for ManySortedSet of I,
  f, f1 for Function;
reserve SF, SG for SubsetFamily of M;
reserve E, T for Element of Bool M;

theorem Th22:
  SF = {A,B} implies union |:SF:| = A (\/) B
proof
  assume
A1: SF = {A,B};
  now
    let i be object;
    assume
A2: i in I;
    hence (union |:SF:|).i = union (|:SF:|.i) by MBOOLEAN:def 2
      .= union { A.i, B.i } by A1,A2,Th20
      .= A.i \/ B.i by ZFMISC_1:75
      .= (A (\/) B).i by A2,PBOOLE:def 4;
  end;
  hence thesis;
end;
