theorem Th15:
  Union Partial_Union A1 = Union A1
proof
  thus Union Partial_Union A1 c= Union A1
  proof
    let x be object;
    assume x in Union Partial_Union A1;
    then consider n being Nat such that
A1: x in (Partial_Union A1).n by PROB_1:12;
    consider k such that
    k <= n and
A2: x in A1.k by A1,Th13;
    thus thesis by A2,PROB_1:12;
  end;
  let x be object;
  assume x in Union A1;
  then consider n being Nat such that
A3: x in A1.n by PROB_1:12;
  x in (Partial_Union A1).n by A3,Th13;
  hence thesis by PROB_1:12;
end;
