reserve T for TopSpace;

theorem Th25:
  for F being Subset-Family of T holds union(Int F) c= union F
proof
  let F be Subset-Family of T;
  now
    let x be object;
    assume x in union(Int F);
    then consider A being set such that
A1: x in A and
A2: A in Int F by TARSKI:def 4;
    reconsider A as Subset of T by A2;
    consider B being Subset of T such that
A3: A = Int B and
A4: B in F by A2,Def1;
    ex B being set st x in B & B in F
    proof
      take B;
      Int B c= B by TOPS_1:16;
      hence thesis by A1,A3,A4;
    end;
    hence x in union F by TARSKI:def 4;
  end;
  hence thesis;
end;
