 reserve U for set,
         X, Y for Subset of U;

theorem Th11:
  for U being non empty set, A being set holds
    A is non empty IntervalSet of U iff
      ex A1, A2 being Subset of U st A1 c= A2 & A = Inter (A1, A2)
  proof
    let U be non empty set,
        A be set;
    hereby assume
A1:   A is non empty IntervalSet of U; then
      consider A1, A2 be Subset of U such that
A2:   A = Inter (A1, A2) by Def2;
      take A1, A2;
      thus A1 c= A2 by A1,A2,Th3;
      thus A = Inter (A1, A2) by A2;
    end;
    given A1, A2 being Subset of U such that
A3: A1 c= A2 & A = Inter (A1, A2);
    A1 in Inter (A1,A2) by A3;
    hence thesis by A3;
  end;
