 reserve U for set,
         X, Y for Subset of U;
 reserve U for non empty set,
         A, B, C for non empty IntervalSet of U;

theorem Th25:
  for A,B being Subset of U st A c= B holds
    Inter (A,B) is non empty ordered Subset-Family of U
  proof
    let A,B be Subset of U;
    assume A1: A c= B;
    consider C,D being set such that
A2: C=A & D=B;
    C in Inter(A,B) & D in Inter(A,B) & for Y being set holds
    Y in Inter(A,B) iff C c= Y & Y c= D by A2,Th1,A1;
    hence thesis by Def8;
  end;
