 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 Th37:
  for U being non empty set, A, B being Subset-Family of U holds
    DIFFERENCE (A,B) is Subset-Family of U
  proof
    let U be non empty set,
        A, B be Subset-Family of U;
    DIFFERENCE (A,B) c= bool U
    proof
      let x be object; assume x in DIFFERENCE (A,B); then
      consider Y,Z being set such that A1: Y in A & Z in B & x = Y \ Z
        by SETFAM_1:def 6;
      Y \ Z c= U by A1,XBOOLE_1:109;
      hence thesis by A1;
    end;
    hence thesis;
  end;
