 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 Th39:
  for U being non empty set, A1, A2, B1, B2 being Subset of U st
        A1 c= A2 & B1 c= B2 holds
      DIFFERENCE (Inter (A1,A2), Inter (B1,B2)) =
      { C where C is Subset of U : A1 \ B2 c= C & C c= A2 \ B1 }
  proof
    let U be non empty set;
    let A1,A2,B1,B2 be Subset of U;
    assume A1 c= A2 & B1 c= B2; then
    reconsider A12 = Inter (A1,A2), B12 = Inter (B1,B2)
      as non empty ordered Subset-Family of U by Th25;
A1: min A12 = A1 & max A12 = A2 by Th27;
    min B12 = B1 & max B12 = B2 by Th27;
    hence thesis by Th38,A1;
  end;
