 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 Th42:
  for X,Y,W,Z being Subset of U st A = Inter (X,Y) & C = Inter (W,Z) holds
    A _\_ C = Inter (X \ Z, Y \ W)
  proof
    let X,Y,W,Z be Subset of U;
    assume A1: A = Inter (X,Y) & C = Inter (W,Z);
    A= Inter (A``1,A``2) & C = Inter (C``1,C``2) by Th15; then
    A``1 = X & A``2 = Y & C``1 =  W & C``2 = Z by A1,Th6;
    hence thesis by Th40;
  end;
