 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 Th41:
  for X,Y being Subset of U st A = Inter (X,Y) holds
    A _\_ C = Inter (X \ C``2, Y \ C``1)
  proof
    let X,Y be Subset of U;
    assume A1: A = Inter (X,Y);
    A = Inter (A``1,A``2) by Th15; then
    X = A``1 & Y = A``2 by A1,Th6;
    hence thesis by Th40;
  end;
