 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 Th46:
  for X,Y being Subset of U st A = Inter (X,Y) & X c= Y holds
    A^ = Inter (Y`,X`)
  proof
    let X,Y be Subset of U;
    assume A1: A = Inter (X,Y) & X c= Y;
    then Inter (X,Y) = Inter (A``1,A``2) by Th15; then
    X = A``1 & Y = A``2 by Th6,A1;
    hence thesis by Th41;
  end;
