 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 Th27:
  for A,B being Subset of U, F being ordered non empty Subset-Family of U st
      F = Inter (A,B) holds
      min F = A & max F = B
  proof
    let A,B be Subset of U;
    let F be ordered non empty Subset-Family of U;
    assume A1: F = Inter (A,B);
    then A is Element of F & for Y being set st Y in F holds A c= Y
      by Th2,Th1; then
A2: A = min F by Lm2;
    B is Element of F & for Y being set st Y in F holds Y c= B
      by Th2,A1,Th1;
    hence thesis by A2,Lm3;
  end;
