 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 Th28:
  for X,Y being set, A being non empty ordered Subset-Family of X holds
    Y in A iff min A c= Y & Y c= max A
  proof
    let X,Y be set;
    let A be non empty ordered Subset-Family of X;
    min A c= Y & Y c= max A implies Y in A
    proof
      assume A1: min A c= Y & Y c= max A;
      consider C,D being set such that
A2:   C in A & D in A & for X being set holds X in A iff C c= X & X c= D
        by Def8;
A3:   min A c= C & D c= max A by Lm2,Lm3,A2;
      C c= min A & max A c= D by A2; then
      min A = C & max A = D by A3;
      hence thesis by A2,A1;
    end;
    hence thesis by Lm2,Lm3;
  end;
