 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 Th14:
  for X being set holds X in A iff A``1 c= X & X c= A``2
  proof
    let X be set;
A1: X in A implies A``1 c= X & X c= A``2
    proof
      assume A2: X in A;
A3:   A``1 c= X
      proof
        consider B being Subset of U such that
A4:     A = Inter (A``1, B) by Def5;
        thus thesis by Th1,A2,A4;
      end;
      X c= A``2
      proof
        consider B being Subset of U such that
A5:     A = Inter (B, A``2) by Def6;
        thus thesis by Th1,A2,A5;
      end;
      hence thesis by A3;
    end;
    A``1 c= X & X c= A``2 implies X in A
    proof
      assume A``1 c= X & X c= A``2; then
A6:   X in Inter (A``1,A``2) by Th1;
      consider B being Subset of U such that A7: A = Inter (A``1, B) by Def5;
      consider C being Subset of U such that A8: A = Inter (C, A``2) by Def6;
      thus thesis by A7,A6,Th6,A8;
    end;
    hence thesis by A1;
  end;
