 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 Th15:
  A = Inter (A``1,A``2)
    proof
A1: Inter (A``1,A``2) c= A
    proof
      let x be object;
     reconsider xx=x as set by TARSKI:1;
      assume x in Inter (A``1,A``2); then
      A``1 c= xx & xx c= A``2 by Th1;
      hence thesis by Th14;
    end;
    A c= Inter (A``1,A``2)
    proof
      let x be object;
     reconsider xx=x as set by TARSKI:1;
      assume x in A; then
      A``1 c= xx & xx c=  A``2 by Th14;
      hence thesis by Th1;
    end;
    hence thesis by A1;
  end;
