 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 Th18:
  A _/\_ B = Inter (A``1 /\ (B``1), A``2 /\ B``2)
    proof
A1: A _/\_ B c= Inter (A``1 /\ (B``1), A``2 /\ B``2)
    proof
      let x be object;
     reconsider xx=x as set by TARSKI:1;
      assume x in A _/\_ B; then
      consider X,Y being set such that
A2:   X in A & Y in B & x = X /\ Y by SETFAM_1:def 5;
      A``1 c= X & B``1 c= Y & X c= A``2 & Y c= B``2 by A2,Th14; then
      A``1 /\ (B``1) c= xx & xx c= A``2 /\ B``2 by A2,XBOOLE_1:27;
      hence thesis by Th1;
    end;
    Inter (A``1 /\ (B``1), A``2 /\ B``2) c= A _/\_ B
    proof
      let x be object;
      assume x in Inter (A``1 /\ (B``1), A``2 /\ B``2); then
      consider Z being Element of bool U such that
A3:   x = Z & A``1 /\ (B``1) c= Z & Z c= A``2 /\ B``2;
      A``1 c= (Z \/ A``1) /\ A``2
      proof
        let x be object;
        assume A4: x in A``1;
        assume A5: not x in (Z \/ A``1) /\ A``2;
        per cases by A5,XBOOLE_0:def 4;
        suppose not x in Z \/ A``1;
          hence thesis by A4,XBOOLE_0:def 3;
        end;
        suppose A6: not x in A``2;
          A``1 c= A``2 & x in A``1 by A4,Th16;
          hence thesis by A6;
        end;
      end; then
      A``1 c= (Z \/ A``1) /\ A``2 & (Z \/ A``1) /\ A``2 c= A``2
      by XBOOLE_1:17; then
A7:   (Z \/ A``1) /\ A``2 in A by Th14;
      B``1 c= (Z \/ B``1) /\ B``2
      proof
        let x be object;
        assume A8: x in B``1; then
A9:     x in Z \/ B``1 by XBOOLE_0:def 3;
        B``1 c= B``2 by Th16;
        hence thesis by A8,A9,XBOOLE_0:def 4;
      end; then
      B``1 c= (Z \/ B``1) /\ B``2 & (Z \/ B``1) /\ B``2 c= B``2
      by XBOOLE_1:17; then
A10:   (Z \/ B``1) /\ B``2 in B by Th14;
      ((Z \/ A``1) /\ A``2) /\ ((Z \/ B``1) /\ B``2)
      = (A``2 /\ (Z \/ A``1)) /\ (Z \/ B``1) /\ B``2 by XBOOLE_1:16
      .=A``2 /\ ((Z \/ A``1) /\ (Z \/ B``1)) /\ B``2 by XBOOLE_1:16
      .= A``2 /\ (Z \/ (A``1 /\ (B``1))) /\ B``2 by XBOOLE_1:24
      .= A``2 /\ Z /\ B``2 by A3,XBOOLE_1:12
      .= Z /\ (A``2 /\ B``2) by XBOOLE_1:16
      .= Z by A3,XBOOLE_1:28;
      hence thesis by A3,A7,A10,SETFAM_1:def 5;
    end;
    hence thesis by A1;
  end;
