 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 Th17:
  A _\/_ B = Inter (A``1 \/ B``1, A``2 \/ B``2)
  proof
    thus 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
A1:   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 4;
A3:   A``1 c= X & X c= A``2 by A2,Th14;
A4:   B``1 c= Y & Y c= B``2 by A2,Th14; then
A5:   xx c= A``2 \/ B``2 by A3,A2,XBOOLE_1:13;
      A``1 \/ B``1 c= xx by A3,A2,A4,XBOOLE_1:13;
      hence thesis by A1,A5;
    end;
    thus 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
A6:   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 A7: x in A``1;
        assume A8: not x in (Z \/ A``1) /\ A``2;
        per cases by A8,XBOOLE_0:def 4;
        suppose not x in Z \/ A``1;
          hence thesis by A7,XBOOLE_0:def 3;
        end;
        suppose A9: not x in A``2;
          A``1 c= A``2 & x in A``1 by A7,Th16;
          hence thesis by A9;
        end;
      end; then
      A``1 c= (Z \/ A``1) /\ A``2 & (Z \/ A``1) /\ A``2 c= A``2
      by XBOOLE_1:17; then
A10:   (Z \/ A``1) /\ A``2 in A by Th14;
      B``1 c= (Z \/ B``1) /\ B``2
      proof
        let x be object;
        assume A11: x in B``1; then
A12:     x in Z \/ B``1 by XBOOLE_0:def 3;
        B``1 c= B``2 by Th16;
        hence thesis by A11,A12,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
A13:   (Z \/ B``1) /\ B``2 in B by Th14;
      ((Z \/ A``1) /\ A``2) \/ ((Z \/ B``1) /\ B``2) = (((Z \/ A``1) /\ A``2)
      \/ (Z \/ B``1)) /\ (((Z \/ A``1) /\ A``2) \/ B``2) by XBOOLE_1:24
      .= (((Z \/ A``1) \/ (Z \/ B``1)) /\ (A``2 \/ (Z \/ B``1)))
      /\ (((Z \/ A``1) /\ A``2) \/ B``2) by XBOOLE_1:24
      .= ((Z \/ (A``1 \/  B``1)) /\ (A``2 \/ (Z \/ B``1)))
      /\ (((Z \/ A``1) /\ A``2) \/ B``2) by XBOOLE_1:5
      .= (Z /\ (A``2 \/ (Z \/ B``1)))
      /\ (((Z \/ A``1) /\ A``2) \/ B``2) by A6,XBOOLE_1:12
      .= ((Z /\ A``2) \/ (Z /\ (Z \/ B``1)))
      /\ (((Z \/ A``1) /\ A``2) \/ B``2) by XBOOLE_1:23
      .= ((Z /\ A``2) \/ Z)
      /\ (((Z \/ A``1) /\ A``2) \/ B``2) by XBOOLE_1:7,28
      .= Z  /\ (B``2 \/ ((Z \/ A``1) /\ A``2)) by XBOOLE_1:12,17
      .= Z /\ (((Z \/ A``1) \/ B``2) /\ (A``2 \/ B``2)) by XBOOLE_1:24
      .= Z /\ ((Z \/ (A``1 \/ B``2)) /\ (A``2 \/ B``2)) by XBOOLE_1:4
      .= Z /\ ((Z /\ (A``2 \/ B``2)) \/ ((A``1 \/ B``2) /\ (A``2 \/ B``2)))
        by XBOOLE_1:23
      .= Z /\ (Z \/ ((A``1 \/ B``2) /\ (A``2 \/ B``2))) by A6,XBOOLE_1:28
      .= Z /\ ((Z \/ (A``1 \/ B``2)) /\ (Z \/ (A``2 \/ B``2))) by XBOOLE_1:24
      .= (Z /\ (Z \/ (A``1 \/ B``2))) /\ (Z \/ (A``2 \/ B``2)) by XBOOLE_1:16
      .= Z /\ (Z \/ (A``2 \/ B``2)) by XBOOLE_1:7,28
      .= Z by XBOOLE_1:7,28; then
      consider X,Y being Subset of U such that
A14:   X in A & Y in B & Z = X \/ Y by A10,A13;
      thus thesis by A14,A6,SETFAM_1:def 4;
    end;
  end;
