 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 Th33:
   for X being set, A, B being non empty ordered Subset-Family of X holds
     INTERSECTION (A, (UNION (A,B))) = A
   proof
     let X be set;
     let A,B be non empty ordered Subset-Family of X;
     set A1 = min A, A2 = max A;
A1:  INTERSECTION (A, (UNION (A,B))) c= A
     proof
       let x be object;
       assume x in INTERSECTION (A, (UNION (A,B))); then
       consider Y,Z being set such that
A2:    Y in A & Z in UNION (A,B) & x = Y /\ Z by SETFAM_1:def 5;
       consider Z1,Z2 being set such that
A3:    Z1 in A & Z2 in B & Z = Z1 \/ Z2 by A2,SETFAM_1:def 4;
A4:    x = (Y /\ Z1) \/ (Y /\ Z2) by A2,A3,XBOOLE_1:23;
A5:    (A1 c= Y & Y c= A2) & (A1 c= Z1 & Z1 c= A2) by Th28,A2,A3; then
       A1 c= Y /\ Z1 & Y /\ Z1 c= A2 /\ A2 by XBOOLE_1:19,27; then
A6:    A1 c= Y /\ Z1 & Y /\ Z1 c= A2 & Y /\ Z1 c= (Y /\ Z1) \/ (Y /\ Z2)
         by XBOOLE_1:7; then
A7:    A1 c= (Y /\ Z1) \/ (Y /\ Z2);
       Y /\ Z2 c= Y by XBOOLE_1:17; then
       Y /\ Z2 c= A2 by A5; then
       (Y /\ Z1) \/ (Y /\ Z2) c= A2 by A6,XBOOLE_1:8;
       hence thesis by Th28,A4,A7;
     end;
     A c= INTERSECTION (A, (UNION (A,B)))
     proof
       let x be object;
     reconsider xx=x as set by TARSKI:1;
       assume A8: x in A;
       set b = the Element of B;
A9:    x = xx /\ (xx \/ b) by XBOOLE_1:21;
       xx \/ b in UNION (A,B) by A8,SETFAM_1:def 4;
       hence thesis by A8,A9,SETFAM_1:def 5;
     end;
     hence thesis by A1;
   end;
