 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 Th34:
   for X being set, A, B being non empty ordered Subset-Family of X holds
     UNION (INTERSECTION(A,B),B) = B
   proof
     let X be set;
     let A,B be non empty ordered Subset-Family of X;
A1:  UNION (INTERSECTION(A,B),B) c= B
     proof
       let x be object;
     reconsider xx=x as set by TARSKI:1;
       set B1 = min B, B2 = max B;
       assume x in UNION (INTERSECTION(A,B),B); then
       consider Y,Z being set such that
A2:    Y in INTERSECTION(A,B) & Z in B & x = Y \/ Z by SETFAM_1:def 4;
       consider Y1,Y2 being set such that
A3:    Y1 in A & Y2 in B & Y = Y1 /\ Y2 by A2,SETFAM_1:def 5;
       B1 c= Y2 & Y2 c= B2 & Y1 /\ Y2 c= Y2 by A3,Th28,XBOOLE_1:17; then
A4:    Y1 /\ Y2 c= B2;
       Z c= B2 by A2,Th28; then
A5:    xx c= B2 by A2,A3,A4,XBOOLE_1:8;
       B1 c= Z & Z c= xx by A2,Th28,XBOOLE_1:7; then
       B1 c= xx & xx c= B2 by A5;
       hence thesis by Th28;
     end;
     B c= UNION (INTERSECTION(A,B),B)
     proof
       let x be object;
     reconsider xx=x as set by TARSKI:1;
       assume A6: x in B;
       set b = the Element of A;
A7:    x = xx \/ (xx /\ b) by XBOOLE_1:22;
       b /\ xx in INTERSECTION (A,B) by A6,SETFAM_1:def 5;
       hence thesis by A7,A6,SETFAM_1:def 4;
     end;
     hence thesis by A1;
   end;
