 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 Th30:
  for X being set, A, B, C being non empty ordered Subset-Family of X holds
    INTERSECTION (A, UNION (B,C)) =
       UNION (INTERSECTION (A,B), INTERSECTION (A,C))
  proof
    let X be set, A, B, C be non empty ordered Subset-Family of X;
A1:  INTERSECTION (A, UNION (B,C)) c=
      UNION (INTERSECTION (A,B), INTERSECTION (A,C)) by Lm5;
    UNION (INTERSECTION (A,B), INTERSECTION (A,C)) c=
      INTERSECTION (A, UNION (B,C))
    proof
      let x be object;
      assume x in UNION (INTERSECTION (A,B), INTERSECTION (A,C)); then
      consider X, Y being set such that
A2:   X in INTERSECTION (A,B) & Y in INTERSECTION (A,C) & x = X \/ Y
      by SETFAM_1:def 4;
      consider X1,X2 being set such that
A3:   X1 in A & X2 in B & X = X1 /\ X2 by A2,SETFAM_1:def 5;
      consider Y1,Y2 being set such that
A4:   Y1 in A & Y2 in C & Y = Y1 /\ Y2 by A2,SETFAM_1:def 5;
A5:   x = ((X1 /\ X2) \/ Y1) /\ ((X1 /\ X2) \/ Y2) by A2,A3,A4,XBOOLE_1:24
       .= (Y1 \/ (X1 /\ X2)) /\ ((Y2 \/ X1) /\ (Y2 \/ X2)) by XBOOLE_1:24
       .= ((Y1 \/ X1) /\ (Y1 \/ X2)) /\ ((Y2 \/ X1) /\ (Y2 \/ X2))
           by XBOOLE_1:24
       .= ((Y1 \/ X1) /\ (Y1 \/ X2) /\ (Y2 \/ X1)) /\ (Y2 \/ X2)
           by XBOOLE_1:16;
       set A1 = min A, A2 = max A;
A6:    A1 c= Y1 & A1 c= X1 by A3,A4,Th28; then
A7:    A1 \/ A1 c= Y1 \/ X1 by XBOOLE_1:13;
       Y1 c= Y1 \/ X2 by XBOOLE_1:7; then
       A1 c= Y1 \/ X2 by A6; then
A8:    A1 /\ A1 c= (Y1 \/ X1) /\ (Y1 \/ X2) by A7,XBOOLE_1:27;
       A1 c= X1 & X1 c= Y2 \/ X1 by Th28,A3,XBOOLE_1:7; then
       A1 c= Y2 \/ X1; then
A9:    A1 c= (Y1 \/ X1) /\ (Y1 \/ X2) /\ (Y2 \/ X1) by A8,XBOOLE_1:19;
       Y1 c= A2 & X1 c= A2 by A3,A4,Th28; then
       (Y1 \/ X1) /\ ((Y1 \/ X2) /\ (Y2 \/ X1)) c= Y1 \/ X1 & Y1 \/ X1 c= A2
         by XBOOLE_1:8,17; then
       A1 c= (Y1 \/ X1) /\ (Y1 \/ X2) /\ (Y2 \/ X1) &
         (Y1 \/ X1) /\ ((Y1 \/ X2) /\ (Y2 \/ X1)) c= A2 by A9;then
       A1 c= (Y1 \/ X1) /\ (Y1 \/ X2) /\ (Y2 \/ X1) &
         (Y1 \/ X1) /\ (Y1 \/ X2) /\ (Y2 \/ X1) c= A2
       by XBOOLE_1:16; then
A10:    (Y1 \/ X1) /\ (Y1 \/ X2) /\ (Y2 \/ X1) in A by Th28;
       Y2 \/ X2 in UNION (B,C) by A3,A4,SETFAM_1:def 4;
      hence thesis by A5,A10,SETFAM_1:def 5;
    end;
    hence thesis by A1;
  end;
