reserve Y,Z for non empty set;
reserve PA,PB for a_partition of Y;
reserve A,B for Subset of Y;
reserve i,j,k for Nat;
reserve x,y,z,x1,x2,y1,z0,X,V,a,b,d,t,SFX,SFY for set;

theorem
  for PA,PB,PC being a_partition of Y holds
  (PA '/\' PB) '/\' PC = PA '/\' (PB '/\' PC)
proof
  let PA,PB,PC be a_partition of Y;
  consider PD,PE being a_partition of Y such that
A1: PD = PA '/\' PB and
A2: PE = PB '/\' PC;
 for u being set st u in PD '/\' PC
  ex v being set st v in PA '/\' PE & u c= v
  proof
    let u be set;
    assume
A3: u in PD '/\' PC;
    then consider u1, u2 being set such that
A4: u1 in PD and
A5: u2 in PC and
A6: u = u1 /\ u2 by SETFAM_1:def 5;
    consider u3, u4 being set such that
A7: u3 in PA and
A8: u4 in PB and
A9: u1 = u3 /\ u4 by A1,A4,SETFAM_1:def 5;
    consider v, v1,v2 being set such that
A10: v1 = u4 /\ u2 and
A11: v2 = u3 and
A12: v = u3 /\ u4 /\ u2;
A13: v = v2 /\ v1 by A10,A11,A12,XBOOLE_1:16;
A14: v1 in INTERSECTION(PB,PC) by A5,A8,A10,SETFAM_1:def 5;
A15: not u in {{}} by A3,XBOOLE_0:def 5;
 u = u3 /\ v1 by A6,A9,A10,XBOOLE_1:16;
then  v1 <> {} by A15,TARSKI:def 1;
then  not v1 in {{}} by TARSKI:def 1;
then  v1 in INTERSECTION(PB,PC) \ {{}} by A14,XBOOLE_0:def 5;
then A16: v in INTERSECTION(PA,PE) by A2,A7,A11,A13,SETFAM_1:def 5;
    take v;
    thus thesis by A6,A9,A12,A15,A16,XBOOLE_0:def 5;
  end;
then A17: PD '/\' PC '<' PA '/\' PE by SETFAM_1:def 2;
 for u being set st u in PA '/\' PE
  ex v being set st v in PD '/\' PC & u c= v
  proof
    let u be set;
    assume
A18: u in PA '/\' PE;
    then consider u1,u2 being set such that
A19: u1 in PA and
A20: u2 in PE and
A21: u = u1 /\ u2 by SETFAM_1:def 5;
    consider u3,u4 being set such that
A22: u3 in PB and
A23: u4 in PC and
A24: u2 = u3 /\ u4 by A2,A20,SETFAM_1:def 5;
A25: u = u1 /\ u3 /\ u4 by A21,A24,XBOOLE_1:16;
    consider v, v1,v2 being set such that
A26: v1 = u1 /\ u3 and v2 = u4 and
A27: v = u1 /\ u3 /\ u4;
A28: v1 in INTERSECTION(PA,PB) by A19,A22,A26,SETFAM_1:def 5;
A29: not u in {{}} by A18,XBOOLE_0:def 5;
then  v1 <> {} by A25,A26,TARSKI:def 1;
then  not v1 in {{}} by TARSKI:def 1;
then  v1 in INTERSECTION(PA,PB) \ {{}} by A28,XBOOLE_0:def 5;
then A30: v in INTERSECTION(PD,PC) by A1,A23,A26,A27,SETFAM_1:def 5;
    take v;
    thus thesis by A25,A27,A29,A30,XBOOLE_0:def 5;
  end;
then  PA '/\' PE '<' PD '/\' PC by SETFAM_1:def 2;
  hence thesis by A1,A2,A17,Th4;
end;
