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 Th24:
  for PA,PB being a_partition of Y holds ERl(PA '/\' PB) = ERl(PA) /\ ERl(PB)
proof
  let PA,PB be a_partition of Y;
A1: PA '>' PA '/\' PB by Th15;
A2: PB '>' PA '/\' PB by Th15;
 for x1,x2 being object holds
  [x1,x2] in ERl(PA '/\' PB) iff [x1,x2] in (ERl(PA) /\ ERl(PB))
  proof
    let x1,x2 be object;
    hereby
      assume [x1,x2] in ERl(PA '/\' PB);
      then consider C being Subset of Y such that
A3:   C in (PA '/\' PB) and
A4:   x1 in C & x2 in C by Def6;
A5:   ex A being set st A in PA & C c= A by A1,A3,SETFAM_1:def 2;
A6:   ex B being set st B in PB & C c= B by A2,A3,SETFAM_1:def 2;
A7:   [x1,x2] in ERl(PA) by A4,A5,Def6;
  [x1,x2] in ERl(PB) by A4,A6,Def6;
      hence [x1,x2] in (ERl(PA) /\ ERl(PB)) by A7,XBOOLE_0:def 4;
    end;
    assume
A8: [x1,x2] in (ERl(PA) /\ ERl(PB));
then A9: [x1,x2] in ERl(PA) by XBOOLE_0:def 4;
A10: [x1,x2] in ERl(PB) by A8,XBOOLE_0:def 4;
    consider A being Subset of Y such that
A11: A in PA and
A12: x1 in A and
A13: x2 in A by A9,Def6;
    consider B being Subset of Y such that
A14: B in PB and
A15: x1 in B and
A16: x2 in B by A10,Def6;
A17: A /\ B <> {} by A12,A15,XBOOLE_0:def 4;
    consider C being Subset of Y such that
A18: C = A /\ B;
A19: C in INTERSECTION(PA,PB) by A11,A14,A18,SETFAM_1:def 5;
 not C in {{}} by A17,A18,TARSKI:def 1;
then A20: C in INTERSECTION(PA,PB) \ {{}} by A19,XBOOLE_0:def 5;
     x1 in C & x2 in C by A12,A13,A15,A16,A18,XBOOLE_0:def 4;
    hence thesis by A20,Def6;
  end;
  hence thesis;
end;
