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 Th15:
  for PA,PB being a_partition of Y holds PA '>' PA '/\' PB
proof
  let PA,PB be a_partition of Y;
 for u being set st u in PA '/\' PB ex v being set st v in PA & u c= v
  proof
    let u be set;
    assume u in PA '/\' PB;
    then consider u1,u2 being set such that
A1: u1 in PA and u2 in PB and
A2: u = u1 /\ u2 by SETFAM_1:def 5;
    take u1;
    thus thesis by A1,A2,XBOOLE_1:17;
  end;
  hence PA '>' PA '/\' PB by SETFAM_1:def 2;
end;
