
theorem Th8: :: Tpart0:
for X being set, P being finite a_partition of X, S being Subset of X
 holds card (P | S) <= card P
proof
 let X be set, P be finite a_partition of X, S be Subset of X;
 per cases;
 suppose X = {}; then
   S = {};
  hence card (P | S) <= card P;
 end;
 suppose X <> {};
   then reconsider Pp1 = P as finite non empty set;
   defpred P[set] means $1 meets S;
   deffunc F(set) = $1 /\ S;
   A3: P | S = {F(x) where x is Element of Pp1: P[x]};
   card (P | S) <= card Pp1 from DILWORTH:sch 1(A3);
  hence thesis;
 end;
end;
