reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;
reserve D for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve L,M for Element of NAT;
reserve f for Function of A,B;
reserve f for Function;
reserve x1,x2,x3,x4,x5 for object;
reserve p for FinSequence;
reserve ND for non empty set;
reserve y1,y2,y3,y4,y5 for Element of ND;
reserve X, A for non empty finite set,
  PX for a_partition of X,
  PA1, PA2 for a_partition of A;

theorem
  card PX <= card X
proof
  assume card PX > card X;
  then card Segm card X in card Segm card PX by NAT_1:41;
  then consider Pi being object such that
A1: Pi in PX and
A2: for x being object st x in X holds (proj PX).x <> Pi by Th66;
  reconsider Pi as Element of PX by A1;
  consider q being Element of X such that
A3: q in Pi by Th85;
  reconsider Pq = (proj PX).q as Element of PX;
A4: Pq = Pi or Pq misses Pi by EQREL_1:def 4;
  q in Pq by EQREL_1:def 9;
  hence contradiction by A2,A3,A4,XBOOLE_0:3;
end;
