reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;

theorem Th18:
  pi(X,x) \ pi(Y,x) c= pi(X \ Y,x)
proof
  let y be object;
  assume
A1: y in pi(X,x) \ pi(Y,x);
  then consider f such that
A2: f in X and
A3: y = f.x by Def6;
  not y in pi(Y,x) by A1,XBOOLE_0:def 5;
  then not f in Y by A3,Def6;
  then f in X \ Y by A2,XBOOLE_0:def 5;
  hence thesis by A3,Def6;
end;
