reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem Th116:
  P c= R implies P.:X c= R.:X
proof
  assume
A1: P c= R;
  let y be object;
  assume y in P.:X;
  then ex x st [x,y] in P & x in X by Def11;
  hence thesis by A1,Def11;
end;
