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 Th19:
  dom(P*R) c= dom P
proof
  let x be object;
  assume x in dom(P*R);
  then consider y being object such that
A1: [x,y] in P*R by XTUPLE_0:def 12;
  ex z st [x,z] in P & [z,y] in R by A1,Def6;
  hence thesis by XTUPLE_0:def 12;
end;
