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 Th33:
  {}*R = {} & R*{} = {}
proof
  thus {}*R = {}
  proof
    let x,y;
    hereby
      assume [x,y] in {}*R;
      then ex z st [x,z] in {} & [z,y] in R by Def6;
      hence [x,y] in {};
    end;
    thus thesis;
  end;
  let x,y;
  hereby
    assume [x,y] in R*{};
    then ex z st [x,z] in R & [z,y] in {} by Def6;
    hence [x,y] in {};
  end;
  thus thesis;
end;
