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
  (P*R)|X = (P|X)*R
proof
  let x,y;
  hereby
    assume
A1: [x,y] in (P*R)|X;
    then [x,y] in P*R by Def9;
    then consider a such that
A2: [x,a] in P and
A3: [a,y] in R by Def6;
    x in X by A1,Def9;
    then [x,a] in P|X by A2,Def9;
    hence [x,y] in (P|X)*R by A3,Def6;
  end;
  assume [x,y] in (P|X)*R;
  then consider a such that
A4: [x,a] in P|X and
A5: [a,y] in R by Def6;
  [x,a] in P by A4,Def9;
  then
A6: [x,y] in P*R by A5,Def6;
  x in X by A4,Def9;
  hence [x,y] in (P*R)|X by A6,Def9;
end;
