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
  Y|`(P*R) = P*(Y|`R)
proof
  let x,y;
  hereby
    assume
A1: [x,y] in Y|`(P*R);
    then [x,y] in P*R by Def10;
    then consider a such that
A2: [x,a] in P and
A3: [a,y] in R by Def6;
    y in Y by A1,Def10;
    then [a,y] in Y|`R by A3,Def10;
    hence [x,y] in P*(Y|`R) by A2,Def6;
  end;
  assume [x,y] in P*(Y|`R);
  then consider a such that
A4: [x,a] in P and
A5: [a,y] in Y|`R by Def6;
  [a,y] in R by A5,Def10;
  then
A6: [x,y] in P*R by A4,Def6;
  y in Y by A5,Def10;
  hence [x,y] in Y|`(P*R) by A6,Def10;
end;
