reserve x,y for object,
  f for Function;

theorem Th41:
  for W being Relation, X being set st dom W c= X holds *graph((W,
  X)*graph) = W
proof
  let W be Relation, X be set such that
A1: dom W c= X;
A2: dom ((W,X)*graph) = X by Def5;
  now
    let x,y be object;
    hereby
      assume [x,y] in *graph((W,X)*graph);
      then x in X & y in ((W,X)*graph).x by A2,Th38;
      then y in Im(W,x) by Def5;
      then ex z being object st [z,y] in W & z in {x} by RELAT_1:def 13;
      hence [x,y] in W by TARSKI:def 1;
    end;
    assume
A3: [x,y] in W;
    then
A4: x in dom W by XTUPLE_0:def 12;
    x in {x} by TARSKI:def 1;
    then y in Im(W,x) by A3,RELAT_1:def 13;
    then y in ((W,X)*graph).x by A1,A4,Def5;
    hence [x,y] in *graph((W,X)*graph) by A1,A2,A4,Th38;
  end;
  hence thesis;
end;
