reserve G for _Graph;
reserve V for non empty set, E for Relation of V;

theorem Th72:
  for v being Vertex of createGraph(V,E) holds Im(E,v) = v.outNeighbors()
proof
  let v be Vertex of createGraph(V,E);
  now
    let y be object;
    hereby
      assume y in Im(E,v);
      then y in E.:{v} by RELAT_1:def 16;
      then consider x being object such that
        A1: [x,y] in E & x in {v} by RELAT_1:def 13;
      x = v by A1, TARSKI:def 1;
      then A2: [x,y] DJoins v,y,createGraph(V,E) by A1, Th63;
      y is set by TARSKI:1;
      hence y in v.outNeighbors() by A2, GLIB_000:70;
    end;
    assume y in v.outNeighbors();
    then consider e being object such that
      A3: e DJoins v,y,createGraph(V,E) by GLIB_000:70;
    e = [v,y] by A3, Th64;
    then A4: [v,y] in E by A3, Th63;
    v in {v} by TARSKI:def 1;
    then y in E.:{v} by A4, RELAT_1:def 13;
    hence y in Im(E,v) by RELAT_1:def 16;
  end;
  hence thesis by TARSKI:2;
end;
