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

theorem Th73:
  for v being Vertex of createGraph(V,E) holds Coim(E,v) = v.inNeighbors()
proof
  let v be Vertex of createGraph(V,E);
  now
    let x be object;
    hereby
      assume x in Coim(E,v);
      then x in E"{v} by RELAT_1:def 17;
      then consider y being object such that
        A1: [x,y] in E & y in {v} by RELAT_1:def 14;
      y = v by A1, TARSKI:def 1;
      then A2: [x,y] DJoins x,v,createGraph(V,E) by A1, Th63;
      x is set by TARSKI:1;
      hence x in v.inNeighbors() by A2, GLIB_000:69;
    end;
    assume x in v.inNeighbors();
    then consider e being object such that
      A3: e DJoins x,v,createGraph(V,E) by GLIB_000:69;
    e = [x,v] by A3, Th64;
    then A4: [x,v] in E by A3, Th63;
    v in {v} by TARSKI:def 1;
    then x in E"{v} by A4, RELAT_1:def 14;
    hence x in Coim(E,v) by RELAT_1:def 17;
  end;
  hence thesis by TARSKI:2;
end;
