reserve E,V for set, G,G1,G2 for _Graph, c,c1,c2 for Cardinal, n for Nat;
reserve f for VColoring of G;

theorem Th10:
  f is proper iff for e,v,w being object st e Joins v,w,G holds f.v <> f.w
proof
  hereby
    assume A1: f is proper;
    let e,v,w be object;
    assume A2: e Joins v,w,G;
    then reconsider v0=v,w0=w as Vertex of G by GLIB_000:13;
    v0,w0 are_adjacent by A2, CHORD:def 3;
    hence f.v <> f.w by A1;
  end;
  assume A3: for e,v,w being object st e Joins v,w,G holds f.v <> f.w;
  let v,w be Vertex of G;
  assume v,w are_adjacent;
  then consider e being object such that
    A4: e Joins v,w,G by CHORD:def 3;
  thus thesis by A3, A4;
end;
