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

theorem Th155:
  for y,e being object, v,w being Vertex of G2, G1 being addEdge of G2,v,e,w
  for t1 being TColoring of G1, t2 being TColoring of G2
  st not e in the_Edges_of G2 & v,w are_adjacent &
    t1_V = t2_V & t1_E = t2_E +* (e .--> y) &
    not y in (rng t2_V)\/rng t2_E & t2 is proper
  holds t1 is proper
proof
  let y,e be object, v,w be Vertex of G2, G1 be addEdge of G2,v,e,w;
  let t1 be TColoring of G1, t2 be TColoring of G2;
  assume that A1: not e in the_Edges_of G2 & v,w are_adjacent and
    A2: t1_V = t2_V & t1_E = t2_E +* (e .--> y) and
    A3: not y in (rng t2_V)\/rng t2_E & t2 is proper;
  A4: t1_V is proper by A1, A2, A3, Th20;
  rng t2_E c= (rng t2_V)\/rng t2_E by XBOOLE_1:7;
  then not y in rng t2_E by A3;
  then A5: t1_E is proper by A1, A2, A3, Th93;
  now
    let e9,v9,w9 be object;
    assume A6: e9 Joins v9,w9,G1;
    then per cases by GLIB_006:72;
    suppose A7: e9 Joins v9,w9,G2;
      then A8: t2_V.v9 <> t2_E.e9 by A3, Th146;
      e9 in the_Edges_of G2 by A7, GLIB_000:def 13;
      then not e9 in dom(e.-->y) by A1, TARSKI:def 1;
      hence t1_V.v9 <> t1_E.e9 by A2, A8, FUNCT_4:11;
    end;
    suppose not e9 in the_Edges_of G2;
      then e9 = e by A1, A6, GLIB_006:106;
      then A9: t1_E.e9 = y by A2, FUNCT_4:113;
      v9 in the_Vertices_of G1 by A6, GLIB_000:13;
      then v9 in dom t1_V by PARTFUN1:def 2;
      then t1_V.v9 in rng t2_V by A2, FUNCT_1:3;
      hence t1_V.v9 <> t1_E.e9 by A3, A9, XBOOLE_0:def 3;
    end;
  end;
  hence thesis by A4, A5, Th146;
end;
