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 Th158:
  for v,e being object, w being Vertex of G2, G1 being addAdjVertex of G2,v,e,w
  for t1 being TColoring of G1, t2 being TColoring of G2, x,y being object
  st not e in the_Edges_of G2 & not v in the_Vertices_of G2 &
    t1_V = t2_V +* (v .--> x) & t1_E = t2_E +* (e .--> y) &
    not y in (rng t2_V)\/rng t2_E & x <> y & x <> t2_V.w & t2 is proper
  holds t1 is proper
proof
  let v,e be object, w be Vertex of G2, G1 be addAdjVertex of G2,v,e,w;
  let t1 be TColoring of G1, t2 be TColoring of G2, x,y be object;
  assume that A1: not e in the_Edges_of G2 & not v in the_Vertices_of G2 and
    A2: t1_V = t2_V +* (v .--> x) & t1_E = t2_E +* (e .--> y) and
    A3: not y in (rng t2_V)\/rng t2_E & x <> y & x <> t2_V.w & t2 is proper;
  A4: t1_V is proper by A1, A2, A3, Th23;
  not y in rng t2_E by A3, XBOOLE_0:def 3;
  then A5: t1_E is proper by A1, A2, A3, Th94;
  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 e9 in the_Edges_of G2 by GLIB_000:def 13;
      then not e9 in dom(e.-->y) by A1, TARSKI:def 1;
      then A8: t1_E.e9 = t2_E.e9 by A2, FUNCT_4:11;
      v9 in the_Vertices_of G2 by A7, GLIB_000:13;
      then not v9 in dom(v.-->x) by A1, TARSKI:def 1;
      then t1_V.v9 = t2_V.v9 by A2, FUNCT_4:11;
      hence t1_V.v9 <> t1_E.e9 by A3, A7, A8, Th146;
    end;
    suppose A9: not e9 in the_Edges_of G2;
      A10: the_Edges_of G1 = the_Edges_of G2 \/ {e} by A1, GLIB_006:def 12;
      e9 in the_Edges_of G1 by A6, GLIB_000:def 13;
      then e9 in {e} by A9, A10, XBOOLE_0:def 3;
      then e9 = e by TARSKI:def 1;
      then A11: t1_E.e9 = y by A2, FUNCT_4:113;
      per cases;
      suppose v9 = v;
        hence t1_V.v9 <> t1_E.e9 by A2, A3, A11, FUNCT_4:113;
      end;
      suppose A12: v9 <> v;
        then not v9 in dom(v.-->x) by TARSKI:def 1;
        then A13: t1_V.v9 = t2_V.v9 by A2, FUNCT_4:11;
        A14: the_Vertices_of G1 = the_Vertices_of G2 \/ {v}
          by A1, GLIB_006:def 12;
        v9 in the_Vertices_of G1 & not v9 in {v}
          by A6, A12, GLIB_000:13, TARSKI:def 1;
        then v9 in the_Vertices_of G2 by A14, XBOOLE_0:def 3;
        then v9 in dom t2_V by PARTFUN1:def 2;
        then t1_V.v9 in rng t2_V by A13, FUNCT_1:3;
        hence t1_V.v9 <> t1_E.e9 by A3, A11, XBOOLE_0:def 3;
      end;
    end;
  end;
  hence thesis by A4, A5, Th146;
end;
