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 Th25:
  for v,x being object, V being Subset of the_Vertices_of G2
  for G1 being addAdjVertexAll of G2,v,V
  for f1 being VColoring of G1, f2 being VColoring of G2
  st not v in the_Vertices_of G2 & f1=f2+*(v.-->x) & not x in rng f2
  holds f2 is proper implies f1 is proper
proof
  let v,x be object, V be Subset of the_Vertices_of G2;
  let G1 be addAdjVertexAll of G2,v,V;
  let f1 be VColoring of G1, f2 be VColoring of G2;
  set h = v .--> x;
  assume A1: not v in the_Vertices_of G2 & f1=f2 +* h & not x in rng f2;
  then A2: the_Vertices_of G1 = the_Vertices_of G2 \/ {v} by GLIB_007:def 4;
  assume A3: f2 is proper;
  now
    let e,u,w be object;
    assume A4: e Joins u,w,G1;
    per cases;
    suppose A5: u <> v & w <> v;
      then A6: e Joins u,w,G2 by A1, A4, GLIB_007:49;
      f1.u = f2.u & f1.w = f2.w by A1, A5, FUNCT_4:83;
      hence f1.u <> f1.w by A3, A6, Th10;
    end;
    suppose A7: u = v;
      then A8: w <> v by A1, A4, GLIB_007:def 4;
      then A9: f1.u = x & f1.w = f2.w by A1, A7, FUNCT_4:83, FUNCT_4:113;
      w in the_Vertices_of G1 by A4, GLIB_000:13;
      then w in the_Vertices_of G2 by A2, A8, ZFMISC_1:136;
      then w in dom f2 by PARTFUN1:def 2;
      hence f1.u <> f1.w by A1, A9, FUNCT_1:3;
    end;
    suppose A10: w = v;
      then A11: u <> v by A1, A4, GLIB_007:def 4;
      then A12: f1.w = x & f1.u = f2.u by A1, A10, FUNCT_4:83, FUNCT_4:113;
      u in the_Vertices_of G1 by A4, GLIB_000:13;
      then u in the_Vertices_of G2 by A2, A11, ZFMISC_1:136;
      then u in dom f2 by PARTFUN1:def 2;
      hence f1.u <> f1.w by A1, A12, FUNCT_1:3;
    end;
  end;
  hence thesis by Th10;
end;
