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 Th7:
  for v,x being object, V being Subset of the_Vertices_of G2
  for G1 being addAdjVertexAll of G2,v,V, f2 being VColoring of G2
  st not v in the_Vertices_of G2 holds f2+*(v.-->x) is VColoring of G1
proof
  let v,x be object, V be Subset of the_Vertices_of G2;
  let G1 be addAdjVertexAll of G2,v,V, f2 be VColoring of G2;
  set f1 = f2 +* (v .--> x);
  assume not v in the_Vertices_of G2;
  then A1: the_Vertices_of G1 = the_Vertices_of G2 \/ {v} by GLIB_007:def 4;
  dom f1 = dom f2 \/ dom(v .--> x) by FUNCT_4:def 1
    .= dom f2 \/ dom{[v,x]} by FUNCT_4:82
    .= dom f2 \/ {v} by RELAT_1:9
    .= the_Vertices_of G1 by A1, PARTFUN1:def 2;
  hence thesis by RELAT_1:def 18, PARTFUN1:def 2;
end;
