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;

theorem Th82:
  for v being object, V being Subset of the_Vertices_of G2
  for G1 being addAdjVertexAll of G2,v,V, g2 being EColoring of G2
  for h being Function
  st not v in the_Vertices_of G2 & dom h = G1.edgesBetween(V,{v})
  holds g2+*h is EColoring of G1
proof
  let v be object, V be Subset of the_Vertices_of G2;
  let G1 be addAdjVertexAll of G2,v,V, g2 be EColoring of G2;
  let h be Function;
  set E = dom h;
  assume A1: not v in the_Vertices_of G2 & dom h = G1.edgesBetween(V,{v});
  dom(g2+*h) = dom g2 \/ dom h by FUNCT_4:def 1
    .= the_Edges_of G2 \/ E by PARTFUN1:def 2
    .= the_Edges_of G1 by A1, GLIB_007:59;
  hence thesis by RELAT_1:def 18, PARTFUN1:def 2;
end;
