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 Th15:
  for H being Subgraph of G, f9 being VColoring of H
  st f9 = f | the_Vertices_of H & f is proper holds f9 is proper
proof
  let H be Subgraph of G, f9 be VColoring of H;
  assume A1: f9 = f | the_Vertices_of H & f is proper;
  now
    let e,v,w be object;
    assume A2: e Joins v,w,H;
    v is set & w is set by TARSKI:1;
    then A3: f.v <> f.w by A1, A2, Th10, GLIB_000:72;
    v in the_Vertices_of H & w in the_Vertices_of H by A2, GLIB_000:13;
    then v in dom f9 & w in dom f9 by PARTFUN1:def 2;
    then f.v = f9.v & f.w = f9.w by A1, FUNCT_1:47;
    hence f9.v <> f9.w by A3;
  end;
  hence thesis by Th10;
end;
