
theorem
  for G, H being _finite chordal _Graph,
      g being perfect VertexScheme of G st G == H holds
    g is perfect VertexScheme of H
proof
  let G,H be _finite chordal _Graph, g be perfect VertexScheme of G such that
A1: G == H;
  reconsider h=g as VertexScheme of H by A1,Th105;
  now
    let n be non zero Nat such that
A2: n <= len h;
    let Hf be inducedSubgraph of H,h.followSet(n);
    let vh be Vertex of Hf such that
A3: vh = h.n;
    G.edgesBetween(g.followSet(n)) = H.edgesBetween(g.followSet(n)) by A1,
GLIB_000:90;
    then reconsider Gf = Hf as inducedSubgraph of G,g.followSet(n) by A1,
GLIB_000:95;
    reconsider vg = vh as Vertex of Gf;
    vg is simplicial by A2,A3,Def13;
    hence vh is simplicial;
  end;
  hence thesis by Def13;
end;
