reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;
reserve V for set;

theorem Th96:
  for G2, V for G1 being addVertices of G2, V
  holds G2 is chordal iff G1 is chordal
proof
  let G2 be _Graph, V be set, G1 be addVertices of G2, V;
  thus G2 is chordal implies G1 is chordal
  proof
    assume A1: G2 is chordal;
    for P being Walk of G1 st
      P.length() > 3 & P is Cycle-like holds P is chordal
    proof
      let P be Walk of G1;
      assume A2: P.length() > 3 & P is Cycle-like;
      per cases by Th94;
      suppose P.vertices() misses V \ the_Vertices_of G2;
        then reconsider W = P as Walk of G2 by Th95;
        A3: W is Cycle-like by A2, Th80;
        P.length() = W.length() by GLIB_001:114;
        then W is chordal by A1, A2, A3, CHORD:def 11;
        then consider m,n being odd Nat such that
          A4: m+2 < n & n <= len W & W.m <> W.n and
          A5: ex e being object st e Joins W.m,W.n,G2 and
          A6: for f being object st f in W.edges() holds not f Joins W.m,W.n,G2
          by CHORD:def 10;
        now
          take m, n;
          thus m+2 < n & n <= len P & P.m <> P.n by A4;
          thus ex e being object st e Joins P.m,P.n,G1 by A5, Th74;
          thus for f being object st f in P.edges() holds
            not f Joins P.m,P.n,G1
          proof
            given f being object such that
              A7: f in P.edges() and
              A8: f Joins P.m,P.n,G1;
            A9: f in W.edges() by A7, GLIB_001:110;
            per cases by A8, GLIB_000:16;
            suppose f DJoins P.m,P.n,G1;
              then per cases by Th75;
              suppose f DJoins P.m,P.n,G2;
                then f Joins W.m,W.n,G2 by GLIB_000:16;
                hence contradiction by A6, A9;
              end;
              suppose not f in the_Edges_of G2;
                hence contradiction by A9;
              end;
            end;
            suppose f DJoins P.n,P.m,G1;
              then per cases by Th75;
              suppose f DJoins P.n,P.m,G2;
                then f Joins W.m,W.n,G2 by GLIB_000:16;
                hence contradiction by A6, A9;
              end;
              suppose not f in the_Edges_of G2;
                hence contradiction by A9;
              end;
            end;
          end;
        end;
        hence P is chordal by CHORD:def 10;
      end;
      suppose P is trivial;
        hence thesis by A2;
      end;
    end;
    hence thesis by CHORD:def 11;
  end;
  assume A10: G1 is chordal;
  G2 is inducedSubgraph of G1, the_Vertices_of G2 by Th88;
  hence thesis by A10;
end;
