
theorem Th86: :: ChordalWalk03
  for G being _Graph,
      S being non empty Subset of the_Vertices_of G,
      H being (inducedSubgraph of G,S),
      W1 being Walk of G, W2 being Walk of H st
    W1 = W2 holds W2 is chordal iff W1 is chordal
proof
  let G be _Graph, S be non empty Subset of the_Vertices_of G,
      H be (inducedSubgraph of G,S), W1 be Walk of G, W2 be Walk of H
    such that
A1: W1 = W2;
  thus W2 is chordal implies W1 is chordal
  proof
    given m, n being odd Nat such that
A2: m+2 < n and
A3: n <= len W2 and
A4: W2.m <> W2.n and
A5: ex e being object st e Joins W2.m,W2.n,H and
A6: for f being object st f in W2.edges() holds not f Joins W2.m,W2.n,H;
    take m,n;
    thus m+2 < n & n <= len W1 & W1.m <> W1.n by A1,A2,A3,A4;
    consider e being object such that
A7: e Joins W2.m,W2.n,H by A5;
    e Joins W1.m,W1.n,G by A1,A7,GLIB_000:72;
    hence ex e being object st e Joins W1.m,W1.n,G;
    let f be object;
    assume f in W1.edges();
    then
A8: f in W2.edges() by A1,GLIB_001:110;
    then not f Joins W1.m,W1.n,H by A1,A6;
    hence thesis by A8,GLIB_000:73;
  end;
A9: S = the_Vertices_of H by GLIB_000:def 37;
  thus W1 is chordal implies W2 is chordal
  proof
    given m, n being odd Nat such that
A10: m+2 < n and
A11: n <= len W1 and
A12: W1.m <> W1.n and
A13: ex e being object st e Joins W1.m,W1.n,G and
A14: for f being object st f in W1.edges() holds not f Joins W1.m,W1.n,G;
    take m,n;
    thus m+2 < n & n <= len W2 & W2.m <> W2.n by A1,A10,A11,A12;
A15: m in NAT by ORDINAL1:def 12;
    n in NAT by ORDINAL1:def 12;
    then
A16: W1.n in the_Vertices_of H by A1,A11,GLIB_001:7;
    m < n by A10,NAT_1:12;
    then W1.m in the_Vertices_of H by A1,A11,A15,GLIB_001:7,XXREAL_0:2;
    hence ex e being object st e Joins W2.m,W2.n,H by A1,A9,A13,A16,Th19;
    let f be object;
    assume f in W2.edges();
    then f in W1.edges() by A1,GLIB_001:110;
    then not f Joins W2.m,W2.n,G by A1,A14;
    hence thesis by GLIB_000:72;
  end;
end;
