
theorem Th85: :: ChordalWalk02
  for G1,G2 being _Graph st G1 == G2
  for W1 being Walk of G1, W2 being Walk of G2 st W1=W2 holds
    W1 is chordal implies W2 is chordal
proof
  let G1,G2 be _Graph such that
A1: G1 == G2;
  let W1 be Walk of G1, W2 be Walk of G2 such that
A2: W1 = W2;
  given m, n being odd Nat such that
A3: m+2 < n and
A4: n <= len W1 and
A5: W1.m <> W1.n and
A6: ex e being object st e Joins W1.m,W1.n,G1 and
A7: for f being object st f in W1.edges() holds not f Joins W1.m,W1.n,G1;
  take m,n;
  thus m+2 < n & n <= len W2 & W2.m <> W2.n by A2,A3,A4,A5;
  consider e being object such that
A8: e Joins W1.m,W1.n,G1 by A6;
  e Joins W2.m,W2.n,G2 by A1,A2,A8;
  hence ex e being object st e Joins W2.m,W2.n,G2;
  let f be object;
  assume f in W2.edges();
  then f in W1.edges() by A2,GLIB_001:110;
  then not f Joins W1.m,W1.n,G1 by A7;
  hence thesis by A1,A2;
end;
