
theorem
  for G being simple _Graph for W1, W2 being Walk of G for k being odd
  Nat st k <= len W1 & k <= len W2 & for j being odd Nat st j <= k holds W1.j =
  W2.j holds for j being Nat st 1 <= j & j <= k holds W1.j = W2.j
proof
  let G be simple _Graph, W1, W2 be Walk of G, k be odd Nat such that
A1: k <= len W1 and
A2: k <= len W2 and
A3: for j being odd Nat st j <= k holds W1.j = W2.j;
  let j be Nat such that
A4: 1 <= j and
A5: j <= k;
  per cases;
  suppose
    j is odd;
    hence thesis by A3,A5;
  end;
  suppose
    j is even;
    then reconsider j9 = j as even Nat;
    1-1 <= j-1 by A4,XREAL_1:9;
    then reconsider j1 = j9-1 as odd Element of NAT by INT_1:3;
A6: j1 < j by XREAL_1:44;
    j <= len W1 by A1,A5,XXREAL_0:2;
    then j1 < len W1 by A6,XXREAL_0:2;
    then
A7: W1.(j1+1) Joins W1.j1, W1.(j1+2), G by GLIB_001:def 3;
    j1+1 < k by A5,XXREAL_0:1;
    then j1+1+1 <= k by NAT_1:13;
    then
A8: W1.(j1+2) = W2.(j1+2) by A3;
    j <= len W2 by A2,A5,XXREAL_0:2;
    then j1 < len W2 by A6,XXREAL_0:2;
    then
A9: W2.(j1+1) Joins W2.j1, W2.(j1+2), G by GLIB_001:def 3;
    W1.j1 = W2.j1 by A3,A5,A6,XXREAL_0:2;
    hence thesis by A7,A9,A8,GLIB_000:def 20;
  end;
end;
