
theorem Th24:
  for G being non-multi _Graph for W1, W2 being Walk of G st W1
  .first() = W2.first() & not W1 is_a_prefix_of W2 & not W2 is_a_prefix_of W1
  holds W1.(len maxPrefix(W1,W2) +2) <> W2.(len maxPrefix(W1,W2) +2)
proof
  let G be non-multi _Graph, W1, W2 be Walk of G such that
A1: W1.first() = W2.first() and
A2: not W1 c= W2 and
A3: not W2 c= W1 and
A4: W1.(len maxPrefix(W1,W2) +2) = W2.(len maxPrefix(W1,W2) +2);
  set dI = len maxPrefix(W1,W2);
A5: dI is odd by A1,Th22;
  dI < len W1 by A2,Th8;
  then
A6: W1.(dI+1) Joins W1.dI,W1.(dI+2), G by A5,GLIB_001:def 3;
A7: W1.dI = W2.dI by Th7;
  dI < len W2 by A3,Th8;
  then
A8: W2.(dI+1) Joins W2.dI,W2.(dI+2), G by A5,GLIB_001:def 3;
  W1.(dI +1) <> W2.(dI +1) by A2,A3,Th9;
  hence contradiction by A4,A7,A6,A8,GLIB_000:def 20;
end;
