
theorem Th51:
  for G being _Graph, W1, W3 being Walk of G, e,u,v being object holds
    W1 is_Walk_from u,v iff W1.replaceEdgeWith(e, W3) is_Walk_from u,v
proof
  let G be _Graph;
  let W1, W3 be Walk of G;
  let e,u,v be object;
  per cases;
  suppose e Joins W3.first(),W3.last(),G &
    G.walkOf(W3.first(),e,W3.last()) is_odd_substring_of W1, 0;
    then consider W2 being Walk of G such that
      A1: W1.replaceEdgeWith(e,W3) = W1.replaceWith(W2,W3) by Th46;
    thus thesis by A1, Th50;
  end;
  suppose not (e Joins W3.first(),W3.last(),G &
    G.walkOf(W3.first(),e,W3.last()) is_odd_substring_of W1, 0);
    hence thesis by Def6;
  end;
end;
