
theorem Th47:
  for G being _Graph, W1, W2 being Walk of G, e being object
  st W2 is_odd_substring_of W1, 0 & e Joins W2.first(),W2.last(),G
  holds ex W3 being Walk of G
    st W1.replaceWithEdge(W2,e) = W1.replaceWith(W2,W3)
proof
  let G be _Graph, W1, W2 be Walk of G, e be object;
  ::per cases;
  ::suppose A1: W2 is_odd_substring_of W1, 0 &
  ::  e Joins W2.first(),W2.last(),G;
  assume A1: W2 is_odd_substring_of W1, 0 &
    e Joins W2.first(),W2.last(),G;
    take G.walkOf(W2.first(),e,W2.last());
    thus thesis by A1, Def7;
  ::end;
  ::suppose not (W2 is_odd_substring_of W1, 0 &
  ::  e Joins W2.first(),W2.last(),G);
  ::  then A2: W1.replaceWithEdge(W2, e) = W1 by Def7;
  ::  take W2;
  ::  thus thesis by A2, Th38;
  ::end;
end;
