
theorem Th44:
  for G being _Graph, T1 being Trail of G, W3 being Walk of G, e being object
  st e Joins W3.first(),W3.last(),G & not e in W3.edges() &
    G.walkOf(W3.first(),e,W3.last()) is_odd_substring_of T1, 0
  holds not e in T1.replaceEdgeWith(e, W3).edges()
proof
  let G be _Graph, T1 be Trail of G, W3 be Walk of G, e be object;
  assume that
    A1: e Joins W3.first(),W3.last(),G and
    A2: not e in W3.edges() and
    A3: G.walkOf(W3.first(),e,W3.last()) is_odd_substring_of T1, 0;
  for n, m being even Nat st n in dom T1 & m in dom T1 & T1.n = e & T1.m = e
    holds n = m
  proof
    let n,m be even Nat;
    assume A4: n in dom T1 & m in dom T1 & T1.n = e & T1.m = e;
    then A5: 1 <= n & n <= len T1 & 1 <= m & m <= len T1 by FINSEQ_3:25;
    reconsider n,m as even Element of NAT by ORDINAL1:def 12;
    per cases by XXREAL_0:1;
    suppose n < m;
      hence thesis by A4, A5, GLIB_001:138;
    end;
    suppose n = m;
      hence thesis;
    end;
    suppose n > m;
      hence thesis by A4, A5, GLIB_001:138;
    end;
  end;
  hence thesis by A1, A2, A3, Th43;
end;
