
theorem
  for G being _Graph, W being Walk of G, e being object
  st e in W.edges() & not e in G.loops() & W is Circuit-like
  ex e0 being object st e0 in W.edges() & e0 <> e
proof
  let G be _Graph, W be Walk of G, e be object;
  assume A1: e in W.edges() & not e in G.loops() & W is Circuit-like;
  then consider n being odd Element of NAT such that
    A2: n < len W & W.(n+1) = e by GLIB_001:100;
  A3: len W > 3
  proof
    assume len W <= 3;
    then len W + 0 <= 3+1 by XREAL_1:7;
    then per cases by CHORD:7;
    suppose len W = 1;
      then 2*W.length()+1 = 2*0+1 by GLIB_001:112;
      hence contradiction by A1, GLIB_001:def 26;
    end;
    suppose len W = 3;
      then 2*W.length()+1 = 2*1+1 by GLIB_001:112;
      then consider e0 being object such that
        A4: e0 Joins W.first(),W.last(),G & W = G.walkOf(W.first(),e0,W.last())
        by Th28;
      W.edges() = {e0} by A4, GLIB_001:108;
      then A5: e = e0 by A1, TARSKI:def 1;
      W.first() = W.last() by A1, GLIB_001:def 24;
      hence contradiction by A1, A4, A5, GLIB_009:def 2;
    end;
  end;
  per cases;
  suppose A6: n = 1;
    reconsider m=n+2 as odd Element of NAT;
    take W.(m+1);
    thus W.(m+1) in W.edges() by A3, A6, GLIB_001:100;
    m+1 <= len W by A3, A6, NAT_1:13;
    then W.(n+1) <> W.(m+1) by A1, A6, GLIB_001:138;
    hence thesis by A2;
  end;
  suppose A7: n <> 1;
    reconsider m=1 as odd Element of NAT by POLYFORM:4;
    take W.(m+1);
    m < len W by A3, XXREAL_0:2;
    hence W.(m+1) in W.edges() by GLIB_001:100;
    1 <= n by CHORD:2;
    then m < n by A7, XXREAL_0:1;
    then A8: m+1 < n+1 by XREAL_1:6;
    n+1 < len W + 1 by A2, XREAL_1:6;
    then n+1 <= len W by NAT_1:13;
    then W.(n+1) <> W.(m+1) by A1, A8, GLIB_001:138;
    hence thesis by A2;
  end;
end;
