
theorem Th22:
  for G being _Graph for W1, W2 being Walk of G st W1.first() = W2
  .first() holds len maxPrefix(W1,W2) is odd
proof
  let G be _Graph, W1, W2 be Walk of G;
  assume that
A1: W1.first() = W2.first() and
A2: len maxPrefix(W1,W2) is even;
  set dI = len maxPrefix(W1,W2);
  reconsider dIp = dI-1 as odd Element of NAT by A1,A2,Th5,INT_1:5;
A3: dIp < dI by XREAL_1:146;
  set mP = maxPrefix(W1,W2);
  set lmP = mP^<*W1.(dI+1)*>;
A4: len lmP = len mP + 1 by FINSEQ_2:16;
A5: now
    let x be object such that
A6: x in dom lmP;
    reconsider n = x as Nat by A6;
A7: 1 <= n by A6,FINSEQ_3:25;
    n <= len lmP by A6,FINSEQ_3:25;
    then
A8: n <= len mP + 1 by FINSEQ_2:16;
    per cases by A8,NAT_1:8;
    suppose
A9:   n <= dI;
      then n in dom mP by A7,FINSEQ_3:25;
      hence lmP.x = mP.x by FINSEQ_1:def 7
        .= W1.x by A9,Th6;
    end;
    suppose
      n = dI + 1;
      hence lmP.x = W1.x by FINSEQ_1:42;
    end;
  end;
A10: dI = dIp+1;
A11: dI <= len W2 by Th3;
  then dIp < len W2 by XREAL_1:146,XXREAL_0:2;
  then
A12: W2.dI Joins W2.dIp, W2.(dIp+2), G by A10,GLIB_001:def 3;
A13: dI <= len W1 by Th3;
  then dIp < len W1 by XREAL_1:146,XXREAL_0:2;
  then
A14: W1.dI Joins W1.dIp, W1.(dIp+2), G by A10,GLIB_001:def 3;
  W1.dI = W2.dI by Th7;
  then
A15: W1.dIp = W2.dIp & W1.(dIp+2) = W2.(dIp+2) or W1.dIp = W2.(dIp+2) & W1.(
  dIp+2) = W2.dIp by A14,A12,GLIB_000:15;
A16: now
    let x be object such that
A17: x in dom lmP;
    reconsider n = x as Nat by A17;
A18: 1 <= n by A17,FINSEQ_3:25;
    n <= len lmP by A17,FINSEQ_3:25;
    then
A19: n <= len mP + 1 by FINSEQ_2:16;
    per cases by A19,NAT_1:8;
    suppose
A20:  n <= dI;
      then n in dom mP by A18,FINSEQ_3:25;
      hence lmP.x = mP.x by FINSEQ_1:def 7
        .= W2.x by A20,Th6;
    end;
    suppose
A21:  n = dI + 1;
      hence lmP.x = W1.(dI+1) by FINSEQ_1:42
        .= W2.x by A3,A15,A21,Th7;
    end;
  end;
  dI < len W2 by A2,A11,XXREAL_0:1;
  then len lmP <= len W2 by A4,NAT_1:13;
  then dom lmP c= dom W2 by FINSEQ_3:30;
  then
A22: lmP c= W2 by A16,GRFUNC_1:2;
  dI < len W1 by A2,A13,XXREAL_0:1;
  then len lmP <= len W1 by A4,NAT_1:13;
  then dom lmP c= dom W1 by FINSEQ_3:30;
  then lmP c= W1 by A5,GRFUNC_1:2;
  then lmP c= mP by A22,Def1;
  then len lmP <= len mP by FINSEQ_1:63;
  then len mP + 1 <= len mP by FINSEQ_2:16;
  hence contradiction by NAT_1:13;
end;
