reserve G,G1,G2 for _Graph;
reserve W,W1,W2 for Walk of G;
reserve e,x,y,z for set;
reserve v for Vertex of G;
reserve n,m for Element of NAT;

theorem Th23:
  n in dom W implies W.n = W.reverse().(len W - n + 1) & (len W -
  n + 1) in dom W.reverse()
proof
  set rn = len W - n + 1;
  assume
A1: n in dom W;
  then n <= len W by FINSEQ_3:25;
  then reconsider rn as Element of NAT by FINSEQ_5:1;
  n in Seg len W by A1,FINSEQ_1:def 3;
  then len W - n + 1 in Seg len W by FINSEQ_5:2;
  then
A2: rn in Seg len W.reverse() by FINSEQ_5:def 3;
  then rn in dom W.reverse() by FINSEQ_1:def 3;
  then W.reverse().rn = W.(len W - rn + 1) by FINSEQ_5:def 3;
  hence thesis by A2,FINSEQ_1:def 3;
end;
