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
  for m,n being odd Element of NAT, i being Element of NAT st m <= n & n
<= len W & i in dom W.cut(m,n) holds W.cut(m,n).i = W.(m+i-1) & m+i-1 in dom W
proof
  let m,n be odd Element of NAT, i be Element of NAT;
  assume that
A1: m <= n and
A2: n <= len W and
A3: i in dom W.cut(m,n);
  1 <= i by A3,FINSEQ_3:25;
  then reconsider iaa1 = i-1 as Element of NAT by INT_1:5;
  i <= len W.cut(m,n) by A3,FINSEQ_3:25;
  then
A4: iaa1 < len W.cut(m,n) - 0 by XREAL_1:15;
  iaa1+1 = i;
  then W.cut(m,n).i = W.(m+iaa1) by A1,A2,A4,Lm15;
  hence thesis by A1,A2,A4,Lm15;
end;
