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 st m <= n & n <= len W holds W.cut(m,
  n).vertices() c= W.vertices()
proof
  let m, n be odd Element of NAT;
  set W2 = W.cut(m,n);
  assume that
A1: m <= n and
A2: n <= len W;
  now
    let x be object;
    assume x in W2.vertices();
    then consider n being odd Element of NAT such that
A3: n <= len W2 and
A4: W2.n = x by Lm45;
    reconsider naa1 = n - 1 as even Element of NAT by ABIAN:12,INT_1:5;
A5: naa1 < len W2 - 0 by A3,XREAL_1:15;
    then m+naa1 in dom W by A1,A2,Lm15;
    then
A6: m+naa1 <= len W by FINSEQ_3:25;
    W2.(naa1+1) = W.(m+naa1) by A1,A2,A5,Lm15;
    hence x in W.vertices() by A4,A6,Lm45;
  end;
  hence thesis by TARSKI:def 3;
end;
