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) is trivial iff m = n
proof
  let m, n be odd Element of NAT;
  assume that
A1: m <= n and
A2: n <= len W;
A3: len W.cut(m,n) + m = n + 1 by A1,A2,Lm15;
  hereby
    assume W.cut(m,n) is trivial;
    then 1 = (n - m) + 1 by A3,Lm55;
    hence m = n;
  end;
  assume m = n;
  hence thesis by A3,Lm55;
end;
