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 W1 being Walk of G1, W2 being Walk of G2, m, n being Element of
  NAT st W1 = W2 holds W1.remove(m,n) = W2.remove(m,n)
proof
  let W1 be Walk of G1, W2 be Walk of G2, m, n be Element of NAT;
  assume
A1: W1 = W2;
  now
    per cases;
    suppose
A2:   m is odd & n is odd & m <= n & n <= len W1 & W1.m = W1.n;
A3:   W1.cut(n,len W1) = W2.cut(n,len W2) by A1,Th46;
A4:   W1.cut(1,m) = W2.cut(1,m) by A1,Th46;
      W1.remove(m,n) = W1.cut(1,m).append(W1.cut(n,len W1)) by A2,Def12;
      then W1.remove(m,n) = W2.cut(1,m).append(W2.cut(n,len W2)) by A4,A3,Th33;
      hence thesis by A1,A2,Def12;
    end;
    suppose
A5:   not (m is odd & n is odd & m <= n & n <= len W1 & W1.m = W1.n);
      hence W1.remove(m,n) = W2 by A1,Def12
        .= W2.remove(m,n) by A1,A5,Def12;
    end;
  end;
  hence thesis;
end;
