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 Th46:
  for W1 being Walk of G1, W2 being Walk of G2, m, n being Element
  of NAT st W1 = W2 holds W1.cut(m,n) = W2.cut(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;
      hence W1.cut(m,n) = (m,n)-cut W2 by A1,Def11
        .= W2.cut(m,n) by A1,A2,Def11;
    end;
    suppose
A3:   not (m is odd & n is odd & m <= n & n <= len W1);
      hence W1.cut(m,n) = W2 by A1,Def11
        .= W2.cut(m,n) by A1,A3,Def11;
    end;
  end;
  hence thesis;
end;
