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
  W.cut(m,n).edges() c= W.edges()
proof
  now
    per cases;
    suppose
A1:   m is odd & n is odd & m <= n & n <= len W;
      then reconsider m9 = m as odd Element of NAT;
      now
        let e be object;
        assume e in W.cut(m,n).edges();
        then consider x being even Element of NAT such that
A2:     1 <= x and
A3:     x <= len W.cut(m,n) and
A4:     W.cut(m,n).x = e by Lm46;
        reconsider xaa1 = x-1 as odd Element of NAT by A2,INT_1:5;
A5:     xaa1 < len W.cut(m,n) - 0 by A3,XREAL_1:15;
        then
A6:     m+xaa1 in dom W by A1,Lm15;
        then
A7:     m9+xaa1 <= len W by FINSEQ_3:25;
        xaa1+1 = x;
        then
A8:     e = W.(m+xaa1) by A1,A4,A5,Lm15;
        1 <= m9+xaa1 by A6,FINSEQ_3:25;
        hence e in W.edges() by A8,A7,Lm46;
      end;
      hence thesis by TARSKI:def 3;
    end;
    suppose
      not (m is odd & n is odd & m <= n & n <= len W);
      hence thesis by Def11;
    end;
  end;
  hence thesis;
end;
