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(1,n).vertexSeq() c= W.vertexSeq()
proof
  now
    per cases;
    suppose
A1:   n is odd & 1 <= n & n <= len W;
      set f = W.cut(1,n).vertexSeq();
      now
        let v be object;
        assume
A2:     v in f;
        then consider x,y being object such that
A3:     v = [x,y] by RELAT_1:def 1;
A4:     y = f.x by A2,A3,FUNCT_1:1;
A5:     x in dom f by A2,A3,FUNCT_1:1;
        then reconsider x as Element of NAT;
A6:     x <= len f by A5,FINSEQ_3:25;
A7:     2*x-1 in dom W.cut(1,n) by A5,Th71;
        then 2*x-1 <= len W.cut(1,n) by FINSEQ_3:25;
        then 2*x-1 <= n by A1,Lm22;
        then
A8:     2*x-1 <= len W by A1,XXREAL_0:2;
        1 <= 2*x-1 by A7,FINSEQ_3:25;
        then 2*x-1 in dom W by A7,A8,FINSEQ_3:25;
        then
A9:     x in dom W.vertexSeq() by Th71;
        then
A10:    x <= len W.vertexSeq() by FINSEQ_3:25;
        1 <= x by A5,FINSEQ_3:25;
        then y = W.cut(1,n).(2*x-1) by A4,A6,Def14;
        then
A11:    y = W.(2*x-1) by A1,A7,Lm23;
        1 <= x by A9,FINSEQ_3:25;
        then W.vertexSeq().x = y by A11,A10,Def14;
        hence v in W.vertexSeq() by A3,A9,FUNCT_1:1;
      end;
      hence thesis by TARSKI:def 3;
    end;
    suppose
      not (n is odd & 1 <= n & n <= len W);
      hence thesis by Def11;
    end;
  end;
  hence thesis;
end;
