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
  x in W.vertices() & y in W.vertices() implies ex W9 being Walk of G st
  W9 is_Walk_from x,y
proof
  assume that
A1: x in W.vertices() and
A2: y in W.vertices();
  consider m being odd Element of NAT such that
A3: m <= len W and
A4: W.m = x by A1,Lm45;
  consider n being odd Element of NAT such that
A5: n <= len W and
A6: W.n = y by A2,Lm45;
  now
    per cases;
    suppose
      m <= n;
      then W.cut(m,n) is_Walk_from x, y by A4,A5,A6,Lm16;
      hence thesis;
    end;
    suppose
      n <= m;
      then W.cut(n,m) is_Walk_from y, x by A3,A4,A6,Lm16;
      then W.cut(n,m).reverse() is_Walk_from x, y by Th22;
      hence thesis;
    end;
  end;
  hence thesis;
end;
