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
  G.walkOf(v).vertices() = {v}
proof
  now
    let x be object;
A1: 1 <= len G.walkOf(v) by ABIAN:12;
    hereby
      assume x in G.walkOf(v).vertices();
      then consider n being odd Element of NAT such that
A2:   n <= len G.walkOf(v) and
A3:   G.walkOf(v).n = x by Lm45;
A4:   1 <= n by ABIAN:12;
      n <= 1 by A2,Th12;
      then x = G.walkOf(v).1 by A3,A4,XXREAL_0:1;
      then x = v;
      hence x in {v} by TARSKI:def 1;
    end;
    assume x in {v};
    then
A5: x = v by TARSKI:def 1;
    G.walkOf(v).1 = v;
    hence x in G.walkOf(v).vertices() by A5,A1,Lm45,JORDAN12:2;
  end;
  hence thesis by TARSKI:2;
end;
