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 Th12:
  len G.walkOf(v) = 1 & G.walkOf(v).1 = v & G.walkOf(v).first() =
  v & G.walkOf(v).last() = v & G.walkOf(v) is_Walk_from v,v
proof
  thus
A1: len G.walkOf(v) = 1 & G.walkOf(v).1 = v by FINSEQ_1:40;
  thus
 G.walkOf(v).first() = v;
  thus G.walkOf(v).last() = v by A1;
  hence thesis;
end;
