reserve GS for GraphStruct;
reserve G,G1,G2,G3 for _Graph;
reserve e,x,x1,x2,y,y1,y2,E,V,X,Y for set;
reserve n,n1,n2 for Nat;
reserve v,v1,v2 for Vertex of G;

theorem
  for G being _Graph, v being Vertex of G, e being object
  st v is endvertex holds not e Joins v,v,G
proof
  let G be _Graph, v be Vertex of G, e be object;
  assume v is endvertex;
  then consider e0 being object such that
    A1: v.edgesInOut() = {e0} & not e0 Joins v,v,G;
  assume A2: e Joins v,v,G;
  then e in v.edgesInOut() by Th62;
  hence contradiction by A1, A2, TARSKI:def 1;
end;
