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 Th18:
  G is loopless iff for v being object holds
    not ex e being object st e Joins v,v,G
proof
  thus G is loopless implies for v being object
   holds not ex e being object st e Joins v,v,G;
  assume
A1: for v being object holds not ex e being object st e Joins v,v,G;
  now
    given e being object such that
A2: e in the_Edges_of G & (the_Source_of G).e = (the_Target_of G).e;
    set v = (the_Source_of G).e;
    e Joins v,v,G by A2;
    hence contradiction by A1;
  end;
  hence thesis;
end;
