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 Th136:
  for G being _Graph holds G is loopless iff
  for v being object holds not ex e being object st e DJoins v,v,G
proof
  let G be _Graph;
  thus G is loopless implies
   for v being object holds not ex e being object st e DJoins v,v,G;
  assume A3: for v being object holds not ex e being object st e DJoins v,v,G;
  for v being object holds not ex e being object st e Joins v,v,G
  proof
    let v be object;
    given e being object such that
      A4: e Joins v,v,G;
    e DJoins v,v,G by A4;
    hence contradiction by A3;
  end;
  hence thesis by Th18;
end;
