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 Th138:
  for G being _Graph holds G is loopless iff
    for v being object holds not ex e being object st e DSJoins {v},{v},G
proof
  let G be _Graph;
  hereby
    assume A1: G is loopless;
    let v be object;
    given e being object such that
      A2: e DSJoins {v},{v},G;
    e SJoins {v},{v},G by A2;
    hence contradiction by A1, Th137;
  end;
  assume A3: for v being object holds
    not ex e being object st e DSJoins {v},{v},G;
  for v being object holds not ex e being object st e SJoins {v},{v},G
  proof
    let v be object;
    given e being object such that
      A4: e SJoins {v},{v},G;
    e DSJoins {v},{v},G by A4;
    hence contradiction by A3;
  end;
  hence thesis by Th137;
end;
