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;
