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 Th137:
  for G being _Graph holds G is loopless iff
    for v being object holds not ex e being object st e SJoins {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 SJoins {v},{v},G by Th131, Th18;
  assume A1: for v being object
    holds not ex e being object st e SJoins {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
      A2: e Joins v,v,G;
    v in {v} by TARSKI:def 1;
    then e SJoins {v},{v},G by A2;
    hence contradiction by A1;
  end;
  hence thesis by Th18;
end;
