
theorem
  for G being _Graph holds G is locally-finite iff
    for v being Vertex of G
    holds v.inDegree() is finite & v.outDegree() is finite
proof
  let G be _Graph;
  hereby
    assume A1: G is locally-finite;
    let v be Vertex of G;
    v.degree() is finite by A1, Th23;
    hence v.inDegree() is finite & v.outDegree() is finite;
  end;
  assume A2: for v being Vertex of G holds
    v.inDegree() is finite & v.outDegree() is finite;
  now
    let v be Vertex of G;
    v.inDegree() is finite & v.outDegree() is finite by A2;
    hence v.degree() is finite;
  end;
  hence thesis by Th23;
end;
