reserve x1,x2,y1,a,b,c for Real;

theorem Th14:
  for p be Real st p >= 1 for lp be non empty NORMSTR st lp =
  NORMSTR (# the_set_of_RealSequences_l^p, Zero_(the_set_of_RealSequences_l^p,
    Linear_Space_of_RealSequences), Add_(the_set_of_RealSequences_l^p,
    Linear_Space_of_RealSequences), Mult_(the_set_of_RealSequences_l^p,
    Linear_Space_of_RealSequences), l_norm^p #) holds lp is
  reflexive discerning RealNormSpace-like
proof
  let p be Real such that
A1: p >=1;
  let lp be non empty NORMSTR;
  assume
A2: lp = NORMSTR (# the_set_of_RealSequences_l^p, Zero_(
    the_set_of_RealSequences_l^p,Linear_Space_of_RealSequences), Add_(
    the_set_of_RealSequences_l^p,Linear_Space_of_RealSequences), Mult_(
    the_set_of_RealSequences_l^p,Linear_Space_of_RealSequences), l_norm^p #);
  hence ||.0.lp.|| = 0 by A1,Th13;
  for x, y being Point of lp, a be Real
   holds ( ||.x.|| = 0 iff x = 0.lp )
  & 0 <= ||.x.|| & ||.x+y.|| <= ||.x.|| + ||.y.|| & ||.(a*x).|| = |.a.| * ||.x
  .|| by A1,Th13,A2;
  hence thesis by NORMSP_1:def 1;
end;
