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

theorem Th9:
  for p be Real st p >= 1 holds 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 #) is Subspace of
  Linear_Space_of_RealSequences
proof
  set V =Linear_Space_of_RealSequences;
  let p be Real such that
A1: 1 <= p;
  set lSpacep = 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 #);
  reconsider lSpacep as RealLinearSpace by A1,Th8;
  set w1= the RLSStruct of lSpacep;
A2: w1 is Subspace of V by A1,Th5;
  then
A3: the Mult of lSpacep = (the Mult of V) | [:REAL, the carrier of lSpacep
  :] by RLSUB_1:def 2;
  0.w1 = 0.V by A2,RLSUB_1:def 2;
  then
A4: 0.lSpacep = 0.V;
  the addF of lSpacep = (the addF of V)||the carrier of lSpacep by A2,
RLSUB_1:def 2;
  hence thesis by A4,A3,RLSUB_1:def 2;
end;
