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

theorem Th13:
  for p be Real st 1 <= p 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 #)
 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.||
proof
  let p be Real such that
A1: 1<= p;
A2: 1/p > 0 by A1,XREAL_1:139;
  let lp be non empty NORMSTR such that
A3: 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 #);
  let x, y be Point of lp;
A4: (seq_id(y) rto_power p) is summable by A1,A3,Th10;
A5: ||.y.|| = ( Sum(seq_id(y) rto_power p) ) to_power (1/p) by A3,Def3;
A6: ||.x.|| = ( Sum(seq_id(x) rto_power p) ) to_power (1/p) by A3,Def3;
A7: now
A8: ||.x.|| = ( Sum(seq_id(x) rto_power p) ) to_power (1/p) by A3,Def3;
A9: x in the_set_of_RealSequences by A1,A3,Def2;
    assume
A10: ||.x.|| = 0;
    seq_id(x) rto_power p is summable by A1,A3,Th10;
    then for n be Nat holds 0 = (seq_id(x)).n by A1,A10,A8,Th12;
    hence x = Zeroseq by A9,RSSPACE:5
      .=0.lp by A1,A3,Th10;
  end;
A11: seq_id(x) rto_power p is summable by A1,A3,Def2;
A12: ( Sum(seq_id(x) rto_power p) ) to_power (1/p) = ||.x.|| by A3,Def3;
A13: now
    assume x=0.lp;
    then x=Zeroseq by A1,A3,Th10;
    then
A14: for n be Nat holds (seq_id(x)).n=0 by RSSPACE:4;
    thus ||.x.|| = ( Sum(seq_id(x) rto_power p) ) to_power (1/p) by A3,Def3
      .= 0 by A1,A14,Th11;
  end;
  let a be Real;
A15: for n be Nat holds 0 <= (seq_id(x) rto_power p).n
  proof
    set xp=seq_id(x) rto_power p;
    let n be Nat;
A16: 0 < |.(seq_id(x)).n.| or 0=|.(seq_id(x)).n.| by COMPLEX1:46;
    xp.n=|.(seq_id(x)).n.| to_power p by Def1;
    hence thesis by A1,A16,POWER:34,def 2;
  end;
  ((seq_id(x))+(seq_id(y))) =seq_id((seq_id(x))+(seq_id(y)))
    .=seq_id(x+y) by A1,A3,Th10;
  then
A17: Sum((((seq_id(x))+(seq_id(y))) rto_power p)) to_power (1/p)
     = ||.x + y.|| by A3,Def3;
  (seq_id(x) rto_power p) is summable by A1,A3,Th10;
  then
A18: ||.x + y.|| <= ||.x.|| + ||.y.|| by A1,A6,A5,A17,A4,Th3;
A19: ||.x.|| = ( Sum(seq_id(x) rto_power p) ) to_power (1/p) by A3,Def3;
  ||.(a*x).|| = ( Sum(seq_id(a*x) rto_power p) ) to_power (1/p) by A3,Def3
    .=(|.a.|)* ||.x.|| by A1,A3,A19,Lm7;
  hence thesis by A2,A7,A13,A15,A11,A12,A18,Lm1,SERIES_1:18;
end;
