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

theorem Th10:
  for p be Real
  st 1 <=p holds 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 the carrier of lp =
the_set_of_RealSequences_l^p & ( for x be set holds x is VECTOR of lp iff x is
Real_Sequence & seq_id(x) rto_power p is summable ) & 0.lp = Zeroseq & ( for x
  be VECTOR of lp holds x =seq_id(x) ) & ( for x,y be VECTOR of lp holds x+y =
seq_id(x)+seq_id(y) ) &
 ( for r be Real for x be VECTOR of lp holds r*x = r(#)
  seq_id(x) ) & ( for x be VECTOR of lp holds -x = -seq_id(x) & seq_id(-x) = -
  seq_id(x) ) & ( for x,y be VECTOR of lp holds x-y =seq_id(x)-seq_id(y) ) & (
  for x be VECTOR of lp holds seq_id(x) rto_power p is summable ) & for x be
  VECTOR of lp holds ||.x.|| = ( Sum(seq_id(x) rto_power p) ) to_power (1/p)
proof
  let p be Real such that
A1: 1<= p;
  let lp be non empty NORMSTR such that
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 #);
A3: for x be VECTOR of lp holds ||.x.|| = ( Sum(seq_id(x) rto_power p) )
  to_power (1/p) by A2,Def3;
A4: for x,y be VECTOR of lp holds x+y =seq_id(x)+seq_id(y)
  proof
    let x,y be VECTOR of lp;
A5: lp is Subspace of Linear_Space_of_RealSequences by A1,A2,Th9;
    then reconsider x1=x as VECTOR of Linear_Space_of_RealSequences by
RLSUB_1:10;
    reconsider y1=y as VECTOR of Linear_Space_of_RealSequences by A5,RLSUB_1:10
;
    set L1=Linear_Space_of_RealSequences;
    set W = the_set_of_RealSequences_l^p;
    dom (the addF of L1) = [:the carrier of L1,the carrier of L1:] by
FUNCT_2:def 1;
    then
A6: dom ((the addF of Linear_Space_of_RealSequences)||W) =[:W,W:] by RELAT_1:62
;
    W is linearly-closed by A1,Th4;
    then x+y= ((the addF of Linear_Space_of_RealSequences)||W).[x,y] by A2,
RSSPACE:def 8
      .=x1+y1 by A2,A6,FUNCT_1:47;
    hence thesis by RSSPACE:2;
  end;
A7: for r be Real for x be VECTOR of lp holds r*x =r(#)seq_id(x)
  proof
    set W = the_set_of_RealSequences_l^p;
    set L1=Linear_Space_of_RealSequences;
    let r be Real;
     reconsider r as Element of REAL by XREAL_0:def 1;
    let x be VECTOR of lp;
    dom (the Mult of L1) = [:REAL,the carrier of L1:] by FUNCT_2:def 1;
    then
A8: dom ((the Mult of Linear_Space_of_RealSequences) | [:REAL,W:]) =[:
    REAL,W:] by RELAT_1:62,ZFMISC_1:96;
    lp is Subspace of Linear_Space_of_RealSequences by A1,A2,Th9;
    then reconsider x1=x as VECTOR of Linear_Space_of_RealSequences by
RLSUB_1:10;
    W is linearly-closed by A1,Th4;
    then r*x =((the Mult of Linear_Space_of_RealSequences)|[:REAL,W:]).[r,x]
    by A2,RSSPACE:def 9
      .=r*x1 by A2,A8,FUNCT_1:47;
    hence thesis by RSSPACE:3;
  end;
  the_set_of_RealSequences_l^p is linearly-closed by A1,Th4;
  then
A9: 0.lp = 0.Linear_Space_of_RealSequences by A2,RSSPACE:def 10
    .= Zeroseq;
A10: for x be set holds x is Element of lp iff x is Real_Sequence & seq_id(x)
  rto_power p is summable
  proof
    let x be set;
    x in the_set_of_RealSequences_l^p iff seq_id(x) rto_power p is
    summable & x in the_set_of_RealSequences by A1,Def2;
    hence thesis by A2,FUNCT_2:8,66;
  end;
A11: for x be set holds x is VECTOR of lp implies x =seq_id(x)
  proof
    let x be set;
    x in the_set_of_RealSequences iff x is Real_Sequence by FUNCT_2:8,66;
    hence thesis by A1,A2,Def2;
  end;
A12: for x be VECTOR of lp holds -x =-seq_id(x) & seq_id(-x)=-seq_id(x)
  proof
    let x be VECTOR of lp;
    lp is Subspace of Linear_Space_of_RealSequences by A1,A2,Th9;
    then -x = (-1)*x by RLVECT_1:16
      .= -seq_id(x) by A7;
    hence thesis;
  end;
  for x,y be VECTOR of lp holds x-y =seq_id(x)-seq_id(y)
  proof
    let x,y be VECTOR of lp;
    thus x-y = seq_id(x)+seq_id(-y) by A4
      .= seq_id(x)-seq_id(y) by A12;
  end;
  hence thesis by A2,A10,A11,A9,A4,A7,A12,A3;
end;
