
theorem Th6:
  the carrier of l1_Space = the_set_of_l1RealSequences & ( for x be
  set holds x is VECTOR of l1_Space iff x is Real_Sequence & seq_id x is
  absolutely_summable ) & 0.l1_Space = Zeroseq & ( for u be VECTOR of l1_Space
  holds u =seq_id u ) & ( for u,v be VECTOR of l1_Space holds u+v =seq_id(u)+
seq_id(v) ) &
 ( for r be Real for u be VECTOR of l1_Space holds r*u =r(#)seq_id
  (u) ) & ( for u be VECTOR of l1_Space holds -u = -seq_id u & seq_id(-u) = -
seq_id(u) ) & ( for u,v be VECTOR of l1_Space holds u-v =seq_id(u)-seq_id v ) &
  ( for v be VECTOR of l1_Space holds seq_id v is absolutely_summable ) & for v
  be VECTOR of l1_Space holds ||.v.|| = Sum abs seq_id v
proof
  set l1 =l1_Space;
A1: for x be set holds x is Element of l1 iff x is Real_Sequence & seq_id x
  is absolutely_summable
  proof
    let x be set;
    x in the_set_of_RealSequences iff x is Real_Sequence by FUNCT_2:8,66;
    hence thesis by Def1;
  end;
A2: for u,v be VECTOR of l1 holds u+v =seq_id(u)+seq_id(v)
  proof
    let u,v be VECTOR of l1;
    reconsider u1=u, v1=v as VECTOR of Linear_Space_of_RealSequences by Lm1,
RLSUB_1:10;
    set L1=Linear_Space_of_RealSequences;
    set W = the_set_of_l1RealSequences;
    dom (the addF of L1) = [:the carrier of L1,the carrier of L1:] by
FUNCT_2:def 1;
    then dom ((the addF of Linear_Space_of_RealSequences)||W) =[:W,W:] by
RELAT_1:62,ZFMISC_1:96;
    then
A3: [u,v] in dom ((the addF of Linear_Space_of_RealSequences)||W) by
ZFMISC_1:87;
    u+v =((the addF of Linear_Space_of_RealSequences)||W).[u,v] by
RSSPACE:def 8
      .=u1+v1 by A3,FUNCT_1:47;
    hence thesis by RSSPACE:2;
  end;
A4: for r be Real for u be VECTOR of l1 holds r*u =r(#)seq_id(u)
  proof
    let r be Real;
    let u be VECTOR of l1;
    reconsider r as Element of REAL by XREAL_0:def 1;
    reconsider u1=u as VECTOR of Linear_Space_of_RealSequences by Lm1,
RLSUB_1:10;
    set L1=Linear_Space_of_RealSequences;
    set W = the_set_of_l1RealSequences;
    dom (the Mult of L1) = [:REAL,the carrier of L1:] by FUNCT_2:def 1;
    then dom ((the Mult of Linear_Space_of_RealSequences) | [:REAL,W:]) =[:
    REAL,W:] by RELAT_1:62,ZFMISC_1:96;
    then
A5: [r,u] in dom ((the Mult of Linear_Space_of_RealSequences)|[:REAL,W :]
    ) by ZFMISC_1:87;
    r*u =((the Mult of Linear_Space_of_RealSequences)|[:REAL,W:]).[r,u]
    by RSSPACE:def 9
      .=r*u1 by A5,FUNCT_1:47;
    hence thesis by RSSPACE:3;
  end;
A6: for u be VECTOR of l1 holds u =seq_id u
  proof
    let u be VECTOR of l1;
    u is VECTOR of Linear_Space_of_RealSequences by Lm1,RLSUB_1:10;
    hence thesis;
  end;
A7: for u be VECTOR of l1 holds -u =-seq_id u & seq_id(-u)=-seq_id u
  proof
    let u be VECTOR of l1;
    -u = (-1)*u by Th5,RLVECT_1:16
      .= (-1)(#)seq_id(u) by A4
      .= -seq_id(u) by SEQ_1:17;
    hence thesis;
  end;
A8: for u,v be VECTOR of l1 holds u-v =seq_id(u)-seq_id(v)
  proof
    let u,v be VECTOR of l1;
    thus u-v = seq_id(u)+seq_id(-v) by A2
      .= seq_id(u)+(-seq_id(v)) by A7
      .= seq_id(u)-seq_id(v) by SEQ_1:11;
  end;
A9: for v be VECTOR of l1 holds ||.v.|| = Sum abs seq_id v
 by Def2;
  0.l1 = 0.Linear_Space_of_RealSequences by RSSPACE:def 10
  .= Zeroseq;
  hence thesis by A1,A6,A2,A4,A7,A8,A9;
end;
