
theorem Th5:
  the carrier of Complex_l1_Space = the_set_of_l1ComplexSequences &
( for x be set holds x is VECTOR of Complex_l1_Space iff x is Complex_Sequence
  & seq_id x is absolutely_summable ) & 0.Complex_l1_Space = CZeroseq & ( for u
  be VECTOR of Complex_l1_Space holds u =seq_id u ) & ( for u,v be VECTOR of
Complex_l1_Space holds u+v =seq_id(u)+seq_id(v) ) & ( for p be Complex for u be
  VECTOR of Complex_l1_Space holds p*u =p(#)seq_id(u) ) & ( for u be VECTOR of
  Complex_l1_Space holds -u = -seq_id u & seq_id(-u) = -seq_id(u) ) & ( for u,v
  be VECTOR of Complex_l1_Space holds u-v =seq_id(u)-seq_id v ) & ( for v be
  VECTOR of Complex_l1_Space holds seq_id v is absolutely_summable ) & for v be
  VECTOR of Complex_l1_Space holds ||.v.|| = Sum |.seq_id v.|
proof
  set l1 =Complex_l1_Space;
A1: for x be set holds x is Element of l1 iff x is Complex_Sequence & seq_id
  x is absolutely_summable
  proof
    let x be set;
    x in the_set_of_ComplexSequences iff x is Complex_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_ComplexSequences by Lm1,
CLVECT_1:29;
    set L1=Linear_Space_of_ComplexSequences;
    set W = the_set_of_l1ComplexSequences;
    dom (the addF of L1) = [:the carrier of L1,the carrier of L1:] by
FUNCT_2:def 1;
    then
A3: dom ((the addF of Linear_Space_of_ComplexSequences)||W) =[:W,W:] by
RELAT_1:62,ZFMISC_1:96;
    u+v =((the addF of Linear_Space_of_ComplexSequences)||W).[u,v] by
CSSPACE:def 8
      .=u1+v1 by A3,FUNCT_1:47;
    hence thesis by CSSPACE:2;
  end;
A4: for p be Complex for u be VECTOR of l1 holds p*u =p(#)seq_id(u)
  proof
    let p be Complex;
    let u be VECTOR of l1;
    reconsider u1=u as VECTOR of Linear_Space_of_ComplexSequences by Lm1,
CLVECT_1:29;
    set L1=Linear_Space_of_ComplexSequences;
    set W = the_set_of_l1ComplexSequences;
    dom (the Mult of L1) = [:COMPLEX,the carrier of L1:] by FUNCT_2:def 1;
    then
A5: dom ((the Mult of Linear_Space_of_ComplexSequences) | [:COMPLEX,W :])
    =[:COMPLEX,W:] by RELAT_1:62,ZFMISC_1:96;
    reconsider p as Element of COMPLEX by XCMPLX_0:def 2;
    p*u =(the Mult of l1).[p,u] by CLVECT_1:def 1
      .=((the Mult of Linear_Space_of_ComplexSequences)|[:COMPLEX,W:]).[p,u]
    by CSSPACE:def 9
      .=(the Mult of Linear_Space_of_ComplexSequences).[p,u] by A5,FUNCT_1:47
      .=p*u1 by CLVECT_1:def 1;
    hence thesis by CSSPACE: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_ComplexSequences by Lm1,CLVECT_1:29;
    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 = (-1r)*u by Th4,CLVECT_1:3
      .= (-1r)(#)seq_id(u) by A4
      .= -seq_id(u) by COMSEQ_1:11;
    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;
  end;
A9: for v be VECTOR of l1 holds ||.v.|| = Sum |.seq_id v.|
      by Def2;
  0.l1 = 0.Linear_Space_of_ComplexSequences by CSSPACE:def 10
    .= CZeroseq;
  hence thesis by A1,A6,A2,A4,A7,A8,A9;
end;
