
theorem Th2:
  the carrier of linfty_Space = the_set_of_BoundedRealSequences & (
for x be set holds x is VECTOR of linfty_Space iff x is Real_Sequence & seq_id
  x is bounded ) & 0.linfty_Space = Zeroseq & ( for u be VECTOR of linfty_Space
holds u = seq_id u ) & ( for u,v be VECTOR of linfty_Space holds u+v =seq_id(u)
+seq_id(v) ) &
( for r be Real for u be VECTOR of linfty_Space holds r*u =r(#)
seq_id(u) ) & ( for u be VECTOR of linfty_Space holds -u = -seq_id u & seq_id(-
  u) = -seq_id u ) & ( for u,v be VECTOR of linfty_Space holds u-v =seq_id(u)-
  seq_id(v) ) & ( for v be VECTOR of linfty_Space holds seq_id v is bounded ) &
  for v be VECTOR of linfty_Space holds ||.v.|| = upper_bound rng abs seq_id v
proof
  set l1 =linfty_Space;
A1: 0.l1 = 0.Linear_Space_of_RealSequences by RSSPACE:def 10
    .= Zeroseq;
A2: for x be set holds x is Element of l1 iff x is Real_Sequence & seq_id x
  is bounded
  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;
A3: 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 Lm3,
RLSUB_1:10;
    set L1=Linear_Space_of_RealSequences;
    set W = the_set_of_BoundedRealSequences;
    dom (the addF of L1) = [:the carrier of L1,the carrier of L1:] by
FUNCT_2:def 1;
    then
A4: dom ((the addF of Linear_Space_of_RealSequences)||W) =[:W,W:] by RELAT_1:62
;
    u+v =((the addF of Linear_Space_of_RealSequences)||W).[u,v] by
RSSPACE:def 8
      .=u1+v1 by A4,FUNCT_1:47;
    hence thesis by RSSPACE:2;
  end;
A5: 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 u1=u as VECTOR of Linear_Space_of_RealSequences by Lm3,
RLSUB_1:10;
    set L1=Linear_Space_of_RealSequences;
    set W = the_set_of_BoundedRealSequences;
    dom (the Mult of L1) = [:REAL,the carrier of L1:] by FUNCT_2:def 1;
    then
A6: dom ((the Mult of Linear_Space_of_RealSequences) | [:REAL,W:]) =[:
    REAL,W:] by RELAT_1:62,ZFMISC_1:96;
    reconsider r as Element of REAL by XREAL_0:def 1;
    r*u =((the Mult of Linear_Space_of_RealSequences)|[:REAL,W:]).[r,u]
    by RSSPACE:def 9
      .=r*u1 by A6,FUNCT_1:47;
    hence thesis by RSSPACE:3;
  end;
A7: 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 Lm3,RLSUB_1:10;
    hence thesis;
  end;
A8: 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 RLVECT_1:16
      .= -seq_id u by A5;
    hence thesis;
  end;
  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 A3
      .= seq_id u-seq_id v by A8;
  end;
  hence thesis by A2,A7,A3,A5,A8,A1,Def2;
end;
