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

theorem Th16:
  for p be Real st 1 <=p for lp be RealNormSpace 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 for vseq be sequence of lp st
  vseq is Cauchy_sequence_by_Norm holds vseq is convergent
proof
  let p be Real such that
A1: 1<= p;
A2: 1/p > 0 by A1,XREAL_1:139;
  let lp be RealNormSpace 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 vseq be sequence of lp such that
A4: vseq is Cauchy_sequence_by_Norm;
  defpred P[object,object] means
   ex i be Nat st $1=i & ex rseqi be
  Real_Sequence st (for n be Nat holds rseqi.n=(seq_id(vseq.n)).i) &
  rseqi is convergent & $2 = lim rseqi;
A5: p * (1/p) = (p*1)/p by XCMPLX_1:74
    .= 1 by A1,XCMPLX_1:60;
A6: for x be object st x in NAT ex y be object st y in REAL & P[x,y]
  proof
    let x be object;
    assume x in NAT;
    then reconsider i=x as Nat;
    deffunc F(Nat) = (seq_id(vseq.$1)).i;
    consider rseqi be Real_Sequence such that
A7: for n be Nat holds rseqi.n= F(n) from SEQ_1:sch 1;
     reconsider lr = lim rseqi as Element of REAL by XREAL_0:def 1;
    take lr;
    now
      let e1 be Real such that
A8:   e1 > 0;
      reconsider e=e1 as Real;
      thus ex k be Nat st for m be Nat st k <= m holds
      |.rseqi.m -rseqi.k.| < e1
      proof
        consider k be Nat such that
A9:     for n, m be Nat st n >= k & m >= k holds ||.(vseq.
        n) - (vseq.m).|| < e by A4,A8,RSSPACE3:8;
        for m being Nat st k <= m holds |.rseqi.m-rseqi.k.| < e
        proof
          let m be Nat such that
A10:      k<=m;
A11:      ||.(vseq.m) - (vseq.k).|| =
          ( Sum(seq_id((vseq.m)-(vseq.k)) rto_power p) ) to_power (1/p)
          by A3,Def3;
          then
          ( Sum(seq_id((vseq.m)-(vseq.k)) rto_power p) ) to_power (1/p) <
          e by A9,A10;
          then
A12:      (( Sum(seq_id((vseq.m)-(vseq.k)) rto_power p) ) to_power (1/p))
          to_power p < e to_power p by A1,A11,Th1,NORMSP_1:4;
A13:      now
            let i be Nat;
            (seq_id((vseq.m)-(vseq.k)) rto_power p).i =|.(seq_id((vseq.
            m)-(vseq.k))).i.| to_power p by Def1;
            hence (seq_id((vseq.m)-(vseq.k)) rto_power p).i >= 0 by A1,Lm1,
COMPLEX1:46;
          end;
          reconsider vsem = (vseq.m), vsek = (vseq.k) as VECTOR of lp;
A14:      now
            let a,b,c be Real such that
A15:        a =0 and
A16:        b>0 and
A17:        c>0;
            b*c >0 by A16,A17,XREAL_1:129;
            hence a to_power (b *c) =0 by A15,POWER:def 2;
          end;
A18:      now
            let a,b,c be Real such that
A19:        a =0 and
A20:        b>0 and
A21:        c>0;
            a to_power b =0 by A19,A20,POWER:def 2;
            hence (a to_power b) to_power c =0 by A21,POWER:def 2;
          end;
A22:      now
            let a,b,c be Real such that
A23:        a =0 and
A24:        b>0 and
A25:        c>0;
            thus (a to_power b) to_power c = 0 by A18,A23,A24,A25
              .= a to_power (b *c) by A14,A23,A24,A25;
          end;
A26:      for a,b,c be Real st a>=0 & b>0 & c>0 holds (a to_power b)
          to_power c = a to_power (b*c)
          proof
            let a,b,c be Real such that
A27:        a>=0 and
A28:        b>0 and
A29:        c>0;
            a >0 or a=0 by A27;
            hence thesis by A22,A28,A29,POWER:33;
          end;
          (seq_id((vseq.m)-(vseq.k)) rto_power p).i =|.(seq_id((vseq.m)
          -(vseq.k))).i.| to_power p by Def1;
          then
A30:      ((seq_id((vseq.m)-(vseq.k)) rto_power p).i) to_power (1/p) =
|.(seq_id((vseq.m)-(vseq.k))).i.| to_power (1) by A1,A2,A5,A26,COMPLEX1:46
            .=|.(seq_id((vseq.m)-(vseq.k))).i.| by POWER:25;
A31:      rseqi.m=(seq_id(vseq.m)).i by A7;
A32:      (seq_id((vseq.m)-(vseq.k)) rto_power p) is summable by A1,A3,Th10;
          then
A33:      (seq_id((vseq.m)-(vseq.k)) rto_power p).i <= Sum(seq_id((vseq.m
          )-(vseq.k)) rto_power p) by A13,RSSPACE2:3;
          (( Sum(seq_id((vseq.m)-(vseq.k)) rto_power p) ) to_power (1/p))
to_power p =( Sum(seq_id((vseq.m)-(vseq.k)) rto_power p) ) to_power ((1/p)*p)
          by A1,A2,A32,A13,HOLDER_1:2,SERIES_1:18
            .= ( Sum(seq_id((vseq.m)-(vseq.k)) rto_power p) ) by A5,POWER:25;
          then
A34:      (seq_id((vseq.m)-(vseq.k)) rto_power p).i <e to_power p by A12,A33,
XXREAL_0:2;
A35:      rseqi.k=(seq_id(vseq.k)).i by A7;
          (vsem)-(vsek) =(seq_id(vsem)) - (seq_id(vsek)) by A1,A3,Th10;
          then
A36:      |.(seq_id((vseq.m)-(vseq.k))).i.|=
          |.(seq_id(vseq.m)).i+(-(seq_id(vseq.k))).i.| by SEQ_1:7
            .=|.(seq_id(vseq.m)).i +-(seq_id(vseq.k)).i.| by SEQ_1:10
            .=|.rseqi.m -rseqi.k.| by A31,A35;
          (e to_power p) to_power (1/p) = e to_power ((1/p) * p) by A8,POWER:33
            .= e to_power(1) by A1,XCMPLX_1:106
            .=e by POWER:25;
          hence thesis by A2,A13,A30,A34,A36,Th1;
        end;
        hence thesis;
      end;
    end;
    then rseqi is convergent by SEQ_4:41;
    hence thesis by A7;
  end;
  consider f be sequence of REAL such that
A37: for x be object st x in NAT holds P[x,f.x] from FUNCT_2:sch 1(A6);
  reconsider tseq=f as Real_Sequence;
A38: now
    let i be Nat;
    reconsider x=i as set;
A39:  i in NAT by ORDINAL1:def 12;
    ex i0 be Nat st x=i0 & ex rseqi be Real_Sequence st ( for
n be Nat holds rseqi.n=(seq_id(vseq.n)).i0 ) & rseqi is convergent &
    f.x=lim rseqi by A37,A39;
    hence
    ex rseqi be Real_Sequence st ( for n be Nat holds rseqi.n=
    (seq_id(vseq.n)).i ) & rseqi is convergent & tseq.i=lim rseqi;
  end;
A40: for e be Real st e >0
  ex k be Nat st for n be Nat
st n >= k holds ((seq_id(tseq)-seq_id(vseq.n)) rto_power p ) is summable & Sum(
  (seq_id(tseq)-seq_id(vseq.n)) rto_power p) < e
  proof
A41: for n be Nat for i be Nat holds for rseq be
Real_Sequence st ( for m be Nat holds rseq.m = Partial_Sums ((seq_id
    ((vseq.m)-(vseq.n))) rto_power p).i ) holds rseq is convergent & lim rseq =
    Partial_Sums ((seq_id(tseq)-seq_id(vseq.n)) rto_power p) .i
    proof
      let n be Nat;
      defpred P[Nat] means
for rseq be Real_Sequence st for m be
      Nat holds rseq.m= Partial_Sums ((seq_id((vseq.m)-(vseq.n)))
rto_power p).$1 holds rseq is convergent & lim rseq = Partial_Sums ((seq_id(
      tseq)-seq_id(vseq.n)) rto_power p).$1;
A42:  for m,k be Nat holds seq_id((vseq.m) - (vseq.k)) =
      seq_id(vseq.m)-seq_id(vseq.k)
      proof
        let m,k be Nat;
        seq_id((vseq.m) - (vseq.k)) = seq_id(seq_id((vseq.m))-seq_id((
        vseq.k))) by A1,A3,Th10;
        hence thesis;
      end;
      now
        let i be Nat such that
A43:    for rseq be Real_Sequence st ( for m be Nat holds
rseq.m= Partial_Sums ((seq_id((vseq.m)-(vseq.n))) rto_power p).i ) holds rseq
is convergent & lim rseq =Partial_Sums ((seq_id(tseq)-seq_id(vseq.n)) rto_power
        p).i;
        thus for rseq be Real_Sequence st ( for m be Nat holds rseq
.m = Partial_Sums ((seq_id((vseq.m)-(vseq.n))) rto_power p).(i+1) ) holds rseq
is convergent & lim rseq =Partial_Sums ((seq_id(tseq)-seq_id(vseq.n)) rto_power
        p).(i+1)
        proof
          deffunc F(Nat) = Partial_Sums ((seq_id((vseq.$1)-(vseq.n)
          )) rto_power p).i;
          consider rseqb be Real_Sequence such that
A44:      for m be Nat holds rseqb.m = F(m) from SEQ_1:
          sch 1;
          consider rseq0 be Real_Sequence such that
A45:      for m be Nat holds rseq0.m=(seq_id(vseq.m)).(i+ 1) and
A46:      rseq0 is convergent and
A47:      tseq.(i+1)=lim rseq0 by A38;
          let rseq be Real_Sequence such that
A48:      for m be Nat holds rseq.m = Partial_Sums ((
          seq_id((vseq.m)-(vseq.n))) rto_power p).(i+1);
A49:      now
            let m be Nat;
            thus rseq.m = Partial_Sums ((seq_id((vseq.m)-(vseq.n))) rto_power
            p).(i+1) by A48
              .=((seq_id((vseq.m)-(vseq.n))) rto_power p).(i+1) +
Partial_Sums((seq_id((vseq.m)-(vseq.n))) rto_power p).i by SERIES_1:def 1
              .=((seq_id(vseq.m)-seq_id(vseq.n)) rto_power p).(i+1) +
            Partial_Sums((seq_id((vseq.m)-(vseq.n))) rto_power p).i by A42
              .=((seq_id(vseq.m)-seq_id(vseq.n)) rto_power p).(i+1) + rseqb.
            m by A44
              .= |.(seq_id(vseq.m)+-seq_id(vseq.n)).(i+1).| to_power p +
            rseqb.m by Def1
              .= |.(seq_id(vseq.m)).(i+1)+(-seq_id(vseq.n)).(i+1).|
            to_power p + rseqb.m by SEQ_1:7
              .= |.(seq_id(vseq.m)).(i+1)+-(seq_id(vseq.n)).(i+1).|
            to_power p + rseqb.m by SEQ_1:10
              .= |.(seq_id(vseq.m)).(i+1)-(seq_id(vseq.n)).(i+1).| to_power
            p + rseqb.m
              .= |.rseq0.m-(seq_id(vseq.n)).(i+1).| to_power p + rseqb.m by
A45;
          end;
A50:      lim rseqb = Partial_Sums ((seq_id(tseq)-seq_id(vseq.n))
          rto_power p).i by A43,A44;
A51:      rseqb is convergent by A43,A44;
          then lim rseq = |.lim rseq0-(seq_id(vseq.n)).(i+1).| to_power p +
          lim rseqb by A1,A46,A49,Lm9
            .= |.tseq.(i+1)+-(seq_id(vseq.n)).(i+1).| to_power p + lim
          rseqb by A47
            .= |.tseq.(i+1)+(-seq_id(vseq.n)).(i+1).| to_power p + lim
          rseqb by SEQ_1:10
            .= (|.(tseq-(seq_id(vseq.n))).(i+1).| to_power p) + lim rseqb
          by SEQ_1:7
            .= ((tseq-(seq_id(vseq.n))) rto_power p).(i+1) + Partial_Sums ((
          seq_id(tseq)-seq_id(vseq.n)) rto_power p).i by A50,Def1
            .=Partial_Sums ((seq_id(tseq)-seq_id(vseq.n)) rto_power p).(i+1)
          by SERIES_1:def 1;
          hence thesis by A1,A51,A46,A49,Lm9;
        end;
      end;
      then
A52:  for i be Nat st P[i] holds P[i+1];
      now
        let rseq be Real_Sequence such that
A53:    for m be Nat holds rseq.m= Partial_Sums ((seq_id((
        vseq.m)-(vseq.n))) rto_power p).0;
        thus rseq is convergent & lim rseq = Partial_Sums ((seq_id(tseq)-
        seq_id(vseq.n)) rto_power p).0
        proof
          consider rseq0 be Real_Sequence such that
A54:      for m be Nat holds rseq0.m=(seq_id(vseq.m)).0 and
A55:      rseq0 is convergent and
A56:      tseq.0=lim rseq0 by A38;
A57:      for m being Nat holds rseq.m = |.rseq0.m -(seq_id(
          vseq.n)).0 .| to_power p
          proof
            let m be Nat;
            rseq.m =Partial_Sums ((seq_id((vseq.m)-(vseq.n))) rto_power p
            ).0 by A53
              .= ((seq_id((vseq.m)-(vseq.n))) rto_power p).0 by SERIES_1:def 1
              .= ((seq_id(vseq.m)-seq_id(vseq.n)) rto_power p).0 by A42
              .=|.(seq_id(vseq.m)+-seq_id(vseq.n)).0 .| to_power p by Def1
              .=|.(seq_id(vseq.m)).0+(-seq_id(vseq.n)).0 .| to_power p by
SEQ_1:7
              .=|.(seq_id(vseq.m)).0+-(seq_id(vseq.n)).0 .| to_power p by
SEQ_1:10
              .=|.(seq_id(vseq.m)).0-(seq_id(vseq.n)).0 .| to_power p;
            hence thesis by A54;
          end;
          then lim rseq = |.lim(rseq0) -(seq_id(vseq.n)).0 .| to_power p by A1
,A55,Lm8
            .= |.tseq.0+-(seq_id(vseq.n)).0 .| to_power p by A56
            .=|.tseq.0+(-(seq_id(vseq.n))).0 .| to_power p by SEQ_1:10
            .=|.(tseq-(seq_id((vseq.n)))).0 .| to_power p by SEQ_1:7
            .=((seq_id(tseq)+-(seq_id(vseq.n))) rto_power p).0 by Def1
            .=Partial_Sums ((seq_id(tseq)-(seq_id(vseq.n))) rto_power p).0
          by SERIES_1:def 1;
          hence thesis by A1,A55,A57,Lm8;
        end;
      end;
      then
A58:  P[0];
      for i be Nat holds P[i] from NAT_1:sch 2(A58,A52);
      hence thesis;
    end;
    let e2 be Real such that
A59: e2 >0;
    set e=e2/2;
    reconsider e1=e to_power (1/p) as Real;
    e >0 by A59,XREAL_1:215;
    then e1>0 by POWER:34;
    then consider k be Nat such that
A60: for n, m be Nat st n >= k & m >= k holds ||.(vseq.n) -
    (vseq.m).|| < e1 by A4,RSSPACE3:8;
A61: for m,n be Nat st n >= k & m >= k holds ((seq_id((vseq.m)-
(vseq.n)) rto_power p) is summable & Sum( (seq_id((vseq.m)-(vseq.n)) rto_power
    p) ) < e & for i be Nat holds 0 <= ((seq_id((vseq.m)-(vseq.n)))
    rto_power p).i)
    proof
      let m,n be Nat such that
A62:  n >= k and
A63:  m >= k;
      ||.(vseq.m) - (vseq.n).|| < e1 by A60,A62,A63;
      then (the normF of lp).((vseq.m)-(vseq.n))<e1;
      then
A64:  (( Sum(seq_id((vseq.m)-(vseq.n)) rto_power p) ) to_power (1/p)) <
      e1 by A3,Def3;
A65:  for i be Nat holds (seq_id((vseq.m)-(vseq.n)) rto_power
      p).i >= 0
      proof
        let i be Nat;
        (seq_id((vseq.m)-(vseq.n)) rto_power p).i =|.(seq_id((vseq.m)-(
        vseq.n))).i.| to_power p by Def1;
        hence thesis by A1,Lm1,COMPLEX1:46;
      end;
A66:  (seq_id((vseq.m)-(vseq.n)) rto_power p) is summable by A1,A3,Th10;
      then
A67:  (( Sum(seq_id((vseq.m)-(vseq.n)) rto_power p) ) to_power (1/p)) >=0
      by A2,A65,Lm1,SERIES_1:18;
A68:  e1 to_power p = e to_power ((1/p) * p) by A59,POWER:33,XREAL_1:215
        .= e to_power(1) by A1,XCMPLX_1:106
        .= e by POWER:25;
      (( Sum(seq_id((vseq.m)-(vseq.n)) rto_power p) ) to_power (1/p))
to_power p =( Sum(seq_id((vseq.m)-(vseq.n)) rto_power p) ) to_power ((1/p)*p)
      by A1,A2,A65,A66,HOLDER_1:2,SERIES_1:18
        .= ( Sum(seq_id((vseq.m)-(vseq.n)) rto_power p) ) by A5,POWER:25;
      hence thesis by A1,A3,A65,A64,A68,A67,Th1,Th10;
    end;
A69: e2 >e by A59,XREAL_1:216;
    for n be Nat st n >= k holds ((seq_id(tseq)-seq_id(vseq.n
)) rto_power p ) is summable & Sum ((seq_id(tseq)-seq_id(vseq.n)) rto_power p )
    < e2
    proof
      let n be Nat such that
A70:  n >= k;
A71:  for i be Nat st 0 <= i holds Partial_Sums ((seq_id(tseq
      )-seq_id(vseq.n)) rto_power p).i <=e
      proof
        let i be Nat such that
        0 <=i;
        deffunc F(Nat)= Partial_Sums ((seq_id((vseq.$1)-(vseq.n)))
        rto_power p).i;
        consider rseq be Real_Sequence such that
A72:    for m be Nat holds rseq.m = F(m) from SEQ_1:sch 1;
A73:    for m be Nat st m >= k holds rseq.m <= e
        proof
          let m be Nat;
A74:      rseq.m = Partial_Sums ((seq_id((vseq.m)-(vseq.n))) rto_power p
          ). i by A72;
          assume
A75:      m >= k;
          then
A76:      for i be Nat holds 0 <= ((seq_id((vseq.m)-(vseq.n))
          ) rto_power p).i by A61,A70;
          (seq_id((vseq.m)-(vseq.n)) rto_power p) is summable by A61,A70,A75;
          then
A77:      Partial_Sums ((seq_id((vseq.m)-(vseq.n))) rto_power p) .i <=
          Sum( (seq_id((vseq.m)-(vseq.n)) rto_power p) ) by A76,RSSPACE2:3;
          Sum( (seq_id((vseq.m)-(vseq.n)) rto_power p) ) < e by A61,A70,A75;
          hence thesis by A77,A74,XXREAL_0:2;
        end;
A78:    lim rseq = Partial_Sums ((seq_id(tseq)-seq_id(vseq.n)) rto_power
        p) .i by A41,A72;
        rseq is convergent by A41,A72;
        hence thesis by A78,A73,RSSPACE2:5;
      end;
      now
        take e2=e+1;
A79:    e2>e by XREAL_1:29;
        let i be Nat;
        Partial_Sums ((seq_id(tseq)-seq_id(vseq.n)) rto_power p) .i <=e
        by A71,NAT_1:2;
        hence Partial_Sums ((seq_id(tseq)-seq_id(vseq.n)) rto_power p) .i <e2
        by A79,XXREAL_0:2;
      end;
      then
A80:  Partial_Sums ((seq_id(tseq)-seq_id(vseq.n)) rto_power p) is
      bounded_above by SEQ_2:def 3;
A81:  Sum ((seq_id(tseq)-seq_id(vseq.n)) rto_power p) = lim Partial_Sums
      ((seq_id(tseq)-seq_id(vseq.n)) rto_power p) by SERIES_1:def 3;
A82:  for i be Nat holds 0 <= ((seq_id(tseq)-seq_id(vseq.n))
      rto_power p) .i
      proof
        let i be Nat;
        ((seq_id(tseq)-seq_id(vseq.n)) rto_power p) .i =(|.(seq_id(
        tseq)-seq_id(vseq.n)).i.|) to_power p by Def1;
        hence thesis by A1,Lm1,COMPLEX1:46;
      end;
      then ((seq_id(tseq)-seq_id(vseq.n)) rto_power p) is summable by A80,
SERIES_1:17;
      then Partial_Sums((seq_id(tseq)-seq_id(vseq.n)) rto_power p) is
      convergent by SERIES_1:def 2;
      then lim Partial_Sums ((seq_id(tseq)-seq_id(vseq.n)) rto_power p) <= e
      by A71,RSSPACE2:5;
      hence thesis by A69,A82,A80,A81,SERIES_1:17,XXREAL_0:2;
    end;
    hence thesis;
  end;
  (seq_id(tseq)) rto_power p is summable
  proof
    consider m be Nat such that
A83: for n be Nat st n >= m holds ((seq_id(tseq)-seq_id(
    vseq.n)) rto_power p ) is summable & Sum((seq_id(tseq)-seq_id(vseq.n))
    rto_power p) < 1 by A40;
A84: ((seq_id(tseq)-seq_id(vseq.m)) rto_power p ) is summable by A83;
    set d=(seq_id(tseq));
    set b=(seq_id(vseq.m));
    set a=(seq_id(tseq) -seq_id(vseq.m));
A85: a +b =(seq_id(tseq)+seq_id(vseq.m))+(-seq_id(vseq.m)) by SEQ_1:13
      .=seq_id(tseq)+seq_id(vseq.m)-seq_id(vseq.m)
      .=d by Lm10;
    seq_id( (vseq.m) ) rto_power p is summable by A1,A3,Def2;
    hence thesis by A1,A84,A85,Lm11;
  end;
  then reconsider tv=tseq as Point of lp by A1,A3,Th10;
  for e be Real st e > 0 ex m be Nat st
   for n be Nat st n >= m holds ||.(vseq.n) - tv.|| < e
  proof
    let e be Real such that
A86: e > 0;
    set e1=e to_power p;
    consider m be Nat such that
A87: for n be Nat st n >= m holds ((seq_id(tseq)-seq_id(
    vseq.n)) rto_power p ) is summable & Sum((seq_id(tseq)-seq_id(vseq.n))
    rto_power p) < e1 by A40,A86,POWER:34;
    now
      let n be Nat such that
A88:  n >= m;
A89:  Sum((seq_id(tseq)-seq_id(vseq.n)) rto_power p) < e1 by A87,A88;
      for i be Nat holds ((seq_id(tseq)-seq_id(vseq.n))
      rto_power p).i >= 0
      proof
        let i be Nat;
        ((seq_id(tseq)-seq_id(vseq.n)) rto_power p).i =|.(seq_id(tseq)
        -seq_id(vseq.n)).i.| to_power p by Def1;
        hence thesis by A1,Lm1,COMPLEX1:46;
      end;
      then
A90:  Sum((seq_id(tseq)-seq_id(vseq.n)) rto_power p) >=0 by A87,A88,SERIES_1:18
;
A91:  e1 to_power(1/p) = e to_power (p * (1/p)) by A86,POWER:33
        .= e to_power 1 by A1,XCMPLX_1:106
        .= e by POWER:25;
      (Sum((seq_id(tv)-seq_id(vseq.n)) rto_power p)) to_power (1/p ) =(
Sum(seq_id(seq_id(tv)-seq_id(vseq.n)) rto_power p)) to_power (1/p )
        .=(Sum(seq_id((tv)-(vseq.n)) rto_power p)) to_power (1/p ) by A1,A3
,Th10
        .= ||.(tv)+(-(vseq.n)).|| by A3,Def3
        .=||.-(vseq.n-tv).|| by RLVECT_1:33
        .=||.vseq.n-tv.|| by NORMSP_1:2;
      hence ||.vseq.n-tv.|| <e by A2,A90,A89,A91,Th1;
    end;
    hence thesis;
  end;
  hence thesis by NORMSP_1:def 6;
end;
