
theorem
  for A be non empty closed_interval Subset of REAL,
      x be Point of
        DualSp R_Normed_Algebra_of_ContinuousFunctions ClstoCmp A
    st 0 < vol A
  holds
    ex rho be Function of A,REAL st
       rho is bounded_variation
      &
       ( for u be continuous PartFunc of REAL,REAL
           st dom u = A holds x.u = integral(u,rho) )
      &
        ||.x.|| = total_vd(rho)
proof
  let A be non empty closed_interval Subset of REAL,
      x be Point of
        DualSp R_Normed_Algebra_of_ContinuousFunctions ClstoCmp(A);
  assume A1: 0 < vol A;
  set X = R_Normed_Algebra_of_ContinuousFunctions ClstoCmp(A);
  set V = R_Normed_Algebra_of_BoundedFunctions the carrier of ClstoCmp(A);
  set AV = the carrier of ClstoCmp(A);
A2: AV = A by Lm1;
  R_Algebra_of_ContinuousFunctions ClstoCmp(A)
    is Subalgebra of R_Algebra_of_BoundedFunctions AV by C0SP2:9; then
A3: the carrier of X c= the carrier of V
    & the addF of X = (the addF of V) || the carrier of X
    & the Mult of X = (the Mult of V) | [:REAL, the carrier of X:]
      by C0SP1:def 9;
A4: 0.X = ClstoCmp(A) -->0 by C0SP2:12
       .= 0.V by C0SP1:25;
B5: X is SubRealNormSpace of V by A3,A4,DUALSP01:def 16;
  reconsider h=x as Lipschitzian linear-Functional of X by DUALSP01:def 10;
  consider f be Lipschitzian linear-Functional of V, F be Point of DualSp V
    such that
A6: f = F & f|(the carrier of X) = h & ||.F.||=||.x.|| by B5,DUALSP01:36;
  consider a,b be Real such that
A7: a <= b & [.a,b.] = A
   & ClstoCmp(A) = Closed-Interval-TSpace(a,b) by Def7ClstoCmp;
  consider m be Function of A, BoundedFunctions(A) such that
A8: for s be Real st s in [.a,b.] holds
        (s = a implies m.s = [.a,b.] --> 0)
      & (s <> a implies m.s = ([.a,s.] --> 1) +* (].s,b.] --> 0)) by A7,LM83;
  the carrier of V = BoundedFunctions(A) by Lm1; then
  reconsider rho = f*m as Function of A,REAL;
A9: for D be Division of A, K be var_volume of rho,D
      st a < D.1 holds Sum(K) <= ||.x.||
  proof
    let D be Division of A, K be var_volume of rho,D;
    assume A10: a < D.1;
    consider s be FinSequence of V such that
A11:  len s = len D and
A12:  for i be Nat st i in dom s holds
        s.i = sgn( Dp2(rho,D,i+1) - Dp2(rho,D,i) )
             * ( Dp1(m,D,i+1) - Dp1(m,D,i) ) by LM84;
    set yD = Sum(s);
    yD in the carrier of V; then
    yD in BoundedFunctions(A) by Lm1; then
    consider g be Function of A,REAL such that
A13:  g=yD & g|A is bounded;
A14: f.yD = Sum(f*s) by LM87;
A15: for t be Element of A holds |. g.t .| <= 1
    proof
      let t be Element of A;
      defpred R1[Nat,set] means
        ex sk be Function of A,REAL st
          sk = s.$1 & $2 = sk.t;
A16:  for k be Nat st k in Seg len D
        ex x be Element of REAL st R1[k,x]
      proof
        let k be Nat;
        assume k in Seg len D; then
        k in dom s by FINSEQ_1:def 3,A11; then
        s.k = sgn( Dp2(rho,D,k+1) - Dp2(rho,D,k) )
             * ( Dp1(m,D,k+1) - Dp1(m,D,k) ) by A12; then
        s.k in the carrier of V; then
        s.k in BoundedFunctions(A) by Lm1; then
        consider sk be Function of A,REAL such that
A17:      sk = s.k & sk|A is bounded;
        take x = sk.t;
        thus thesis by A17;
      end;
      consider z be FinSequence of REAL such that
A18:    dom z = Seg len D
      & for k be Nat st k in Seg len D holds R1[k,z.k]
          from FINSEQ_1:sch 5(A16);
A20:  len s = len z by A11,A18,FINSEQ_1:def 3;
A22:  dom z = dom D by FINSEQ_1:def 3,A18;
      A = [.lower_bound A,upper_bound A.] by INTEGRA1:4; then
      lower_bound A = a by A7,INTEGRA1:5; then
      consider i be Element of NAT such that
A50:    i in dom D and
A51:    t in divset(D,i) and
A52:    i = 1 or
          (lower_bound divset(D,i) < t
           & t <= (upper_bound divset(D,i)) ) by A10,LM94;
      t in [.(lower_bound divset(D,i)),(upper_bound divset(D,i)).]
        by A51,INTEGRA1:4; then
A53:  (lower_bound divset(D,i)) <= t
        & t <= (upper_bound divset(D,i)) by XXREAL_1:1;
      reconsider i as Nat;
A54:  i in Seg len D by A50,FINSEQ_1:def 3; then
      consider si be Function of A,REAL such that
A55:    si = s.i & z.i = si.t by A18;
      i in Seg len s by A11,A50,FINSEQ_1:def 3; then
B56:  i in dom s by FINSEQ_1:def 3;
      set r = sgn( Dp2(rho,D,i+1) - Dp2(rho,D,i) );
A57:  1 <= i <= len D by A50,FINSEQ_3:25; then
      i + 0 <= len D + 1 by XREAL_1:7; then
A58:  i in Seg (len D + 1) by A57;
b:    1 + 0 <= i + 1 & i <= len D by A54,FINSEQ_1:1,XREAL_1:7; then
      i + 1 <= len D + 1 by XREAL_1:7; then
A59:  i + 1 in Seg (len D + 1) by b;
      z.i = r
      proof
        set f0 = [.a,b.] --> 0;
        set f1 = ([.a,D.i .] --> 1) +* (]. D.i,b.] --> 0);
        set g1 = [.a,D.i .] --> 1;
        set g2 = ]. D.i,b.] --> 0;
        set f2 = ([.a,D.(i-1) .] --> 1) +* (]. D.(i-1),b.] --> 0);
        set h1 = [.a,D.(i-1) .] --> 1;
        set h2 = ]. D.(i-1),b.] --> 0;
B0:     dom f0 = [.a,b.] &
        dom g1 = [.a,D.i .] & dom g2 = ]. D.i,b.] &
        dom h1 = [.a,D.(i-1) .] & dom h2 = ]. D.(i-1),b.] by FUNCOP_1:13;
        per cases;
        suppose A62: i = 1;
          A = [.lower_bound A,upper_bound A.] by INTEGRA1:4; then
A63:      lower_bound A = a by A7,INTEGRA1:5;
A64:      a in [.a,b.] by A7;
A65:      D.i in [.a,b.] by A7,A50,INTEGRA1:6;
A66:      lower_bound divset(D,i) = lower_bound A &
          upper_bound divset(D,i) = D.i by A50,A62,INTEGRA1:def 4;
A69:      Dp1(m,D,(i+1)) = m.(D.(i+1 -1)) by A59,defDp1,A62
                        .= ([.a,D.i .] --> 1) +* (]. D.i,b.] --> 0)
                              by A8,A65,A10,A62;
A70:      Dp1(m,D,i) = m.(lower_bound A) by A58,A62,defDp1
                    .= [.a,b.] -->0 by A8,A63,A64;
A72:      dom f0 = A by A7,FUNCOP_1:13;
A73:      a <= D.i & D.i <= b by A65,XXREAL_1:1;
A74:      dom f1 = dom g1 \/ dom g2 by FUNCT_4:def 1
                .= A by A7,B0,A73,XXREAL_1:167;
          rng f0 c= REAL; then
          reconsider f0 as Function of A,REAL by A72,FUNCT_2:2;
          rng f1 c= REAL; then
          reconsider f1 as Function of A,REAL by A74,FUNCT_2:2;
A76:      si.t = r * (f1.t - f0.t)
          proof
            reconsider H = Dp1(m,D,i+1) - Dp1(m,D,i)
              as Element of R_Normed_Algebra_of_BoundedFunctions(A) by Lm1;
            reconsider h = H as Function of A,REAL by LM88;
            si = r * (Dp1(m,D,i+1) - Dp1(m,D,i)) by A55,B56,A12; then
            si = r * H by Lm1; then
            si.t = r * (h.t) by C0SP1:30
                .= r * (f1.t - f0.t) by A2,A69,A70,C0SP1:34;
            hence thesis;
          end;
A77:      t in [.a,D.i .] by A63,A66,INTEGRA1:4,A51; then
A79:      f1.t = ([.a,D.i .] --> 1).t by FUNCT_4:16,B0,XXREAL_1:90
              .= 1 by A77,FUNCOP_1:7;
          f0.t = 0 by A7,FUNCOP_1:7;
          hence thesis by A55,A76,A79;
        end;
        suppose A80: i <> 1; then
A82:      D.(i-1) in [.a,b.] by A7,A50,INTEGRA1:7;
A81:      D.i in [.a,b.] by A7,A50,INTEGRA1:6;
A83:      lower_bound divset(D,i) = D.(i-1) &
          upper_bound divset(D,i) = D.i by A50,A80,INTEGRA1:def 4;
          i-1 in dom D by A50,A80,INTEGRA1:7; then
A86:      D.(i-1) < D.i by A50,XREAL_1:146,VALUED_0:def 13;
          D.(i-1) = a or D.(i-1) in ].a,b.]
            by A80,A7,A50,INTEGRA1:7,XXREAL_1:6; then
          per cases by XXREAL_1:2;
          suppose A88: a = D.(i-1);
            1 + 0 < i + 1 by XREAL_1:8,A57; then
A90:        Dp1(m,D,(i+1)) = m.(D.(i+1 -1)) by A59,defDp1
                          .= ([.a,D.i .] --> 1) +* (]. D.i,b.] --> 0)
                                by A8,A81,A88,A86;
A91:        Dp1(m,D,i) = m.(D.(i-1)) by A58,A80,defDp1
                      .= [.a,b.] -->0 by A8,A82,A88;
A93:        dom f0 = A by A7,FUNCOP_1:13;
A94:        a <= D.i & D.i <= b by A81,XXREAL_1:1;
A96:        dom f1 = dom g1 \/ dom g2 by FUNCT_4:def 1
                  .= A by A7,B0,A94,XXREAL_1:167;
            rng f0 c= REAL; then
            reconsider f0 as Function of A,REAL by A93,FUNCT_2:2;
            rng f1 c= REAL; then
            reconsider f1 as Function of A,REAL by A96,FUNCT_2:2;
A98:        si.t = r * (f1.t - f0.t)
            proof
              reconsider H = Dp1(m,D,i+1) - Dp1(m,D,i)
                as Element of R_Normed_Algebra_of_BoundedFunctions(A) by Lm1;
              reconsider h = H as Function of A,REAL by LM88;
              si = r * (Dp1(m,D,i+1) - Dp1(m,D,i)) by A55,B56,A12; then
              si = r * H by Lm1; then
              si.t = r * (h.t) by C0SP1:30
                  .= r * (f1.t - f0.t) by A2,A90,A91,C0SP1:34;
              hence thesis;
            end;
A99:        t in [. D.(i-1),D.i .] by A83,INTEGRA1:4,A51;
            a <= D.(i-1) & D.(i-1) <= b by A82,XXREAL_1:1; then
B100:       [. D.(i-1),D.i .] c= [.a,D.i .] by XXREAL_1:34; then
A102:       f1.t = ([.a,D.i .] --> 1).t
                      by A99,FUNCT_4:16,B0,XXREAL_1:90
                .= 1 by B100,A99,FUNCOP_1:7;
            f0.t = 0 by A7,FUNCOP_1:7;
            hence thesis by A55,A98,A102;
          end;
          suppose A103: a < D.(i-1);
            1 + 0 < i + 1 by XREAL_1:8,A57; then
A105:       Dp1(m,D,(i+1)) = m.(D.(i+1 -1)) by A59,defDp1
                          .= ([.a,D.i .] --> 1) +* (]. D.i,b.] --> 0)
                                by A8,A81,A103,A86;
A106:       Dp1(m,D,i) = m.(D.(i-1)) by A58,A80,defDp1
                      .= ([.a,D.(i-1) .] --> 1) +* (]. D.(i-1),b.] --> 0)
                            by A8,A82,A103;
A108:       a <= D.i <= b by A81,XXREAL_1:1;
A109:       a <= D.(i-1) <= b by A82,XXREAL_1:1;
A110:       dom f1 = dom g1 \/ dom g2 by FUNCT_4:def 1
                        .= A by A7,B0,A108,XXREAL_1:167;
A111:       dom f2 = dom h1 \/ dom h2 by FUNCT_4:def 1
                  .= A by A7,B0,A109,XXREAL_1:167;
            rng f1 c= REAL; then
            reconsider f1 as Function of A,REAL by A110,FUNCT_2:2;
            rng f2 c= REAL; then
            reconsider f2 as Function of A,REAL by A111,FUNCT_2:2;
A112:       a <= t & t <= b by XXREAL_1:1,A7;
A113:       si.t = r * (f1.t - f2.t)
            proof
              reconsider H = Dp1(m,D,i+1) - Dp1(m,D,i)
                as Element of R_Normed_Algebra_of_BoundedFunctions(A) by Lm1;
              reconsider h = H as Function of A,REAL by LM88;
              si = r * (Dp1(m,D,i+1) - Dp1(m,D,i)) by A55,B56,A12; then
              si = r * H by Lm1; then
              si.t = r * (h.t) by C0SP1:30
                  .= r * (f1.t - f2.t) by A2,A105,A106,C0SP1:34;
              hence thesis;
            end;
A114:       t in [. D.(i-1),D.i .] by A83,INTEGRA1:4,A51;
B115:       [. D.(i-1),D.i .] c= [.a,D.i .] by A109,XXREAL_1:34; then
A117:       f1.t = ([.a,D.i .] --> 1).t
                      by A114,FUNCT_4:16,B0,XXREAL_1:90
                .= 1 by B115,A114,FUNCOP_1:7;
            D.(i-1) < t & t <= D.i by A50,A80,INTEGRA1:def 4,A52; then
A118:       t in ]. D.(i-1),b.] by A112; then
            f2.t = (]. D.(i-1),b.] --> 0).t by B0,FUNCT_4:13
                .= 0 by A118,FUNCOP_1:7;
            hence thesis by A55,A113,A117;
          end;
        end;
      end; then
A119: |.z.i.| <= 1 by LM91;
      for k be Nat st k in dom z & k <> i holds z.k = 0
      proof
        let k be Nat;
        assume that
A120:   k in dom z and
A121:   k <> i;
        consider sk be Function of A,REAL such that
A124:     sk = s.k & z.k = sk.t by A18,A120;
B125:   k in dom s by FINSEQ_1:def 3,A11,A18,A120; then
A125:   s.k = sgn( Dp2(rho,D,k+1) - Dp2(rho,D,k) )
             * ( Dp1(m,D,k+1) - Dp1(m,D,k) ) by A12;
        set r = sgn( Dp2(rho,D,k+1) - Dp2(rho,D,k) );
A126:   k in dom D by A18,A120,FINSEQ_1:def 3; then
A127:   1 <= k <= len D by FINSEQ_3:25; then
        k + 0 <= len D + 1 by XREAL_1:7; then
A128:   k in Seg (len D + 1) by A127;
a:      1 + 0 <= k + 1 by XREAL_1:7;
        k + 1 <= len D + 1 by A127,XREAL_1:7; then
A129:   k + 1 in Seg (len D + 1) by a;
        set f0 = [.a,b.] --> 0;
        set f1 = ([.a,D.k .] --> 1) +* (]. D.k,b.] --> 0);
        set g1 = [.a,D.k .] --> 1;
        set g2 = ]. D.k,b.] --> 0;
        set f2 = ([.a,D.(k-1) .] --> 1) +* (]. D.(k-1),b.] --> 0);
        set h1 = [.a,D.(k-1) .] --> 1;
        set h2 = ]. D.(k-1),b.] --> 0;
B10:    dom f0 = [.a,b.] &
        dom g1 = [.a,D.k .] & dom g2 = ]. D.k,b.] &
        dom h1 = [.a,D.(k-1) .] & dom h2 = ]. D.(k-1),b.] by FUNCOP_1:13;
        per cases;
        suppose A130: k = 1; then
A134:     lower_bound divset(D,i) = D.(i-1) &
          upper_bound divset(D,i) = D.i by A50,INTEGRA1:def 4,A121;
A136:     i-1 in dom D by A50,INTEGRA1:7,A130,A121;
A141:     D.k in [.a,b.] by A7,A126,INTEGRA1:6; then
A147:     a <= D.k & D.k <= b by XXREAL_1:1;
A146:     dom f0 = A by A7,FUNCOP_1:13;
A148:     dom f1 = dom g1 \/ dom g2 by FUNCT_4:def 1
                .= A by A7,B10,A147,XXREAL_1:167;
          rng f0 c= REAL; then
          reconsider f0 as Function of A,REAL by A146,FUNCT_2:2;
          rng f1 c= REAL; then
          reconsider f1 as Function of A,REAL by A148,FUNCT_2:2;
A153:     Dp1(m,D,(k+1)) = m.(D.(k+1 -1)) by A129,defDp1,A130
                        .= ([.a,D.k .] --> 1) +* (]. D.k,b.] --> 0)
                              by A8,A141,A10,A130;
          A = [.lower_bound A,upper_bound A.] by INTEGRA1:4; then
A139:     lower_bound A = a by A7,INTEGRA1:5;
A140:     a in [.a,b.] by A7;
A144:     Dp1(m,D,k) = m.(lower_bound A) by A128,A130,defDp1
                    .= [.a,b.] -->0 by A8,A139,A140;
A154:     sk.t = r * (f1.t - f0.t)
          proof
            reconsider H = Dp1(m,D,k+1) - Dp1(m,D,k)
              as Element of R_Normed_Algebra_of_BoundedFunctions(A) by Lm1;
            reconsider h = H as Function of A,REAL by LM88;
            sk = r * (Dp1(m,D,k+1) - Dp1(m,D,k)) by A124,B125,A12; then
            sk = r * H by Lm1; then
            sk.t = r * (h.t) by C0SP1:30
                .= r * (f1.t - f0.t) by A2,A144,A153,C0SP1:34;
            hence thesis;
          end;
          k < i by A130,A121,A57,XXREAL_0:1; then
          k + 1 <= i by NAT_1:13; then
A157:     k+1 -1 <= i-1 by XREAL_1:13;
          D.k <= D.(i-1)
          proof
            k = i-1 or k < i-1 by A157,XXREAL_0:1;
            hence thesis by A126,A136,VALUED_0:def 13;
          end; then
A158:     D.k < t by A52,A130,A121,XXREAL_0:2,A134;
          a <= t & t <= b by XXREAL_1:1,A7; then
A160:     t in ].D.k,b .] by A158; then
A162:     f1.t = (].D.k,b .] --> 0).t by B10,FUNCT_4:13
              .= 0 by A160,FUNCOP_1:7;
          f0.t = 0 by A7,FUNCOP_1:7;
          hence thesis by A124,A154,A162;
        end;
        suppose A163: k <> 1; then
A165:     D.(k-1) in [.a,b.] by A7,A126,INTEGRA1:7;
A164:     D.k in [.a,b.] by A7,A126,INTEGRA1:6;
A168:     k-1 in dom D by A126,A163,INTEGRA1:7; then
A169:     D.(k-1) < D.k by A126,XREAL_1:146,VALUED_0:def 13;
          1 < k by A127,A163,XXREAL_0:1; then
          1+1 <= k by NAT_1:13; then
A171:     2-1 <= k -1 by XREAL_1:13;
          1 <= len D by A127,XXREAL_0:2; then
A172:     1 in dom D by FINSEQ_3:25;
B173:     D.1 <= D.(k-1)
          proof
            1 = k -1 or 1 < k -1 by A171,XXREAL_0:1;
            hence thesis by A168,A172,VALUED_0:def 13;
          end; then
A173:     a < D.(k-1) by A10,XXREAL_0:2;
          1 + 0 < k + 1 by XREAL_1:8,A127; then
A175:     Dp1(m,D,(k+1)) = m.(D.(k+1 -1)) by A129,defDp1
                        .= ([.a,D.k .] --> 1) +* (]. D.k,b.] --> 0)
                              by A8,A164,A173,A169;
A176:     Dp1(m,D,k) = m.(D.(k-1)) by A128,A163,defDp1
                    .= ([.a,D.(k-1) .] --> 1) +* (]. D.(k-1),b.] --> 0)
                          by A8,A165,B173,A10;
A178:     a <= D.k & D.k <= b by A164,XXREAL_1:1;
A179:     a <= D.(k-1) & D.(k-1) <= b by A165,XXREAL_1:1;
A180:     dom f1 = dom g1 \/ dom g2 by FUNCT_4:def 1
                .= A by A7,B10,A178,XXREAL_1:167;
A181:     dom f2 = dom h1 \/ dom h2 by FUNCT_4:def 1
                .= A by A7,B10,A179,XXREAL_1:167;
          rng f1 c= REAL; then
          reconsider f1 as Function of A,REAL by A180,FUNCT_2:2;
          rng f2 c= REAL; then
          reconsider f2 as Function of A,REAL by A181,FUNCT_2:2;
A183:     sk.t = r * (f1.t - f2.t)
          proof
            reconsider H = Dp1(m,D,k+1) - Dp1(m,D,k)
              as Element of R_Normed_Algebra_of_BoundedFunctions(A) by Lm1;
            reconsider h = H as Function of A,REAL by LM88;
            sk = r * H by Lm1,A124,A125; then
            sk.t = r * (h.t) by C0SP1:30
                .= r * (f1.t - f2.t) by A2,A175,A176,C0SP1:34;
            hence thesis;
          end;
          per cases by A121,XXREAL_0:1;
          suppose i < k; then
            i+1 <= k by NAT_1:13; then
A186:       i+1 -1 <= k-1 by XREAL_1:13;
A187:       D.i <= D.(k-1)
            proof
              i = k -1 or i < k -1 by A186,XXREAL_0:1;
              hence thesis by A50,A168,VALUED_0:def 13;
            end;
A189:       (upper_bound divset(D,i)) <=D.(k-1)
            proof
              per cases;
              suppose i = 1;
                hence (upper_bound divset(D,i)) <=D.(k-1)
                        by A187,A50,INTEGRA1:def 4;
              end;
              suppose i <> 1;
                hence (upper_bound divset(D,i)) <=D.(k-1)
                        by A187,A50,INTEGRA1:def 4;
              end;
            end;
A191:       a <= t <= D.(k-1)
              by A189,XXREAL_0:2,A53,XXREAL_1:1,A7; then
A192:       t in [.a,D.(k-1) .];
            a <= t <= D.k by A169,A191,XXREAL_0:2; then
A193:       t in [.a,D.k .]; then
A196:       f1.t = ([.a,D.k .] --> 1).t
                      by FUNCT_4:16,B10,XXREAL_1:90
                .= 1 by A193,FUNCOP_1:7;
            f2.t = ([.a,D.(k-1) .] --> 1).t
                      by B10,A192,FUNCT_4:16,XXREAL_1:90
                .= 1 by A192,FUNCOP_1:7;
            hence thesis by A124,A183,A196;
          end;
          suppose A198: k < i; then
            k+1 <= i by NAT_1:13; then
A201:       k+1 -1 <= i-1 by XREAL_1:13;
A202:       i-1 in dom D by A50,A198,A127,INTEGRA1:7;
A203:       lower_bound divset(D,i) = D.(i-1) &
            upper_bound divset(D,i) = D.i by A50,A198,A127,INTEGRA1:def 4;
            D.k <= D.(i-1)
            proof
              k = i-1 or k < i-1 by A201,XXREAL_0:1;
              hence thesis by A126,A202,VALUED_0:def 13;
            end; then
A206:       D.k < t
              by A198,A52,A18,A120,FINSEQ_1:1,XXREAL_0:2,A203; then
A207:       D.(k-1) < t by A169,XXREAL_0:2;
A208:       a <= t & t <= b by XXREAL_1:1,A7; then
A209:       t in ]. D.k,b.] by A206; then
A211:       f1.t = (]. D.k,b.] --> 0).t by B10,FUNCT_4:13
                .= 0 by A209,FUNCOP_1:7;
A210:       t in ]. D.(k-1),b.] by A207,A208; then
            f2.t = (]. D.(k-1),b.]--> 0).t by B10,FUNCT_4:13
                .= 0 by A210,FUNCOP_1:7;
            hence thesis by A124,A183,A211;
          end;
        end;
      end; then
      |. Sum z .| <= 1 by A22,A50,A119,INTEGR23:6;
      hence |. g.t .| <= 1 by A13,A20,LM89,A18;
    end;
    now
      let k be Real;
      assume k in PreNorms g; then
      consider t be Element of A such that
A213:   k = |.g.t.|;
      thus k <= 1 by A15,A213;
    end; then
    upper_bound PreNorms g <= 1 by SEQ_4:45; then
    (BoundedFunctionsNorm A).g <= 1 by A13,C0SP1:20; then
A214: ||.yD.|| <= 1 by A13,Lm1;
A215: len K = len D
    & for i be Nat st i in dom D holds K.i = |. vol (divset(D,i),rho) .|
        by INTEGR22:def 2;
    dom f = the carrier of V by FUNCT_2:def 1; then
    rng s c= dom f; then
A216: dom (f*s) = dom s by RELAT_1:27
               .= Seg len s by FINSEQ_1:def 3
               .= dom K by A11,A215,FINSEQ_1:def 3;
A217: for i be Nat st i in dom D
        holds (f*s).i = |. vol (divset(D,i),rho) .|
    proof
      let i be Nat;
      assume A218: i in dom D; then
A220: 1 <= i <= len D by FINSEQ_3:25; then
      i + 0 <= len D + 1 by XREAL_1:7; then
A221: i in Seg (len D + 1) by A220;
b:    1 + 0 <= i + 1 by XREAL_1:7;
      i + 1 <= len D + 1 by A220,XREAL_1:7; then
A222: i + 1 in Seg (len D + 1) by b;
      i in Seg (len s) by A11,A218,FINSEQ_1:def 3; then
A223: i in dom s by FINSEQ_1:def 3;
      set r = sgn( Dp2(rho,D,i+1) - Dp2(rho,D,i) );
      A = [.lower_bound A,upper_bound A.] by INTEGRA1:4; then
A224: lower_bound A = a by A7,INTEGRA1:5;
      D.i in A by A218,INTEGRA1:6; then
A225: D.i in dom m by FUNCT_2:def 1;
      lower_bound A in A by A7,A224; then
A226: lower_bound A in dom m by FUNCT_2:def 1;
      per cases;
      suppose A227: i = 1; then
A228:   lower_bound divset(D,i) = lower_bound A &
        upper_bound divset(D,i) = D.i by A218,INTEGRA1:def 4;
A229:   Dp1(m,D,i+1) = m.(D.(i+1 -1) ) by A222,defDp1,A227
                    .= m.(D.i);
A231:   (f*s).i = f.(s.i) by A223,FUNCT_1:13
               .= f.( r * ( Dp1(m,D,i+1) - Dp1(m,D,i) ) ) by A12,A223
               .= r * f.( Dp1(m,D,i+1) - Dp1(m,D,i) ) by HAHNBAN:def 3
               .= r * ( f.( Dp1(m,D,i+1) ) - f.( Dp1(m,D,i) ) )
                      by HAHNBAN:19
               .= r * ( f.( m.(D.i) ) - f.( m.(lower_bound A) ))
                      by A229,A221,A227,defDp1
               .= r * ( (f*m).(D.i) - f.( m.(lower_bound A) ) )
                      by A225,FUNCT_1:13
               .= r * ( rho.(upper_bound divset(D,i))
                                  - rho.(lower_bound divset(D,i)) )
                      by A228,A226,FUNCT_1:13
               .= r * vol( divset(D,i),rho ) by INTEGR22:def 1;
A232:   Dp2(rho,D,i+1) = rho.(D.(i+1 -1) ) by defDp2,A227
                      .= rho.(D.i);
        r = sgn( rho.(upper_bound divset(D,i))
                            - rho.(lower_bound divset(D,i)) )
              by A228,A232,A227,defDp2
         .= sgn(vol( divset(D,i),rho )) by INTEGR22:def 1;
        hence (f*s).i = |. vol (divset(D,i),rho) .| by A231,LM86;
      end;
      suppose A234: i <> 1;
        D.i in A by A218,INTEGRA1:6; then
A235:   D.i in dom m by FUNCT_2:def 1;
        D.(i-1) in A by A218,A234,INTEGRA1:7; then
A236:   D.(i-1) in dom m by FUNCT_2:def 1;
A237:   lower_bound divset(D,i) = D.(i-1) &
        upper_bound divset(D,i) = D.i by A218,A234,INTEGRA1:def 4;
A238:   1 + 0 < i + 1 by XREAL_1:8,A220; then
A239:   Dp1(m,D,(i+1)) = m.(D.(i+1 -1) ) by A222,defDp1
                      .= m.(D.i);
A241:   (f*s).i = f.(s.i) by A223,FUNCT_1:13
               .= f.(r * ( Dp1(m,D,i+1) - Dp1(m,D,i) ) ) by A12,A223
               .= r * f.( Dp1(m,D,i+1) - Dp1(m,D,i) ) by HAHNBAN:def 3
               .= r * ( f.( Dp1(m,D,i+1) ) - f.( Dp1(m,D,i) ) )
                      by HAHNBAN:19
               .= r * ( f.( m.(D.i)) - f.( m.(D.(i-1) ) ) )
                      by A239,A221,A234,defDp1
               .= r * ( (f*m).(D.i) - f.( m.(D.(i-1)) ) )
                      by A235,FUNCT_1:13
               .= r * ( rho.(upper_bound divset(D,i))
                                - rho.(lower_bound divset(D,i)) )
                      by A237,A236,FUNCT_1:13
               .= r * vol( divset(D,i),rho ) by INTEGR22:def 1;
A242:   Dp2(rho,D,i+1) = rho.(D.(i+1 -1) ) by A238,defDp2
                      .= rho.(D.i);
        r = sgn( rho.(upper_bound divset(D,i))
                             - rho.(lower_bound divset(D,i)) )
              by A237,A242,A234,defDp2
         .= sgn(vol( divset(D,i),rho )) by INTEGR22:def 1;
        hence (f*s).i = |. vol (divset(D,i),rho) .| by A241,LM86;
      end;
    end;
    for i be Nat st i in dom K holds (f*s).i = K.i
    proof
      let i be Nat;
      assume i in dom K; then
      i in Seg len D by A215,FINSEQ_1:def 3; then
A244: i in dom D by FINSEQ_1:def 3; then
      K.i = |. vol (divset(D,i),rho) .| by INTEGR22:def 2;
      hence thesis by A217,A244;
    end; then
A245: f.yD = Sum(K) by A14,A216,FINSEQ_1:13;
A246: f.yD <= |.f.yD.| by ABSVALUE:4;
A247: |.f.yD.| <= ||.F.|| * ||.yD.|| by A6,DUALSP01:26;
    ||.F.|| * ||.yD.|| <= ||.F.|| * 1 by A214,XREAL_1:64; then
    |.f.yD.| <= ||.F.|| by A247,XXREAL_0:2;
    hence Sum(K) <= ||. x .|| by A6,A245,A246,XXREAL_0:2;
  end;
  reconsider d = ||.x.|| + 1 as Real;
  A = [.lower_bound A, upper_bound A.] by INTEGRA1:4; then
A249: lower_bound A = a by A7,INTEGRA1:5; then
  for D be Division of A, K be var_volume of rho,D
    holds Sum(K) <= ||.x.|| by A1,A9,LM95; then
B250: for t be Division of A, F0 be var_volume of rho,t
        holds Sum(F0) <= d by XREAL_1:39; then
A251: rho is bounded_variation; then
  consider VD be non empty Subset of REAL such that
    VD is bounded_above and
A253: VD = { r where r is Real:
             ex t be Division of A, F0 be var_volume of rho,t st
               r = Sum(F0) } and
A254: total_vd(rho) = upper_bound VD by INTEGR22:def 4;
  now
    let k be Real;
    assume k in VD; then
    consider r be Real such that
A255: k = r and
A256: ex t be Division of A, F0 be var_volume of rho,t st
        r = Sum(F0) by A253;
    consider t be Division of A, F0 be var_volume of rho,t such that
A257: r = Sum(F0) by A256;
    thus k <= ||.x.|| by A249,A1,A9,LM95,A255,A257;
  end; then
A258: total_vd(rho) <= ||.x.|| by A254,SEQ_4:45;
A259: for u be continuous PartFunc of REAL,REAL
        st dom u = A holds x.u = integral(u,rho)
  proof
    let u be continuous PartFunc of REAL,REAL;
    assume A260: dom u = A; then
A261: for r be Real st 0 < r
        ex s be Real st
          0 < s &
      for x1,x2 be Real st x1 in dom(u|A) & x2 in dom(u|A) & |.x1 - x2.| < s
        holds |.(u|A).x1 - (u|A).x2.| < r by FCONT_2:def 1,FCONT_2:11;
B262: u is_RiemannStieltjes_integrable_with rho by A251,A260,INTEGR23:21;
    consider T being DivSequence of A such that
A263: delta T is convergent & lim (delta T) = 0
     & ( for n being Element of NAT ex Tn being Division of A st
         ( Tn divide_into_equal n + 1 & T.n = Tn ) )
        by A1,INTEGRA6:16;
A264: for n be Nat holds a < (T.n).1
    proof
      let n be Nat;
      n is Element of NAT by ORDINAL1:def 12; then
      consider Tn being Division of A such that
A265:   Tn divide_into_equal n + 1 & T.n = Tn by A263;
A266: len Tn = n+1 by A265,INTEGRA4:def 1;
      n+1 in Seg (n+1) by FINSEQ_1:4; then
      1 <= n+1 <= len Tn by FINSEQ_1:1,A265,INTEGRA4:def 1; then
      1 <= len Tn by XXREAL_0:2; then
B268: 1 in dom Tn by FINSEQ_3:25;
      0 < (vol A) / (len Tn) by A1,A266,XREAL_1:139; then
      (lower_bound A) + 0 <
        (lower_bound A) + (( (vol A) / (len Tn) ) * 1) by XREAL_1:8;
      hence thesis by A265,A249,B268,INTEGRA4:def 1;
    end;
    set S = the middle_volume_Sequence of rho,u,T;
A269: u is Point of
        R_Normed_Algebra_of_ContinuousFunctions(ClstoCmp(A)) by A260,Th80;
    defpred P[Element of NAT, set] means
     ex p being FinSequence of REAL st
       p = $2 & len p = len (T.$1) & for i be Nat st i in dom (T.$1) holds
       p.i in dom (u|divset((T.$1),i)) & ex z be Real st
       z = (u|divset((T.$1),i)).(p.i) &
       (S.$1).i = z * (vol (divset((T.$1),i),rho));
A270: for k be Element of NAT ex p be Element of (REAL)* st P[k, p]
    proof
      let k be Element of NAT;
      defpred P1[Nat,set] means
        $2 in dom (u|divset((T.k),$1)) &
        ex c be Real st c = (u|divset((T.k),$1)).$2 &
        (S.k).$1 = c * (vol (divset((T.k),$1),rho));
A271: Seg len(T.k) = dom(T.k) by FINSEQ_1:def 3;
A272: for i be Nat st i in Seg len (T.k)
            ex x be Element of REAL st P1[i,x]
      proof
        let i be Nat;
        assume i in Seg len(T.k); then
        i in dom (T.k) by FINSEQ_1:def 3; then
        consider c be Real such that
A273:     c in rng (u|divset((T.k),i)) &
            (S.k).i = c * (vol(divset((T.k),i),rho))
              by A251,A260,INTEGR22:def 5;
        consider x be object such that
A274:     x in dom (u|divset(T.k,i)) &
          c = (u|divset(T.k,i)).x by A273,FUNCT_1:def 3;
        reconsider x as Element of REAL by A274;
        take x;
        thus thesis by A273,A274;
      end;
      consider p be FinSequence of REAL such that
A275:   dom p = Seg len(T.k)
       & for i be Nat st i in Seg len(T.k) holds P1[i,p.i]
          from FINSEQ_1:sch 5(A272);
      take p;
      len p = len (T.k) by A275,FINSEQ_1:def 3;
      hence thesis by A271,A275,FINSEQ_1:def 11;
    end;
    consider F be sequence of (REAL)* such that
A276: for x be Element of NAT holds P[x, F.x] from FUNCT_2:sch 3(A270);
    defpred Q[Element of NAT, object] means
      ex q be FinSequence of V st
        len q = len (T.$1) & $2=Sum q &
        for i be Nat st i in dom (T.$1) holds
           ex r be Real st
             (F.$1).i in dom (u|divset((T.$1),i))
           & r = (u|divset((T.$1),i)).((F.$1).i)
           & q.i = r * ( Dp1(m,(T.$1),i+1) - Dp1(m,(T.$1),i) );
A277: for k be Element of NAT
        ex x be Element of the carrier of V st Q[k,x]
    proof
      let k be Element of NAT;
      defpred Q1[Nat,object] means
        ex r be Real st
          (F.k).$1 in dom (u|divset((T.k),$1))
        & r = (u|divset((T.k),$1)).((F.k).$1)
        & $2 = r * ( Dp1(m,(T.k),$1+1) - Dp1(m,(T.k),$1) );
A278: for i be Nat st i in Seg len (T.k) ex y be Element of
        the carrier of V st Q1[i,y]
      proof
        let i be Nat;
        assume i in Seg len (T.k); then
A279:   i in dom(T.k) by FINSEQ_1:def 3;
        consider p being FinSequence of REAL such that
A280:     p = F.k & len p = len (T.k)
         & for i be Nat st i in dom (T.k) holds
             p.i in dom (u|divset((T.k),i)) & ex z be Real st
             z = (u|divset((T.k),i)).(p.i) &
             (S.k).i = z * (vol (divset((T.k),i),rho)) by A276;
        p.i in dom (u|divset((T.k),i)) by A279,A280; then
        (u|divset((T.k),i)).(p.i) in rng (u|divset((T.k),i)) by FUNCT_1:3; then
        reconsider r = (u|divset((T.k),i)).(p.i) as Element of REAL;
        r * ( Dp1(m,(T.k),i+1) - Dp1(m,(T.k),i) ) is
          Element of the carrier of V;
        hence thesis by A280,A279;
      end;
      consider q be FinSequence of V such that
A282:   dom q = Seg len(T.k)
       & for i be Nat st i in Seg len(T.k) holds Q1[i,q.i]
           from FINSEQ_1:sch 5(A278);
      take x=Sum q;
A283: dom(T.k) = Seg len(T.k) by FINSEQ_1:def 3;
      len q = len(T.k) by A282,FINSEQ_1:def 3;
      hence thesis by A282,A283;
    end;
    consider xD be sequence of V such that
A284: for z be Element of NAT holds Q[z,xD.z] from FUNCT_2:sch 3(A277);
B314: for n be Element of NAT holds (f*xD).n = (middle_sum(S)).n
    proof
      let n be Element of NAT;
      consider p1 be FinSequence of REAL such that
A285:   p1 = F.n & len p1 = len (T.n)
       & for i be Nat st i in dom (T.n)
           holds p1.i in dom (u|divset(T.n,i))
       & ex z be Real st z = (u|divset(T.n,i)).(p1.i)
       & (S.n).i = z * (vol (divset(T.n,i),rho)) by A276;
      consider q1 be FinSequence of V such that
A286:   len q1 = len (T.n) & xD.n = Sum q1 &
        for i be Nat st i in dom (T.n) holds
          ex r be Real st
            (F.n).i in dom (u|divset((T.n),i))
           & r = (u|divset((T.n),i)).((F.n).i)
           & q1.i = r * ( Dp1(m,(T.n),i+1) - Dp1(m,(T.n),i) ) by A284;
A287: (middle_sum(S)).n = Sum(S.n) by INTEGR22:def 7;
      dom xD = NAT by FUNCT_2:def 1; then
A288: (f*xD).n = f.(Sum q1) by A286,FUNCT_1:13
              .= Sum(f*q1) by LM87;
      dom f = the carrier of V by FUNCT_2:def 1; then
      rng q1 c= dom f; then
      dom (f*q1) = dom q1 by RELAT_1:27
                .= Seg len q1 by FINSEQ_1:def 3
                .= Seg len (S.n) by A251,A260,A286,INTEGR22:def 5; then
A289: len (f*q1) = len (S.n) by FINSEQ_1:def 3;
      for k be Nat st 1 <= k & k <= len (S.n) holds (f*q1).k = (S.n).k
      proof
        let k be Nat;
        assume A290: 1 <= k & k <= len (S.n); then
B291:   k in Seg len (S.n); then
        k in Seg len (T.n) by A251,A260,INTEGR22:def 5; then
A292:   k in dom (T.n) by FINSEQ_1:def 3; then
        consider z be Real such that
A293:     z = (u|divset(T.n,k)).((F.n).k)
         & (S.n).k = z * (vol (divset(T.n,k),rho)) by A285;
        consider r be Real such that
A294:     (F.n).k in dom (u|divset((T.n),k))
          & r = (u|divset((T.n),k)).((F.n).k)
          & q1.k = r * ( Dp1(m,(T.n),k+1) - Dp1(m,(T.n),k) ) by A286,A292;
        1 <= k <= len(T.n) by A251,A260,A290,INTEGR22:def 5; then
        k + 0 <= len(T.n) + 1 by XREAL_1:7; then
A296:   k in Seg (len(T.n) + 1) by A290;
d:      1 + 0 <= k + 1 & k <= len(T.n)
          by A251,A260,A290,INTEGR22:def 5,XREAL_1:7; then
        k + 1 <= len(T.n) + 1 by XREAL_1:7; then
A297:   k + 1 in Seg (len(T.n) + 1) by d;
        k in Seg len q1 by A286,B291,A251,A260,INTEGR22:def 5; then
A298:   k in dom q1 by FINSEQ_1:def 3;
        per cases;
        suppose A299: k = 1;
          A = [.lower_bound A,upper_bound A.] by INTEGRA1:4; then
          lower_bound A = a by A7,INTEGRA1:5; then
          lower_bound A in A by A7; then
A301:     lower_bound A in dom m by FUNCT_2:def 1;
          (T.n).k in A by A292,INTEGRA1:6; then
A302:     (T.n).k in dom m by FUNCT_2:def 1;
A303:     lower_bound divset((T.n),k) = lower_bound A &
          upper_bound divset((T.n),k) = (T.n).k
             by A292,A299,INTEGRA1:def 4;
A304:     Dp1(m,(T.n),(k+1)) = m.((T.n).(k+1 -1)) by A297,defDp1,A299
                            .= m.((T.n).k);
A306:     f.( Dp1(m,(T.n),k+1) - Dp1(m,(T.n),k) )
             = f.( Dp1(m,(T.n),k+1) ) - f.( Dp1(m,(T.n),k) ) by HAHNBAN:19
            .= f.( m.((T.n).k) ) - f.( m.(lower_bound A) )
                 by A304,A296,A299,defDp1
            .= (f*m).((T.n).k) - f.( m.(lower_bound A) ) by A302,FUNCT_1:13
            .= rho.(upper_bound divset((T.n),k))
                              - rho.(lower_bound divset((T.n),k))
                 by A303,A301,FUNCT_1:13
            .= vol( divset((T.n),k),rho ) by INTEGR22:def 1;
          thus (f*q1).k = f.( r * ( Dp1(m,(T.n),k+1) - Dp1(m,(T.n),k) ) )
                       by A294,A298,FUNCT_1:13
                  .= (S.n).k by A293,A294,A306,HAHNBAN:def 3;
        end;
        suppose A307: k <> 1;
          (T.n).k in A by A292,INTEGRA1:6; then
A308:     (T.n).k in dom m by FUNCT_2:def 1;
          (T.n).(k-1) in A by A292,A307,INTEGRA1:7; then
A309:     (T.n).(k-1) in dom m by FUNCT_2:def 1;
A310:     lower_bound divset((T.n),k) = (T.n).(k-1) &
          upper_bound divset((T.n),k) = (T.n).k
            by A292,A307,INTEGRA1:def 4;
          1 + 0 < k + 1 by XREAL_1:8,A290; then
A311:     Dp1(m,(T.n),(k+1)) = m.((T.n).(k+1 -1)) by A297,defDp1
                            .= m.((T.n).k);
A313:     f.( Dp1(m,(T.n),k+1) - Dp1(m,(T.n),k) )
             = f.( Dp1(m,(T.n),k+1) ) - f.( Dp1(m,(T.n),k) ) by HAHNBAN:19
            .= f.( m.((T.n).k) ) - f.( m.((T.n).(k-1)) )
                    by A311,A296,A307,defDp1
            .= (f*m).((T.n).k) - f.( m.((T.n).(k-1)) ) by A308,FUNCT_1:13
            .= rho.(upper_bound divset((T.n),k))
                               - rho.(lower_bound divset((T.n),k))
                    by A310,A309,FUNCT_1:13
            .= vol( divset((T.n),k),rho ) by INTEGR22:def 1;
          thus (f*q1).k = f.(r * ( Dp1(m,(T.n),k+1) - Dp1(m,(T.n),k) ) )
                       by A294,A298,FUNCT_1:13
                  .= (S.n).k by A293,A294,A313,HAHNBAN:def 3;
        end;
      end;
      hence thesis by A287,A288,A289,FINSEQ_1:14;
    end;
A315: middle_sum(S) is convergent &
      lim (middle_sum(S)) = integral(u,rho)
        by B262,A251,A260,INTEGR22:def 9,A263;
A316: u in BoundedFunctions(the carrier of ClstoCmp(A)) by A269,Lm2;
    reconsider v=u as Point of V by A269,Lm2;
    v in BoundedFunctions(A) by A316,Lm1; then
    consider g be Function of A,REAL such that
A317: g=v & g|A is bounded;
    reconsider v0=v as Point of V;
A318: for r be Real st 0 < r ex m0 be Nat st
          for n be Nat st m0 <= n holds ||. xD.n - v0 .|| < r
    proof
      let r be Real;
      assume A319: 0 < r; then
A321: r/2 < r by XREAL_1:216;
      consider s be Real such that
A322:   0 < s and
A323:   for x1,x2 be Real st
          x1 in dom(u|A) & x2 in dom(u|A) & |.x1 - x2.| < s
         holds |.(u|A).x1 - (u|A).x2.| < r/2 by A261,A319,XREAL_1:215;
      consider m0 be Nat such that
A324:   for i be Nat st m0 <= i holds |. (delta T).i - 0 .| < s
          by A263,A322,SEQ_2:def 7;
A325: for n be Nat st m0 <= n holds ||. xD.n - v0 .|| < r
      proof
        let n be Nat;
A326:   n in NAT by ORDINAL1:def 12;
        consider p2 be FinSequence of REAL such that
A327:     p2 = F.n & len p2 = len (T.n)
         & for i be Nat st i in dom (T.n)
             holds p2.i in dom (u|divset((T.n),i))
         & ex z be Real st z = (u|divset((T.n),i)).(p2.i)
         & (S.n).i = z * (vol (divset((T.n),i),rho)) by A276,A326;
        consider q2 be FinSequence of V such that
A328:     len q2 = len (T.n) & xD.n = Sum q2
         & for i be Nat st i in dom (T.n) holds
             ex r be Real st
               (F.n).i in dom (u|divset((T.n),i))
              & r = (u|divset((T.n),i)).((F.n).i)
              & q2.i = r * ( Dp1(m,(T.n),i+1) - Dp1(m,(T.n),i) )
                by A284,A326;
        assume m0 <= n; then
        |. (delta T).n - 0 .| < s by A324; then
        |. delta(T.n) .| < s by A326,INTEGRA3:def 2; then
A329:   delta(T.n) < s by ABSVALUE:def 1,INTEGRA3:9;
        xD.n in the carrier of V; then
        xD.n in BoundedFunctions(A) by Lm1; then
        consider g1 be Function of A,REAL such that
A330:     g1=xD.n & g1|A is bounded;
A331:   for t be Element of A, i be Nat
          st i in dom (T.n) & t in divset(T.n,i)
            & ( i = 1 or
                  (lower_bound divset(T.n,i) < t
                    & t <= ( upper_bound divset(T.n,i) )) )
            holds g1.t = g.(p2.i)
        proof
          let t be Element of A, i be Nat;
          assume that
A332:     i in dom (T.n) and
A333:     t in divset((T.n),i) and
A334:     i = 1 or
            (lower_bound divset(T.n,i) < t
              & t <= (upper_bound divset(T.n,i)) );
          consider r be Real such that
A336:       p2.i in dom (u|divset((T.n),i)) and
A337:       r = (u|divset((T.n),i)).(p2.i) and
A338:       q2.i = r * ( Dp1(m,(T.n),i+1) - Dp1(m,(T.n),i) )
                         by A327,A328,A332;
          defpred R2[Nat,set] means
            ex qi be Function of A,REAL st
              qi = q2.$1 & $2 = qi.t;
A340:     for i be Nat st i in Seg len (T.n)
            ex x be Element of REAL st R2[i,x]
          proof
            let i be Nat;
            assume i in Seg len (T.n); then
            i in dom (T.n) by FINSEQ_1:def 3; then
            consider r be Real such that
A342:         p2.i in dom (u|divset((T.n),i))
             & r = (u|divset((T.n),i)).(p2.i)
             & q2.i = r * ( Dp1(m,(T.n),i+1) - Dp1(m,(T.n),i) )
                    by A327,A328;
            q2.i in the carrier of V by A342; then
            q2.i in BoundedFunctions(A) by Lm1; then
            consider qi be Function of A,REAL such that
A343:         qi = q2.i & qi|A is bounded;
            take x = qi.t;
            thus thesis by A343;
          end;
          consider z be FinSequence of REAL such that
A344:       dom z = Seg len (T.n)
           & for i be Nat st i in Seg len (T.n) holds R2[i,z.i]
              from FINSEQ_1:sch 5(A340);
A346:     len q2 = len z by A328,A344,FINSEQ_1:def 3;
A347:     i in dom z by A344,A332,FINSEQ_1:def 3;
A348:     g1.t = Sum(z) by A328,A330,A346,LM89,A344;
A349:     dom z = dom (T.n) by FINSEQ_1:def 3,A344;
A350:     i in Seg len (T.n) by A332,FINSEQ_1:def 3; then
          consider qi be Function of A,REAL such that
A351:       qi = q2.i & z.i = qi.t by A344;
          set D = T.n;
B352:     t in [.(lower_bound divset(D,i)),(upper_bound divset(D,i)).]
                  by A333,INTEGRA1:4;
A354:     1 <= i & i <= len D by A350,FINSEQ_1:1; then
          1 <= i & i + 0 <= len D + 1 by XREAL_1:7; then
A355:     i in Seg (len D + 1);
          1 + 0 <= i + 1 & i <= len D by A350,FINSEQ_1:1,XREAL_1:7; then
          1 + 0 <= i + 1 <= len D + 1 by XREAL_1:7; then
A356:     i + 1 in Seg (len D + 1);
A357:     z.i = r
          proof
            set f0 = [.a,b.] --> 0;
            set f1 = ([.a,D.i .] --> 1) +* (]. D.i,b.] --> 0);
            set g1 = [.a,D.i .] --> 1;
            set g2 = ]. D.i,b.] --> 0;
            set f2 = ([.a,D.(i-1) .] --> 1) +* (]. D.(i-1),b.] --> 0);
            set h1 = [.a,D.(i-1) .] --> 1;
            set h2 = ]. D.(i-1),b.] --> 0;
B20:        dom f0 = [.a,b.] &
            dom g1 = [.a,D.i .] & dom g2 = ]. D.i,b.] &
            dom h1 = [.a,D.(i-1) .] & dom h2 = ]. D.(i-1),b.]
              by FUNCOP_1:13;
            per cases;
            suppose A358: i = 1;
              A = [.lower_bound A,upper_bound A.] by INTEGRA1:4; then
A359:         lower_bound A = a by A7,INTEGRA1:5;
A360:         a in [.a,b.] by A7;
A361:         D.i in [.a,b.] by A7,A332,INTEGRA1:6;
A362:         lower_bound divset(D,i) = lower_bound A &
              upper_bound divset(D,i) = D.i by A332,A358,INTEGRA1:def 4;
A364:         a < D.i by A264,A358;
A365:         Dp1(m,D,(i+1)) = m.(D.(i+1 -1)) by A356,defDp1,A358
                            .= ([.a,D.i .] --> 1) +* (]. D.i,b.] --> 0)
                                  by A8,A361,A364;
A366:         Dp1(m,D,i) = m.(lower_bound A) by A355,A358,defDp1
                        .= [.a,b.] -->0 by A8,A359,A360;
A368:         dom f0 = A by A7,FUNCOP_1:13;
A369:         a <= D.i & D.i <= b by A361,XXREAL_1:1;
A370:         dom f1 = dom g1 \/ dom g2 by FUNCT_4:def 1
                    .= A by A7,B20,A369,XXREAL_1:167;
              rng f0 c= REAL; then
              reconsider f0 as Function of A,REAL by A368,FUNCT_2:2;
              rng f1 c= REAL; then
              reconsider f1 as Function of A,REAL by A370,FUNCT_2:2;
A372:         qi.t = r * (f1.t - f0.t)
              proof
                reconsider H = Dp1(m,D,i+1) - Dp1(m,D,i)
                  as Element of R_Normed_Algebra_of_BoundedFunctions(A) by Lm1;
                reconsider h = H as Function of A,REAL by LM88;
                qi = r * H by Lm1,A351,A338; then
                qi.t = r * (h.t) by C0SP1:30
                    .= r * (f1.t - f0.t) by A2,A365,A366,C0SP1:34;
                hence thesis;
              end;
A373:         t in [.a,D.i .] by A359,A362,INTEGRA1:4,A333; then
A375:         f1.t = ([.a,D.i .] --> 1).t
                        by FUNCT_4:16,B20,XXREAL_1:90
                  .= 1 by A373,FUNCOP_1:7;
              f0.t = 0 by A7,FUNCOP_1:7;
              hence thesis by A351,A372,A375;
            end;
            suppose A376: i <> 1; then
A378:         D.(i-1) in [.a,b.] by A7,A332,INTEGRA1:7;
A377:         D.i in [.a,b.] by A7,A332,INTEGRA1:6;
A379:         lower_bound divset(D,i) = D.(i-1) &
              upper_bound divset(D,i) = D.i by A332,A376,INTEGRA1:def 4;
              i-1 in dom D by A332,A376,INTEGRA1:7; then
A382:         D.(i-1) < D.i by A332,XREAL_1:146,VALUED_0:def 13;
              D.(i-1) = a or D.(i-1) in ].a,b.]
                      by A376,A7,A332,INTEGRA1:7,XXREAL_1:6; then
              per cases by XXREAL_1:2;
              suppose A384: a = D.(i-1);
                1 + 0 < i + 1 by XREAL_1:8,A354; then
A386:           Dp1(m,D,(i+1)) = m.(D.(i+1 -1)) by A356,defDp1
                              .= ([.a,D.i .] --> 1) +* (]. D.i,b.] --> 0)
                                    by A8,A377,A384,A382;
A387:           Dp1(m,D,i) = m.(D.(i-1)) by A355,A376,defDp1
                          .= [.a,b.] -->0 by A8,A378,A384;
A389:           dom f0 = A by A7,FUNCOP_1:13;
A390:           a <= D.i & D.i <= b by A377,XXREAL_1:1;
A392:           dom f1 = dom g1 \/ dom g2 by FUNCT_4:def 1
                      .= A by A7,B20,A390,XXREAL_1:167;
                rng f0 c= REAL; then
                reconsider f0 as Function of A,REAL by A389,FUNCT_2:2;
                rng f1 c= REAL; then
                reconsider f1 as Function of A,REAL by A392,FUNCT_2:2;
A394:           qi.t = r * (f1.t - f0.t)
                proof
                  reconsider H = Dp1(m,D,i+1) - Dp1(m,D,i)
                    as Element of R_Normed_Algebra_of_BoundedFunctions(A)
                      by Lm1;
                  reconsider h = H as Function of A,REAL by LM88;
                  qi = r * H by Lm1,A351,A338; then
                  qi.t = r * (h.t) by C0SP1:30
                      .= r * (f1.t - f0.t) by A2,A386,A387,C0SP1:34;
                  hence thesis;
                end;
A395:           t in [. D.(i-1),D.i .] by A379,INTEGRA1:4,A333;
                a <= D.(i-1) & D.(i-1) <= b by A378,XXREAL_1:1; then
B396:           [. D.(i-1),D.i .] c= [.a,D.i .] by XXREAL_1:34; then
A398:           f1.t = ([.a,D.i .] --> 1).t
                          by A395,FUNCT_4:16,B20,XXREAL_1:90
                    .= 1 by B396,A395,FUNCOP_1:7;
                f0.t = 0 by A7,FUNCOP_1:7;
                hence thesis by A351,A394,A398;
              end;
              suppose A399: a < D.(i-1);
                1 + 0 < i + 1 by XREAL_1:8,A354; then
A401:           Dp1(m,D,(i+1)) = m.(D.(i+1 -1)) by A356,defDp1
                              .= ([.a,D.i .] --> 1) +* (]. D.i,b.] --> 0)
                                    by A8,A377,A399,A382;
A402:           Dp1(m,D,i) = m.(D.(i-1)) by A355,A376,defDp1
                          .= ([.a,D.(i-1) .] --> 1) +* (]. D.(i-1),b.] --> 0)
                                by A8,A378,A399;
A404:           a <= D.i & D.i <= b by A377,XXREAL_1:1;
A405:           a <= D.(i-1) & D.(i-1) <= b by A378,XXREAL_1:1;
A406:           dom f1 = dom g1 \/ dom g2 by FUNCT_4:def 1
                      .= A by A7,B20,A404,XXREAL_1:167;
A407:           dom f2 = dom h1 \/ dom h2 by FUNCT_4:def 1
                      .= A by A7,B20,A405,XXREAL_1:167;
                rng f1 c= REAL; then
                reconsider f1 as Function of A,REAL by A406,FUNCT_2:2;
                rng f2 c= REAL; then
                reconsider f2 as Function of A,REAL by A407,FUNCT_2:2;
A408:           a <= t & t <= b by XXREAL_1:1,A7;
A409:           qi.t = r * (f1.t - f2.t)
                proof
                  reconsider H = Dp1(m,D,i+1) - Dp1(m,D,i)
                    as Element of R_Normed_Algebra_of_BoundedFunctions(A)
                      by Lm1;
                  reconsider h = H as Function of A,REAL by LM88;
                  qi = r * H by Lm1,A351,A338; then
                  qi.t = r * (h.t) by C0SP1:30
                      .= r * (f1.t - f2.t) by A2,A401,A402,C0SP1:34;
                  hence thesis;
                end;
A410:           t in [. D.(i-1),D.i .] by A379,INTEGRA1:4,A333;
B411:           [. D.(i-1),D.i .] c= [.a,D.i .] by A405,XXREAL_1:34; then
A413:           f1.t = ([.a,D.i .] --> 1).t
                          by A410,FUNCT_4:16,B20,XXREAL_1:90
                    .= 1 by B411,A410,FUNCOP_1:7;
                D.(i-1) < t & t <= D.i by A332,INTEGRA1:def 4,A334,A376; then
A414:           t in ]. D.(i-1),b.] by A408; then
                f2.t = (]. D.(i-1),b.] --> 0).t by B20,FUNCT_4:13
                    .= 0 by A414,FUNCOP_1:7;
                hence thesis by A351,A409,A413;
              end;
            end;
          end;
          for k be Nat st k in dom z & k <> i holds z.k = 0
          proof
            let k be Nat;
            assume that
A415:       k in dom z and
A416:       k <> i;
            consider r be Real such that
              p2.k in dom (u|divset((T.n),k)) and
              r = (u|divset((T.n),k)).(p2.k) and
A420:         q2.k = r * ( Dp1(m,(T.n),k+1) - Dp1(m,(T.n),k) )
                       by A327,A328,A349,A415;
            consider qk be Function of A,REAL such that
A423:         qk = q2.k & z.k = qk.t by A344,A415;
A425:       k in dom D by A344,A415,FINSEQ_1:def 3;
A426:       1 <= k <= len D by A344,A415,FINSEQ_1:1; then
            k + 0 <= len D + 1 by XREAL_1:7; then
A427:       k in Seg (len D + 1) by A426;
e:          1 + 0 <= k + 1 by XREAL_1:7;
            k + 1 <= len D + 1 by XREAL_1:7,A426; then
A428:       k + 1 in Seg (len D + 1) by e;
            set f0 = [.a,b.] --> 0;
            set f1 = ([.a,D.k .] --> 1) +* (]. D.k,b.] --> 0);
            set g1 = [.a,D.k .] --> 1;
            set g2 = ]. D.k,b.] --> 0;
            set f2 = ([.a,D.(k-1) .] --> 1) +* (]. D.(k-1),b.] --> 0);
            set h1 = [.a,D.(k-1) .] --> 1;
            set h2 = ]. D.(k-1),b.] --> 0;
B30:        dom f0 = [.a,b.] &
            dom g1 = [.a,D.k .] & dom g2 = ]. D.k,b.] &
            dom h1 = [.a,D.(k-1) .] & dom h2 = ]. D.(k-1),b.] by FUNCOP_1:13;
            per cases;
            suppose A429: k = 1; then
A433:         lower_bound divset(D,i) = D.(i-1) &
              upper_bound divset(D,i) = D.i by A332,INTEGRA1:def 4,A416;
A435:         i-1 in dom D by A332,A429,A416,INTEGRA1:7;
A440:         D.k in [.a,b.] by A7,A425,INTEGRA1:6; then
A446:         a <= D.k & D.k <= b by XXREAL_1:1;
A445:         dom f0 = A by A7,FUNCOP_1:13;
A447:         dom f1 = dom g1 \/ dom g2 by FUNCT_4:def 1
                    .= A by A7,B30,A446,XXREAL_1:167;
              rng f0 c= REAL; then
              reconsider f0 as Function of A,REAL by A445,FUNCT_2:2;
              rng f1 c= REAL; then
              reconsider f1 as Function of A,REAL by A447,FUNCT_2:2;
A451:         a < D.k by A264,A429;
A452:         Dp1(m,D,(k+1)) = m.(D.(k+1 -1)) by A428,defDp1,A429
                            .= ([.a,D.k .] --> 1) +* (]. D.k,b.] --> 0)
                                  by A8,A440,A451;
              A = [.lower_bound A,upper_bound A.] by INTEGRA1:4; then
A438:         lower_bound A = a by A7,INTEGRA1:5;
A439:         a in [.a,b.] by A7;
A443:         Dp1(m,D,k) = m.(lower_bound A) by A427,A429,defDp1
                        .= [.a,b.] -->0 by A8,A438,A439;
A453:         qk.t = r * (f1.t - f0.t)
              proof
                reconsider H = Dp1(m,D,k+1) - Dp1(m,D,k)
                  as Element of R_Normed_Algebra_of_BoundedFunctions(A) by Lm1;
                reconsider h = H as Function of A,REAL by LM88;
                qk = r * H by Lm1,A423,A420; then
                qk.t = r * (h.t) by C0SP1:30
                    .= r * (f1.t - f0.t) by A2,A443,A452,C0SP1:34;
                hence thesis;
              end;
              k < i by A429,A416,A354,XXREAL_0:1; then
              k+1 <= i by NAT_1:13; then
A456:         k+1 -1 <= i-1 by XREAL_1:13;
              D.k <= D.(i-1)
              proof
                k = i-1 or k < i-1 by A456,XXREAL_0:1;
                hence thesis by A425,A435,VALUED_0:def 13;
              end; then
A457:         D.k < t by A334,A429,A416,XXREAL_0:2,A433;
              a <= t <= b by XXREAL_1:1,A7; then
A459:         t in ].D.k,b .] by A457; then
A461:         f1.t = (].D.k,b .] --> 0).t by B30,FUNCT_4:13
                  .= 0 by A459,FUNCOP_1:7;
              f0.t = 0 by A7,FUNCOP_1:7;
              hence thesis by A423,A453,A461;
            end;
            suppose A462: k <> 1; then
A464:         D.(k-1) in [.a,b.] by A7,A425,INTEGRA1:7;
A463:         D.k in [.a,b.] by A7,A425,INTEGRA1:6;
A467:         k-1 in dom D by A425,A462,INTEGRA1:7; then
A468:         D.(k-1) < D.k by A425,XREAL_1:146,VALUED_0:def 13;
              1 < k by A426,A462,XXREAL_0:1; then
              1+1 <= k by NAT_1:13; then
A470:         2-1 <= k -1 by XREAL_1:13;
              1 <= len D by A426,XXREAL_0:2; then
A471:         1 in dom D by FINSEQ_3:25;
              D.1 <= D.(k-1)
              proof
                1 = k -1 or 1 < k -1 by A470,XXREAL_0:1;
                hence thesis by A467,A471,VALUED_0:def 13;
              end; then
A472:         a < D.(k-1) by A264,XXREAL_0:2;
              1 + 0 < k + 1 by XREAL_1:8,A426; then
A474:         Dp1(m,D,(k+1)) = m.(D.(k+1 -1)) by A428,defDp1
                            .= ([.a,D.k .] --> 1) +* (]. D.k,b.] --> 0)
                                  by A8,A463,A472,A468;
A475:         Dp1(m,D,k) = m.(D.(k-1)) by A427,A462,defDp1
                        .= ([.a,D.(k-1) .] --> 1) +* (]. D.(k-1),b.] --> 0)
                              by A8,A464,A472;
A477:         a <= D.k & D.k <= b by A463,XXREAL_1:1;
A478:         a <= D.(k-1) & D.(k-1) <= b by A464,XXREAL_1:1;
A479:         dom f1 = dom g1 \/ dom g2 by FUNCT_4:def 1
                    .= A by A7,B30,A477,XXREAL_1:167;
A480:         dom f2 = dom h1 \/ dom h2 by FUNCT_4:def 1
                    .= A by A7,B30,A478,XXREAL_1:167;
              rng f1 c= REAL; then
              reconsider f1 as Function of A,REAL by A479,FUNCT_2:2;
              rng f2 c= REAL; then
              reconsider f2 as Function of A,REAL by A480,FUNCT_2:2;
A482:         qk.t = r * (f1.t - f2.t)
              proof
                reconsider H = Dp1(m,D,k+1) - Dp1(m,D,k)
                  as Element of R_Normed_Algebra_of_BoundedFunctions(A)
                    by Lm1;
                reconsider h = H as Function of A,REAL by LM88;
                qk = r * H by Lm1,A423,A420; then
                qk.t = r * (h.t) by C0SP1:30
                    .= r * (f1.t - f2.t) by A2,A474,A475,C0SP1:34;
                hence thesis;
              end;
              per cases by A416,XXREAL_0:1;
              suppose i < k; then
                i +1 <= k by NAT_1:13; then
A485:           i+1 -1 <= k-1 by XREAL_1:13;
A486:           D.i <= D.(k-1)
                proof
                  i = k -1 or i < k -1 by A485,XXREAL_0:1;
                  hence thesis by A332,A467,VALUED_0:def 13;
                end;
A488:           (upper_bound divset(D,i)) <= D.(k-1)
                proof
                  per cases;
                  suppose i = 1;
                    hence (upper_bound divset(D,i)) <=D.(k-1)
                            by A486,A332,INTEGRA1:def 4;
                  end;
                  suppose i <> 1;
                    hence (upper_bound divset(D,i)) <=D.(k-1)
                            by A486,A332,INTEGRA1:def 4;
                  end;
                end;
                a <= t & t <= (upper_bound divset(D,i))
                   by B352,XXREAL_1:1,A7; then
A489:           a <= t & t <= D.(k-1) by A488,XXREAL_0:2; then
A490:           t in [.a,D.(k-1) .];
                a <= t & t <= D.k by A468,A489,XXREAL_0:2; then
A491:           t in [.a,D.k .]; then
A494:           f1.t = ([.a,D.k .] --> 1).t
                                by FUNCT_4:16,B30,XXREAL_1:90
                    .= 1 by A491,FUNCOP_1:7;
                f2.t = ([.a,D.(k-1) .] --> 1).t
                          by A490,FUNCT_4:16,B30,XXREAL_1:90
                    .= 1 by A490,FUNCOP_1:7;
                hence thesis by A423,A482,A494;
              end;
              suppose A496: k < i; then
                k +1 <= i by NAT_1:13; then
A499:           k+1 -1 <= i-1 by XREAL_1:13;
A500:           i-1 in dom D by A332,A496,A426,INTEGRA1:7;
A502:           D.k <= D.(i-1)
                proof
                  k = i-1 or k < i-1 by A499,XXREAL_0:1;
                  hence thesis by A425,A500,VALUED_0:def 13;
                end;
                D.(i-1) < t by A496,A426,A334,A332,INTEGRA1:def 4; then
A504:           D.k < t by A502,XXREAL_0:2; then
A505:           D.(k-1) < t by A468,XXREAL_0:2;
A506:           a <= t & t <= b by XXREAL_1:1,A7; then
A507:           t in ]. D.k,b.] by A504; then
A509:           f1.t = (]. D.k,b.] --> 0).t by B30,FUNCT_4:13
                    .= 0 by A507,FUNCOP_1:7;
A508:           t in ]. D.(k-1),b.] by A505,A506; then
                f2.t = (]. D.(k-1),b.]--> 0).t by B30,FUNCT_4:13
                    .= 0 by A508,FUNCOP_1:7;
                hence thesis by A423,A482,A509;
              end;
            end;
          end;
          hence g1.t = g.(p2.i)
                  by A317,A336,FUNCT_1:47,A348,A337,A347,A357,INTEGR23:6;
        end;
A511:   for t be Element of A holds |.g1.t - g.t.| < r/2
        proof
          let t be Element of A;
          A = [.lower_bound A,upper_bound A.] by INTEGRA1:4; then
A512:     lower_bound A = a by A7,INTEGRA1:5;
          a < (T.n).1 by A264; then
          consider j be Element of NAT such that
A540:       j in dom (T.n) and
A541:       t in divset((T.n),j) and
A542:       ( j=1 or
                (lower_bound divset(T.n,j) < t
                & t <= ( upper_bound divset(T.n,j) )) ) by A512,LM94;
          reconsider i=j as Nat;
          set tD = p2.i;
A543:     for tD,t be Real st tD in dom g & t in dom g & |.tD - t.| < s
            holds |.g.tD - g.t.| < r/2
          proof
            let tD,t be Real;
            assume A544: tD in dom g & t in dom g & |.tD - t.| < s;
B545:       dom g = dom(u|A) by A260,RELAT_1:62,A317; then
            (u|A).tD = u.tD & (u|A).t = u.t by A544,FUNCT_1:47;
            hence |.g.tD - g.t.| < r/2 by A317,B545,A323,A544;
          end;
          p2.i in dom (u|divset((T.n),i)) by A327,A540; then
          p2.i in dom u /\ divset((T.n),i) by RELAT_1:61; then
A549:     p2.i in dom u & p2.i in divset((T.n),i) by XBOOLE_0:def 4; then
B551:     |.tD - t.| < s by A329,A540,A541,INTEGR20:12;
          g1.t = g.tD by A331,A540,A541,A542;
          hence |.g1.t - g.t.| < r/2 by B551,A543,A549,A260,A317;
        end;
        reconsider z = xD.n - v0 as Element of
          R_Normed_Algebra_of_BoundedFunctions(A) by Lm1;
        reconsider g0 = z as Function of A,REAL by LM88;
        g0 in BoundedFunctions(A); then
        consider g2 be Function of A,REAL such that
A552:     g2=g0 & g2|A is bounded;
        now
          let k be Real;
          assume k in PreNorms g0; then
          consider t be Element of A such that
A553:     k = |.g0.t.|;
          |.g1.t - g.t.| < r/2 by A511;
          hence k <= r/2 by A553,A2,A317,A330,C0SP1:34;
        end; then
        upper_bound PreNorms g0 <= r/2 by SEQ_4:45; then
        ||.z.|| <= r/2 by A552,C0SP1:20; then
        ||. xD.n - v0 .|| <= r/2 by Lm1;
        hence thesis by A321,XXREAL_0:2;
      end;
      take m0;
      thus thesis by A325;
    end; then
A555: xD is convergent by NORMSP_1:def 6; then
A556: lim xD = v by A318,NORMSP_1:def 7;
    dom f = the carrier of V by FUNCT_2:def 1; then
A557: rng xD c= dom f;
    v in the carrier of V; then
A558: v in dom f by FUNCT_2:def 1;
    consider K be Real such that
A559: 0 <= K &
      for x be Point of V holds |. f.x .| <= K * ||. x .||
        by DUALSP01:def 9;
    for r be Real st 0 < r ex s be Real st 0 < s &
      for x1 be Point of V st x1 in dom f & ||. x1 - v .|| < s holds
        |. f/.x1 - f/.v .| < r
    proof
      let r be Real;
      assume A561: 0 < r;
      reconsider s = r/(K+1) as Real;
A563: for x1 be Point of V st x1 in dom f & ||. x1 - v .|| < s holds
        |. f/.x1 - f/.v .| < r
      proof
        let x1 be Point of V;
        assume that
        x1 in dom f and
A565:   ||. x1 - v .|| < s;
        |. f/.x1 - f/.v .| = |. f.(x1 - v) .| by HAHNBAN:19; then
A567:   |. f/.x1 - f/.v .| <= K * ||. x1 - v .|| by A559;
        K < K+1 by XREAL_1:145; then
A569:   K * ||. x1 - v .|| <= (K+1) * ||. x1 - v .|| by XREAL_1:64;
        (K+1) * ||. x1 - v .|| < (K+1) * s by A559,A565,XREAL_1:68; then
        K * ||. x1 - v .|| < (K+1) * s by A569,XXREAL_0:2; then
        |. f/.x1 - f/.v .| < (K+1) * s by A567,XXREAL_0:2;
        hence |. f/.x1 - f/.v .| < r by A559,XCMPLX_1:87;
      end;
      take s;
      thus thesis by A559,A561,XREAL_1:139,A563;
    end; then
    f is_continuous_in v by A558,NFCONT_1:8; then
    f/.v = lim (f/*xD) by A555,A556,A557,NFCONT_1:def 6; then
    lim(f*xD) = f.v by A557,FUNCT_2:def 11; then
    f.u = integral(u,rho) by B314,FUNCT_2:def 8,A315;
    hence x.u = integral(u,rho) by A6,FUNCT_1:49,A269;
  end; then
B573: ||.x.|| <= total_vd(rho) by A251,Lm83;
  take rho;
  thus thesis by B250,A259,B573,A258,XXREAL_0:1;
end;
