reserve X,G for RealNormSpace-Sequence,
          Y for RealNormSpace;
reserve f for MultilinearOperator of X,Y;

theorem LM01:
  for X be RealNormSpace-Sequence,
    Y be RealNormSpace,
    f be MultilinearOperator of X,Y,
    K being Real
  st (0 <= K &
      for x being Point of product X
      holds ||. f.x .|| <= K * NrProduct x) holds
  for v0 being Point of product X
  ex M be Real
  st 0 <= M
   & for v1 be Point of product X
     st ||.v1-v0.|| <= 1
     holds
     ex F be FinSequence of REAL
     st dom F = dom X
      & ||.f/.v1 - f/.v0.|| <= M * K * Sum F
      & for i be Element of dom X
        holds F.i = ||.(v1-v0).i.||
  proof
    let X be RealNormSpace-Sequence,
        Y be RealNormSpace,
        f be MultilinearOperator of X,Y,
        K be Real;
    assume that
    A1: 0 <= K and
    A2: for x being Point of product X
         holds ||. f.x .|| <= K * NrProduct x;
    A3: product X = NORMSTR(# (product (carr X)),(zeros X),
      [:(addop X):],[:(multop X):],(productnorm X) #) by PRVECT_2:6;
    let v0 be Point of product X;
    consider g be Function such that
    A4: v0 = g & dom g = dom carr X
        & for i be object st i in dom carr X
          holds g.i in (carr X).i by A3,CARD_3:def 5;
    dom g = Seg len (carr X) by A4,FINSEQ_1:def 3; then
    reconsider Cv0 = v0 as FinSequence by A4,FINSEQ_1:def 2;
    set L = ||.v0.|| + 3;
    set M = L |^ len X;
    take M;
    thus 0 <= M by POWER:3;
    defpred P[Nat] means
    for v0, v1 be Point of product X,
        Cv0,Cv1 be FinSequence
    st ||.v1-v0.|| <= 1
     & v0 = Cv0 & v1 = Cv1 & $1 <=len X
     & Cv1 | (len X -'$1 ) = Cv0 | (len X -'$1 ) holds
    ex F be FinSequence of REAL
    st dom F = Seg $1
     & ||.f/.v1 - f/.v0.|| <= (||.v0.|| + 3) |^ len X * K * Sum F
     & for n be Nat st n in Seg $1 holds
       ex i be Element of dom X
       st i = len X -'$1 + n & F.n = ||.(v1-v0).i.||;
    A6: P[0]
    proof
      let v0,v1 be Point of product X,
        Cv0,Cv1 be FinSequence;
      assume
      A7: ||.v1-v0.|| <= 1
       & v0 = Cv0 & v1 = Cv1 & 0 <=len X
       & Cv1 | (len X -'0 ) = Cv0 | (len X -'0 );
      A8: len X -'0 = (len X + 0) -'0
      .= len X by NAT_D:34;
      reconsider F = <*>REAL as FinSequence of REAL;
      take F;
      thus dom F = Seg 0;
      consider g be Function such that
      A9: v0 = g & dom g = dom carr X
        & for i be object st i in dom carr X
          holds g.i in (carr X).i by A3,CARD_3:def 5;
      A10: dom g = Seg len (carr X) by A9,FINSEQ_1:def 3; then
      reconsider Cv0 = v0 as FinSequence by A9,FINSEQ_1:def 2;
      A11: len Cv0 = len carr X by A9,A10,FINSEQ_1:def 3
      .= len X by PRVECT_1:def 11;
      consider g be Function such that
      A12: v1 = g & dom g = dom carr X
        & for i be object st i in dom carr X
          holds g.i in (carr X).i by A3,CARD_3:def 5;
      dom Cv1 = Seg len carr X by A7,A12,FINSEQ_1:def 3; then
      A14: len Cv1 = len carr X by FINSEQ_1:def 3
      .= len X by PRVECT_1:def 11;
      Cv1 = Cv0 | len X by A7,A8,A14,FINSEQ_1:58
      .= Cv0 by A11,FINSEQ_1:58; then
      f/.v1 - f/.v0 = 0.Y by A7,RLVECT_1:15;
      hence ||.f/.v1 - f/.v0.|| <= (||.v0.||+3) |^len X * K * Sum F
        by RVSUM_1:72;
      thus for n be Nat st n in Seg 0 holds
      ex i be Element of dom X
      st i = len X -'0 + n & F.n = ||.(v1-v0).i.||;
    end;
    A16: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume
      A17: P[k];
      let v0,v1 be Point of product X,
        Cv0,Cv1 be FinSequence;
      assume
      A18: ||.v1-v0.|| <= 1
        & v0 = Cv0 & v1 = Cv1 & k+1 <=len X
        & Cv1 | (len X -'(k+1) ) = Cv0 | (len X -'(k+1) );
      consider g be Function such that
      A19: v0 = g & dom g = dom carr X
          & for i be object st i in dom carr X
            holds g.i in (carr X).i by A3,CARD_3:def 5;
      dom g = Seg len (carr X) by A19,FINSEQ_1:def 3; then
      reconsider Cv0 = v0 as FinSequence by A19,FINSEQ_1:def 2;
      dom Cv0 = Seg len carr X by A19,FINSEQ_1:def 3; then
      A21: len Cv0 = len carr X by FINSEQ_1:def 3
      .= len X by PRVECT_1:def 11; then
      A22: dom Cv0 = Seg len X by FINSEQ_1:def 3
      .= dom X by FINSEQ_1:def 3;
      consider g1 be Function such that
      A23: v1 = g1 & dom g1 = dom carr X
        & for i be object st i in dom carr X
          holds g1.i in (carr X).i by A3,CARD_3:def 5;
      dom g1 = Seg len (carr X) by A23,FINSEQ_1:def 3; then
      reconsider Cv1 = v1 as FinSequence by A23,FINSEQ_1:def 2;
      1 <= 1+k by NAT_1:11; then
      1 <= len X by A18,XXREAL_0:2; then
      len X in Seg len X; then
      reconsider i = len X as Element of dom X by FINSEQ_1:def 3;
      per cases;
      suppose
        A25: k = 0; then
        A26: len X -'(k+1) = len X - 1 by A18,XREAL_0:def 2;
        for j be Element of dom X st i <> j holds Cv1.j = Cv0.j
        proof
          let j be Element of dom X;
          j in dom X; then
          j in Seg len X by FINSEQ_1:def 3; then
          A27: 1 <= j & j <= len X by FINSEQ_1:1;
          assume i <> j; then
          j < len X by A27,XXREAL_0:1; then
          j+1 <= len X by NAT_1:13; then
          j +1 -1 <= len X - 1 by XREAL_1:9; then
          A28: j in Seg ( len X -'(k+1)) by A26,A27;
          thus Cv1.j = (Cv0 | (len X -'(k+1))).j by A18,A28,FUNCT_1:49
          .= Cv0.j by A28,FUNCT_1:49;
        end; then
        A29: ||.f/.v1 - f/.v0.|| <= (||.v0.||+1) |^ len X
          * K * ||.(v1-v0).i.|| by A1,A2,A18,LM02;
        set F = <* ||. (v1-v0).i .|| *>;
        rng F c= REAL; then
        reconsider F as FinSequence of REAL by FINSEQ_1:def 4;
        take F;
        thus dom F = Seg len F by FINSEQ_1:def 3
        .= Seg (k+1) by A25,FINSEQ_1:40;
        A31: ||.f/.v1 - f/.v0.|| <= (||.v0.||+1) |^ len X
         * K * Sum F by A29,RVSUM_1:73;
        ||.v0.|| + 1 + 0 <= ||.v0.|| + 1 + 2 by XREAL_1:7; then
        A32: ( ||.v0.||+1) |^ len X <= ( ||.v0.||+3) |^ len X
          by PREPOWER:9;
        0 <= Sum F by RVSUM_1:73; then
        ( ||.v0.||+1) |^ len X * (K * Sum F)
        <= ( ||.v0.||+3) |^ len X * (K * Sum F) by A1,A32,XREAL_1:64;
        hence ||.f/.v1 -f/.v0 .|| <= (||.v0.||+3) |^len X * K * Sum F
          by A31,XXREAL_0:2;
        thus for n be Nat st n in Seg (k+1) holds
          ex j be Element of dom X st j = len X -'(k+1) + n
           & F.n = ||.(v1-v0).j.||
        proof
          let n be Nat;
          assume
          A33: n in Seg (k+1); then
          A34: len X -'(k+1) + n
           = len X -1 + 1 by A25,A26,FINSEQ_1:2,TARSKI:def 1
          .= i;
          take i;
          n = 1 by A25,A33,FINSEQ_1:2,TARSKI:def 1;
          hence thesis by A34;
        end;
      end;
      suppose k <> 0;
        A35: k+1 - k <= len X - k by A18,XREAL_1:13;
        A36: len X - k <= len X - 0 by XREAL_1:13;
        len X-k in NAT by A35,INT_1:3; then
        len X-k in Seg len X by A35,A36; then
        reconsider k1 =len X -k as Element of dom X by FINSEQ_1:def 3;
        Cv0.k1 = v0.k1; then
        reconsider Cv0k1 = Cv0.k1 as Point of X.k1;
        k <= k+1 by NAT_1:11; then
        A38: k <= len X by A18,XXREAL_0:2;
        reconsider v2= (reproj (k1,v1)). Cv0k1 as Point of product X;
        consider g be Function such that
        A39: v2 = g & dom g = dom carr X
          & for i be object st i in dom carr X
            holds g.i in (carr X).i by A3,CARD_3:def 5;
        A40: dom g =Seg len (carr X) by A39,FINSEQ_1:def 3; then
        reconsider Cv2 = v2 as FinSequence by A39,FINSEQ_1:def 2;
        A41: len Cv2 = len carr X by A39,A40,FINSEQ_1:def 3
        .= len X by PRVECT_1:def 11;
        reconsider w12 = v1-v2 as Element of product carr X by A3;
        reconsider w02 = v2-v0 as Element of product carr X by A3;
        reconsider w10 = v1-v0 as Element of product carr X by A3;
        ||.v2-v0.|| <= ||.v1-v0.||
        proof
          A42: ||.v2-v0.|| = |. normsequence (X,w02) .| by A3,PRVECT_2:def 12;
          A43: ||.v1-v0.|| = |. normsequence (X,w10) .| by A3,PRVECT_2:def 12;
          A44: 0 <= Sum(sqr normsequence (X,w02)) by RVSUM_1:86;
          for j being Nat st j in Seg len X holds
          (sqr normsequence (X,w02)).j <= (sqr normsequence (X,w10)).j
          proof
            let j be Nat;
            assume
            A45: j in Seg len X;
            reconsider i = j as Element of dom X by A45,FINSEQ_1: def 3;
            A46: (sqr normsequence(X,w02)).j
             = (normsequence(X,w02).j) ^2 by VALUED_1:11
            .= (||.(v2-v0).i.||) ^2 by PRVECT_2:def 11;
            A47: (sqr normsequence (X,w10)).j
             = (normsequence (X,w10).j) ^2 by VALUED_1:11
            .= (||.(v1-v0).i.||) ^2 by PRVECT_2:def 11;
            A48: (v2-v0).i = v2.i - v0.i by LOPBAN10:26;
            A49: (v1-v0).i = v1.i - v0.i by LOPBAN10:26;
            ||.(v2-v0).i.|| <=||.(v1-v0).i.||
            proof
              per cases;
              suppose
                A50: i = k1;
                v2.i = v0.i by A50,LOPBAN10:15; then
                v2.i - v0.i = 0.(X.i) by RLVECT_1:15;
                hence ||.(v2-v0).i.|| <= ||.(v1-v0).i.|| by A48;
              end;
              suppose
                i <> k1;
                hence ||.(v2-v0).i.|| <= ||.(v1-v0).i.||
                  by A48,A49,LOPBAN10:16;
              end;
            end;
            hence thesis by A46,A47,SQUARE_1:15;
          end;
          hence thesis by A42,A43,A44,SQUARE_1:26,RVSUM_1:82;
        end; then
        A51: ||.v2-v0.|| <= 1 by A18,XXREAL_0:2;
        A52: len X -'k = k1 by XREAL_0:def 2;
        len ( Cv0 | (len X -'k) ) = k1 by A21,A36,A52,FINSEQ_1:59; then
        A53: dom ( Cv0 | (len X -'k) ) = Seg k1 by FINSEQ_1:def 3;
        len ( Cv2 | (len X -'k) ) = k1 by A36,A41,A52,FINSEQ_1:59; then
        A54: dom ( Cv2 | (len X -'k) ) = Seg k1 by FINSEQ_1:def 3;
        A55: len X -'(k+1) = len X -(k+1) by A18,XREAL_0:def 2,XREAL_1:48;
        for j be Nat st j in dom (Cv0 | (len X -'k)) holds
        ( Cv0 | (len X -'k )).j = (Cv2 | (len X -'k)).j
        proof
          let j be Nat;
          assume
          A56: j in dom(Cv0 | (len X -'k)); then
          A57: (Cv0 | (len X -'k)).j = Cv0.j by FUNCT_1:47;
          A59: 1 <= j & j <= k1 by A53,A56,FINSEQ_1:1;
          per cases;
          suppose
            j = k1; then
            Cv0.j = Cv2.j by LOPBAN10:15;
            hence (Cv0 | (len X -'k)).j = (Cv2 | (len X -'k)).j
              by A53,A54,A56,A57,FUNCT_1:47;
          end;
          suppose
            A61: j <> k1; then
            j < k1 by A59,XXREAL_0:1; then
            j+1 <= k1 by NAT_1:13; then
            j + 1 - 1 <= k1 - 1 by XREAL_1:13; then
            A62: j in Seg (len X -'(k+1)) by A55,A59;
            j in dom X by A22,A56,RELAT_1:60,TARSKI:def 3; then
            v2.j = Cv1.j by A61,LOPBAN10:16
            .= (Cv0 | (len X -'(k+1))).j by A18,A62,FUNCT_1:49
            .= v0.j by A62,FUNCT_1:49;
            hence (Cv0 | (len X -'k)).j = (Cv2 | (len X -'k)).j
              by A53,A54,A56,A57,FUNCT_1:47;
          end;
        end; then
        A63: Cv0 | (len X -'k) = Cv2 | (len X -'k) by A53,A54,FINSEQ_1:13;
        consider F1 be FinSequence of REAL such that
        A64: dom F1 = Seg k
        & ||.f/.v2 - f/.v0.|| <= (||.v0.|| + 3) |^len X * K * Sum F1
        & for n be Nat st n in Seg k
          holds ex i be Element of dom X
          st i = len X -'k + n & F1.n = ||.(v2-v0).i.|| by A17,A38,A51,A63;
        ||.v1-v2.|| <= ||.v1-v0.||
        proof
          A65: ||.v1-v2.|| = |. normsequence(X,w12) .| by A3,PRVECT_2:def 12;
          A66: ||.v1-v0.|| = |. normsequence(X,w10) .| by A3,PRVECT_2:def 12;
          A67: 0 <= Sum(sqr normsequence(X,w12)) by RVSUM_1:86;
          for j being Nat st j in Seg len X holds
          (sqr normsequence(X,w12)).j <= (sqr normsequence(X,w10)).j
          proof
            let j be Nat;
            assume
            A68: j in Seg len X;
            reconsider i = j as Element of dom X by A68,FINSEQ_1: def 3;
            A69: (sqr normsequence(X,w12)).j
             = (normsequence (X,w12).j ) ^2 by VALUED_1:11
            .= (||.(v1-v2).i.||) ^2 by PRVECT_2:def 11;
            A70: ( sqr normsequence (X,w10)).j
             = (normsequence(X,w10).j) ^2 by VALUED_1:11
            .= (||.(v1-v0).i.||) ^2 by PRVECT_2:def 11;
            A71: (v1-v2).i = v1.i - v2.i by LOPBAN10:26;
            ||.(v1-v2).i.|| <= ||.(v1-v0).i.||
            proof
              per cases;
              suppose
                i = k1; then
                v2.i = v0.i by LOPBAN10:15;
                hence ||.(v1-v2).i.|| <= ||.(v1-v0).i.|| by A71,LOPBAN10:26;
              end;
              suppose
                i <> k1; then
                v2.i = v1.i by LOPBAN10:16; then
                v1.i - v2.i = 0.(X.i) by RLVECT_1:15;
                hence ||.(v1-v2).i.|| <= ||.(v1-v0).i.|| by A71;
              end;
            end;
            hence thesis by A69,A70,SQUARE_1:15;
          end;
          hence thesis by A65,A66,A67,RVSUM_1:82,SQUARE_1:26;
        end; then
        A74: ||.v1-v2.|| <= 1 by A18,XXREAL_0:2;
        for j be Element of dom X st k1 <> j
        holds Cv1.j = Cv2.j by LOPBAN10:16; then
        A75: ||.f/.v1 -f/.v2 .||
          <= ( ||.v2.||+1) |^ len X * K * ||.(v1-v2).k1.||
            by A1,A2,A74,LM02;
        v2 = v1+(v2-v1) by RLVECT_4:1; then
        A76: ||.v2.|| <= ||.v1.|| + ||.v2-v1.|| by NORMSP_1:def 1;
        ||.v2-v1.|| <= 1 by A74,NORMSP_1:7; then
        ||.v1.|| + ||.v2-v1.|| <= ||.v1.|| + 1 by XREAL_1:7; then
        A77: ||.v2.|| <= ||.v1.|| + 1 by A76,XXREAL_0:2;
        v1 = (v1-v0) + v0 by RLVECT_4:1; then
        ||.v1.|| <= ||.v0.|| + ||.v1-v0.|| by NORMSP_1:def 1; then
        A78: ||.v1.|| <= ||.v0.|| + ||.v0-v1.|| by NORMSP_1:7;
        ||.v0-v1.|| <= 1 by A18,NORMSP_1:7; then
        ||.v0.|| + ||.v0-v1.|| <= ||.v0.|| + 1 by XREAL_1:7; then
        ||.v1.|| <= ||.v0.|| + 1 by A78,XXREAL_0:2; then
        ||.v1.|| + 1 <= ||.v0.|| + 1 + 1 by XREAL_1:7; then
        ||.v2.|| <= ||.v0.|| + 2 by A77,XXREAL_0:2; then
        A79: ||.v2.|| + 1 <= ||.v0.|| + 2 + 1 by XREAL_1:7;
        A80: ( ||.v2.||+1) |^ len X <= ( ||.v0.||+3) |^ len X
          by A79,PREPOWER:9;
        A81: 0 < ( ||.v2.||+1) |^ len X by PREPOWER:6;
        (v1-v2).k1 = v1.k1 - v2.k1 by LOPBAN10:26
        .= v1.k1 - v0.k1 by LOPBAN10:15
        .= (v1-v0).k1 by LOPBAN10:26; then
        ( ||.v2.||+1) |^ len X * (K* ||.(v1-v2).k1.||)
          <= ( ||.v0.||+3) |^ len X * (K * ||.(v1-v0).k1.||)
          by A1,A80,A81,XREAL_1:66; then
        A84: ||.f/.v1 -f/.v2 .||
          <= ( ||.v0.||+3) |^ len X * K * ||.(v1-v0).k1.||
          by A75,XXREAL_0:2;
        set F = <* ||. (v1-v0).k1 .|| *> ^ F1;
        rng F c= REAL; then
        reconsider F as FinSequence of REAL by FINSEQ_1:def 4;
        k is Element of NAT by ORDINAL1:def 12; then
        A85: len F1 = k by A64,FINSEQ_1:def 3;
        len F = len F1 + len <* ||. (v1-v0).k1 .|| *> by FINSEQ_1:22
        .= k + 1 by A85,FINSEQ_1:40; then
        A86: dom F = Seg (k+1) by FINSEQ_1:def 3;
        A87: for n be Nat st n in Seg (k+1) holds
            ex i be Element of dom X
            st i = len X -'(k+1) + n & F.n = ||.(v1-v0).i.||
        proof
          let n be Nat;
          assume n in Seg(k+1); then
          A88: 1 <= n & n <= k+1 by FINSEQ_1:1;
          per cases;
          suppose
            A89: n = 1; then
            A90:len X -'(k+1) + n
             = len X -(k+1) + 1 by A18,XREAL_0:def 2,XREAL_1:48
            .= k1;
            take k1;
            thus thesis by A89,A90,FINSEQ_1:41;
          end;
          suppose
            n <> 1; then
            1 < n by A88,XXREAL_0:1; then
            A91: 1+1 <= n by NAT_1:13;
            A93: len X -(k+1) + n <= len X -(k+1) + (k+1) by A88,XREAL_1:7;
            A94: len X -(k+1) +2 <= len X -(k+1) + n by A91,XREAL_1:7;
            1 + 0 <= 1 + (len X - k) by A35,XREAL_1:7; then
            1 <= len X -'(k+1) + n by A55,A94,XXREAL_0:2; then
            len X -'(k+1) + n in Seg len X by A55,A93; then
            reconsider i = len X -'(k+1) + n as Element of dom X
              by FINSEQ_1:def 3;
            take i;
            thus i = len X -'(k+1) + n;
            A95: 2 -1 <= n -1 by A91,XREAL_1:9;
            A96: n-1 <= k+1 -1 by A88,XREAL_1:9;
            reconsider n1 = n-1 as Element of NAT by A88,INT_1:3;
            A97: n1 in Seg k by A95,A96;
            A98: len <* ||. (v1-v0).k1 .|| *> = 1 by FINSEQ_1:40;
            A100: F.n = F.(1+n1)
            .= F1.n1 by A85,A95,A96,A98,FINSEQ_1:65;
            consider i1 be Element of dom X such that
            A101: i1 = len X -'k + n1 & F1.n1 = ||.(v2-v0).i1.|| by A64,A97;
            A102: i1 = len X -(k+1) + n by A52,A101
            .= i by A18,XREAL_0:def 2,XREAL_1:48;
            A105: k1 + 0 < k1 + 1 by XREAL_1:8;
            A106: k1 + 1 <= k1+n1 by A95,XREAL_1:7;
            (v2-v0).i = v2.i-v0.i by LOPBAN10:26
            .= v1.i -v0.i by A52,A101,A102,A105,A106,LOPBAN10:16
            .= (v1-v0).i by LOPBAN10:26;
            hence F.n = ||.(v1-v0).i.|| by A100,A101,A102;
          end;
        end;
        f/.v1 -f/.v0 = f/.v1 -f/.v2 + f/.v2 - f/.v0 by RLVECT_4:1
        .= (f/.v1 -f/.v2) + (f/.v2 - f/.v0) by RLVECT_1:28; then
        A107: ||. f/.v1 -f/.v0 .||
        <= ||.f/.v1 -f/.v2.|| + ||.f/.v2 - f/.v0.|| by NORMSP_1:def 1;
        A108: ||.f/.v1 -f/.v2.|| + ||.f/.v2 - f/.v0.||
        <= ((||.v0.||+3) |^ len X * K * ||.(v1-v0).k1.||)
        + ((||.v0.||+3) |^len X * K * Sum F1) by A64,A84,XREAL_1:7;
        ((||.v0.||+3) |^ len X * K * ||.(v1-v0).k1.||)
        + ((||.v0.||+3) |^len X * K * Sum F1)
         = ((||.v0.||+3) |^ len X * K) * (||.(v1-v0).k1.|| + Sum F1)
        .= ((||.v0.||+3) |^ len X * K) * Sum F by RVSUM_1:76;
        hence ex F be FinSequence of REAL
        st dom F = Seg (k+1)
         & ||.f/.v1 -f/.v0 .|| <= (||.v0.||+3) |^len X * K * Sum F
         & for n be Nat st n in Seg (k+1)
           holds ex i be Element of dom X
           st i = len X -'(k+1) + n
         & F.n = ||.(v1-v0).i.|| by A86,A87,A107,A108,XXREAL_0:2;
      end;
    end;
    A109: for n be Nat holds P[n] from NAT_1:sch 2(A6,A16);
    let v1 be Point of product X;
    assume
    A110: ||.v1-v0.|| <= 1;
    consider g be Function such that
    A111: v1 = g & dom g = dom carr X
        & for i be object st i in dom carr X
          holds g.i in (carr X).i by A3,CARD_3:def 5;
    dom g = Seg len (carr X) by A111,FINSEQ_1:def 3; then
    reconsider Cv1 = v1 as FinSequence by A111,FINSEQ_1:def 2;
    A112: len X -' len X = ( 0 + len X ) -' len X
    .= 0 by NAT_D:34;
    Cv1 | (len X -' len X) = {} by A112
    .= Cv0 | (len X -' len X ) by A112; then
    consider F be FinSequence of REAL such that
    A113: dom F = Seg len X
    & ||.f/.v1 - f/.v0.|| <= (||.v0.||+3) |^len X * K * Sum F
    & for n be Nat st n in Seg len X holds
      ex i be Element of dom X
      st i = len X -'len X + n & F.n = ||.(v1-v0).i.|| by A109,A110;
    for i be Element of dom X holds F.i = ||.(v1-v0).i.||
    proof
      let i be Element of dom X;
      i in dom X; then
      A116: i in Seg len X by FINSEQ_1:def 3;
      reconsider n = i as Nat;
      consider j be Element of dom X such that
      A117: j = len X -'len X + n & F.n = ||.(v1-v0).j.|| by A113,A116;
      thus thesis by A112,A117;
    end;
    hence thesis by A113,FINSEQ_1:def 3;
  end;
