ndiff_5.miz



    begin

    reserve j for set;

    reserve p,r for Real;

    reserve S,T,F for RealNormSpace;

    reserve x0 for Point of S;

    reserve g for PartFunc of S, T;

    reserve c for constant sequence of S;

    reserve R for RestFunc of S, T;

    reserve G for RealNormSpace-Sequence;

    reserve i for Element of ( dom G);

    reserve f for PartFunc of ( product G), F;

    reserve x for Element of ( product G);

    theorem :: NDIFF_5:1

    

     Th1: for R be Function of REAL , S holds R is RestFunc-like iff for r be Real st r > 0 holds ex d be Real st d > 0 & for z be Real st z <> 0 & |.z.| < d holds (( |.z.| " ) * ||.(R /. z).||) < r

    proof

      let R be Function of REAL , S;

      

       A1: ( dom R) = REAL by PARTFUN1:def 2;

       A2:

      now

        assume

         A3: R is RestFunc-like;

        assume not (for r be Real st r > 0 holds ex d be Real st d > 0 & for z be Real st z <> 0 & |.z.| < d holds (( |.z.| " ) * ||.(R /. z).||) < r);

        then

        consider r be Real such that

         A4: r > 0 and

         A5: for d be Real st d > 0 holds ex z be Real st z <> 0 & |.z.| < d & not (( |.z.| " ) * ||.(R /. z).||) < r;

        defpred P[ Nat, Real] means $2 <> 0 & |.$2.| < (1 / ($1 + 1)) & not (( |.$2.| " ) * ||.(R /. $2).||) < r;

        

         A6: for n be Element of NAT holds ex z be Element of REAL st P[n, z]

        proof

          let n be Element of NAT ;

          set d = (1 / (n + 1));

          consider z be Real such that

           A7: z <> 0 & |.z.| < d & not (( |.z.| " ) * ||.(R /. z).||) < r by A5;

          reconsider z as Element of REAL by XREAL_0:def 1;

          take z;

          thus thesis by A7;

        end;

        consider s be Real_Sequence such that

         A8: for n be Element of NAT holds P[n, (s . n)] from FUNCT_2:sch 3( A6);

        

         A9: for n be Nat holds P[n, (s . n)]

        proof

          let n be Nat;

          n in NAT by ORDINAL1:def 12;

          hence thesis by A8;

        end;

         A10:

        now

          let p be Real;

          assume

           A11: 0 < p;

          consider n be Nat such that

           A12: (p " ) < n by SEQ_4: 3;

          reconsider q0 = 0 , q1 = 1 as Real;

          ((p " ) + q0) < (n + q1) by A12, XREAL_1: 8;

          then

           A13: (1 / (n + 1)) < (1 / (p " )) by A11, XREAL_1: 76;

          take n;

          let m be Nat;

          assume n <= m;

          then (n + 1) <= (m + 1) by XREAL_1: 6;

          then (1 / (m + 1)) <= (1 / (n + 1)) by XREAL_1: 118;

          then |.((s . m) - 0 ).| < (1 / (n + 1)) by A9, XXREAL_0: 2;

          hence |.((s . m) - 0 ).| < p by A13, XXREAL_0: 2;

        end;

        then s is convergent by SEQ_2:def 6;

        then ( lim s) = 0 by A10, SEQ_2:def 7;

        then

        reconsider s as 0 -convergent non-zero Real_Sequence by A10, A9, SEQ_1: 5, SEQ_2:def 6, FDIFF_1:def 1;

        ((s " ) (#) (R /* s)) is convergent & ( lim ((s " ) (#) (R /* s))) = ( 0. S) by A3, NDIFF_3:def 1;

        then

        consider n0 be Nat such that

         A16: for m be Nat st n0 <= m holds ||.((((s " ) (#) (R /* s)) . m) - ( 0. S)).|| < r by A4, NORMSP_1:def 7;

        

         A17: n0 in NAT by ORDINAL1:def 12;

        

         A19: ||.(((s . n0) " ) * (R /. (s . n0))).|| = ( |.((s . n0) " ).| * ||.(R /. (s . n0)).||) by NORMSP_1:def 1

        .= (( |.(s . n0).| " ) * ||.(R /. (s . n0)).||) by COMPLEX1: 66;

        

         A20: ( rng s) c= ( dom R) by A1;

         ||.((((s " ) (#) (R /* s)) . n0) - ( 0. S)).|| = ||.(((s " ) (#) (R /* s)) . n0).|| by RLVECT_1: 13

        .= ||.(((s " ) . n0) * ((R /* s) . n0)).|| by NDIFF_1:def 2

        .= ||.(((s . n0) " ) * ((R /* s) . n0)).|| by VALUED_1: 10

        .= ||.(((s . n0) " ) * (R /. (s . n0))).|| by A20, FUNCT_2: 109, A17;

        hence for r be Real st r > 0 holds ex d be Real st d > 0 & for z be Real st z <> 0 & |.z.| < d holds (( |.z.| " ) * ||.(R /. z).||) < r by A9, A16, A19;

      end;

      now

        assume

         A21: for r be Real st r > 0 holds ex d be Real st d > 0 & for z be Real st z <> 0 & |.z.| < d holds (( |.z.| " ) * ||.(R /. z).||) < r;

        now

          let s be 0 -convergent non-zero Real_Sequence;

          

           A22: s is convergent & ( lim s) = 0 ;

           A23:

          now

            let r be Real;

            assume r > 0 ;

            then

            consider d be Real such that

             A24: d > 0 and

             A25: for z be Real st z <> 0 & |.z.| < d holds (( |.z.| " ) * ||.(R /. z).||) < r by A21;

            consider n0 be Nat such that

             A26: for m be Nat st n0 <= m holds |.((s . m) - 0 ).| < d by A22, A24, SEQ_2:def 7;

            take n0;

            thus for m be Nat st n0 <= m holds ||.((((s " ) (#) (R /* s)) . m) - ( 0. S)).|| < r

            proof

              

               A27: ( rng s) c= ( dom R) by A1;

              let m be Nat;

              assume n0 <= m;

              then

               A28: |.((s . m) - 0 ).| < d by A26;

              

               A30: m in NAT by ORDINAL1:def 12;

              (( |.(s . m).| " ) * ||.(R /. (s . m)).||) = ( |.((s . m) " ).| * ||.(R /. (s . m)).||) by COMPLEX1: 66

              .= ||.(((s . m) " ) * (R /. (s . m))).|| by NORMSP_1:def 1

              .= ||.(((s . m) " ) * ((R /* s) . m)).|| by A27, FUNCT_2: 109, A30

              .= ||.(((s " ) . m) * ((R /* s) . m)).|| by VALUED_1: 10

              .= ||.(((s " ) (#) (R /* s)) . m).|| by NDIFF_1:def 2

              .= ||.((((s " ) (#) (R /* s)) . m) - ( 0. S)).|| by RLVECT_1: 13;

              hence thesis by A25, A28, SEQ_1: 5;

            end;

          end;

          hence ((s " ) (#) (R /* s)) is convergent by NORMSP_1:def 6;

          hence ( lim ((s " ) (#) (R /* s))) = ( 0. S) by A23, NORMSP_1:def 7;

        end;

        hence R is RestFunc-like by NDIFF_3:def 1;

      end;

      hence thesis by A2;

    end;

    theorem :: NDIFF_5:2

    

     Th2: for R be RestFunc of S st (R /. 0 ) = ( 0. S) holds for e be Real st e > 0 holds ex d be Real st d > 0 & for h be Real st |.h.| < d holds ||.(R /. h).|| <= (e * |.h.|)

    proof

      let R be RestFunc of S such that

       A1: (R /. 0 ) = ( 0. S);

      let e be Real such that

       A2: e > 0 ;

      R is total by NDIFF_3:def 1;

      then

      consider d be Real such that

       A3: d > 0 and

       A4: for z be Real st z <> 0 & |.z.| < d holds (( |.z.| " ) * ||.(R /. z).||) < e by A2, Th1;

      take d;

      now

        let h be Real such that

         A5: |.h.| < d;

        

         A6: 0 <= |.h.| by COMPLEX1: 46;

        per cases ;

          suppose

           A7: h <> 0 ;

          then (( |.h.| " ) * ||.(R /. h).||) <= e by A4, A5;

          then ( |.h.| * (( |.h.| " ) * ||.(R /. h).||)) <= ( |.h.| * e) by A6, XREAL_1: 64;

          then

           A8: (( |.h.| * ( |.h.| " )) * ||.(R /. h).||) <= (e * |.h.|);

           |.h.| <> 0 by A7, COMPLEX1: 45;

          then (1 * ||.(R /. h).||) <= (e * |.h.|) by A8, XCMPLX_0:def 7;

          hence ||.(R /. h).|| <= (e * |.h.|);

        end;

          suppose

           A9: h = 0 ;

          reconsider p0 = 0 as Real;

          (p0 * |.h.|) <= (e * |.h.|) by A2, A6;

          hence ||.(R /. h).|| <= (e * |.h.|) by A1, A9;

        end;

      end;

      hence thesis by A3;

    end;

    theorem :: NDIFF_5:3

    

     Th3: for R be RestFunc of S holds for L be Lipschitzian LinearOperator of S, T holds (L * R) is RestFunc of T

    proof

      let R be RestFunc of S;

      let L be Lipschitzian LinearOperator of S, T;

      consider K be Real such that

       A1: 0 <= K and

       A2: for z be Point of S holds ||.(L . z).|| <= (K * ||.z.||) by LOPBAN_1:def 8;

      ( dom L) = the carrier of S by FUNCT_2:def 1;

      then

       A3: ( rng R) c= ( dom L);

      

       A4: R is total by NDIFF_3:def 1;

      then

       A5: ( dom R) = REAL by PARTFUN1:def 2;

      now

        let e be Real such that

         A6: e > 0 ;

        set e1 = ((e / 2) / (1 + K));

        consider d be Real such that

         A7: 0 < d and

         A8: for h be Real st h <> 0 & |.h.| < d holds (( |.h.| " ) * ||.(R /. h).||) < e1 by A1, A4, A6, Th1;

        

         A9: (e / 2) < e by A6, XREAL_1: 216;

        now

          let h be Real;

          reconsider hh = h as Element of REAL by XREAL_0:def 1;

          assume

           A10: h <> 0 & |.h.| < d;

          then (( |.h.| " ) * ||.(R /. h).||) < e1 by A8;

          then ((K + 1) * (( |.h.| " ) * ||.(R /. h).||)) <= ((K + 1) * e1) by A1, XREAL_1: 64;

          then

           A11: ((K + 1) * (( |.h.| " ) * ||.(R /. h).||)) <= (e / 2) by A1, XCMPLX_1: 87;

           |.h.| <> 0 by A10, COMPLEX1: 45;

          then

           A12: |.h.| > 0 by COMPLEX1: 46;

          reconsider p0 = 0 , p1 = 1 as Element of REAL by XREAL_0:def 1;

          (p0 + K) < (p1 + K) by XREAL_1: 8;

          then

           A13: (K * ||.(R /. h).||) <= ((K + 1) * ||.(R /. h).||) by XREAL_1: 64;

           ||.(L . (R /. h)).|| <= (K * ||.(R /. h).||) by A2;

          then ||.(L . (R /. h)).|| <= ((K + 1) * ||.(R /. h).||) by A13, XXREAL_0: 2;

          then (( |.h.| " ) * ||.(L . (R /. h)).||) <= (( |.h.| " ) * ((K + 1) * ||.(R /. h).||)) by A12, XREAL_1: 64;

          then

           A14: (( |.h.| " ) * ||.(L . (R /. h)).||) <= (e / 2) by A11, XXREAL_0: 2;

          (L . (R /. h)) = (L /. (R /. h));

          then (L . (R /. hh)) = ((L * R) /. hh) by A5, A3, PARTFUN2: 5;

          hence (( |.h.| " ) * ||.((L * R) /. h).||) < e by A9, A14, XXREAL_0: 2;

        end;

        hence ex d be Real st d > 0 & for h be Real st h <> 0 & |.h.| < d holds (( |.h.| " ) * ||.((L * R) /. h).||) < e by A7;

      end;

      hence thesis by A4, Th1;

    end;

    theorem :: NDIFF_5:4

    

     Th4: for R1 be RestFunc of S st (R1 /. 0 ) = ( 0. S) holds for R2 be RestFunc of S, T st (R2 /. ( 0. S)) = ( 0. T) holds for L be LinearFunc of S holds (R2 * (L + R1)) is RestFunc of T

    proof

      let R1 be RestFunc of S;

      assume (R1 /. 0 ) = ( 0. S);

      then

      consider d0 be Real such that

       A1: 0 < d0 and

       A2: for h be Real st |.h.| < d0 holds ||.(R1 /. h).|| <= (1 * |.h.|) by Th2;

      let R2 be RestFunc of S, T such that

       A3: (R2 /. ( 0. S)) = ( 0. T);

      let L be LinearFunc of S;

      consider r be Point of S such that

       A4: for h be Real holds (L /. h) = (h * r) by NDIFF_3:def 2;

      reconsider K = ||.r.|| as Real;

      R2 is total by NDIFF_1:def 5;

      then ( dom R2) = the carrier of S by PARTFUN1:def 2;

      then

       A5: ( rng (L + R1)) c= ( dom R2);

      R1 is total by NDIFF_3:def 1;

      then (L + R1) is total by VFUNCT_1: 32;

      then

       A6: ( dom (L + R1)) = REAL by PARTFUN1:def 2;

      then ( dom (R2 * (L + R1))) = REAL by A5, RELAT_1: 27;

      then

       A7: (R2 * (L + R1)) is total by PARTFUN1:def 2;

      now

        let e be Real such that

         A8: e > 0 ;

        

         A9: (e / 2) < e by A8, XREAL_1: 216;

        set e1 = ((e / 2) / (1 + K));

        consider d be Real such that

         A10: 0 < d and

         A11: for z be Point of S st ||.z.|| < d holds ||.(R2 /. z).|| <= (e1 * ||.z.||) by A3, A8, NDIFF_2: 7;

        set d1 = (d / (1 + K));

        set dd1 = ( min (d0,d1));

        

         A12: dd1 <= d1 & dd1 <= d0 by XXREAL_0: 17;

         A13:

        now

          let hh be Real such that

           A14: hh <> 0 and

           A15: |.hh.| < dd1;

          reconsider h = hh as Element of REAL by XREAL_0:def 1;

           |.h.| < d0 by A12, A15, XXREAL_0: 2;

          then

           A16: ||.(R1 /. h).|| <= (1 * |.h.|) by A2;

          reconsider p0 = 0 as Element of REAL by XREAL_0:def 1;

          (L . h) = (L /. h)

          .= (h * r) by A4;

          then (( ||.(L . h).|| - (K * |.h.|)) + (K * |.h.|)) <= (p0 + (K * |.h.|)) by NORMSP_1:def 1;

          then ||.((L . h) + (R1 /. h)).|| <= ( ||.(L . h).|| + ||.(R1 /. h).||) & ( ||.(L . h).|| + ||.(R1 /. h).||) <= ((K * |.h.|) + (1 * |.h.|)) by A16, NORMSP_1:def 1, XREAL_1: 7;

          then

           A17: ||.((L . h) + (R1 /. h)).|| <= ((K + 1) * |.h.|) by XXREAL_0: 2;

          then

           A18: (e1 * ||.((L . h) + (R1 /. h)).||) <= (e1 * ((K + 1) * |.h.|)) by A8, XREAL_1: 64;

           |.h.| < d1 by A12, A15, XXREAL_0: 2;

          then ((K + 1) * |.h.|) < ((K + 1) * d1) by XREAL_1: 68;

          then ||.((L . h) + (R1 /. h)).|| < ((K + 1) * d1) by A17, XXREAL_0: 2;

          then ||.((L . h) + (R1 /. h)).|| < d by XCMPLX_1: 87;

          then ||.(R2 /. ((L . h) + (R1 /. h))).|| <= (e1 * ||.((L . h) + (R1 /. h)).||) by A11;

          then

           A19: ||.(R2 /. ((L . h) + (R1 /. h))).|| <= (e1 * ((K + 1) * |.h.|)) by A18, XXREAL_0: 2;

          

           A20: (R2 /. ((L . h) + (R1 /. h))) = (R2 /. ((L /. h) + (R1 /. h)))

          .= (R2 /. ((L + R1) /. h)) by A6, VFUNCT_1:def 1

          .= ((R2 * (L + R1)) /. h) by A6, A5, PARTFUN2: 5;

          

           A21: |.h.| <> 0 by A14, COMPLEX1: 45;

          then |.h.| > 0 by COMPLEX1: 46;

          then (( |.h.| " ) * ||.((R2 * (L + R1)) /. h).||) <= (( |.h.| " ) * ((e1 * (K + 1)) * |.h.|)) by A20, A19, XREAL_1: 64;

          then (( |.h.| " ) * ||.((R2 * (L + R1)) /. h).||) <= ((( |.h.| * ( |.h.| " )) * e1) * (K + 1));

          then (( |.h.| " ) * ||.((R2 * (L + R1)) /. h).||) <= ((1 * e1) * (K + 1)) by A21, XCMPLX_0:def 7;

          then (( |.h.| " ) * ||.((R2 * (L + R1)) /. h).||) <= (e / 2) by XCMPLX_1: 87;

          hence (( |.hh.| " ) * ||.((R2 * (L + R1)) /. hh).||) < e by A9, XXREAL_0: 2;

        end;

         0 < dd1 by A1, A10, XXREAL_0: 15;

        hence ex dd1 be Real st dd1 > 0 & for h be Real st h <> 0 & |.h.| < dd1 holds (( |.h.| " ) * ||.((R2 * (L + R1)) /. h).||) < e by A13;

      end;

      hence thesis by A7, Th1;

    end;

    theorem :: NDIFF_5:5

    

     Th5: for R1 be RestFunc of S st (R1 /. 0 ) = ( 0. S) holds for R2 be RestFunc of S, T st (R2 /. ( 0. S)) = ( 0. T) holds for L1 be LinearFunc of S holds for L2 be Lipschitzian LinearOperator of S, T holds ((L2 * R1) + (R2 * (L1 + R1))) is RestFunc of T

    proof

      let R1 be RestFunc of S such that

       A1: (R1 /. 0 ) = ( 0. S);

      let R2 be RestFunc of S, T such that

       A2: (R2 /. ( 0. S)) = ( 0. T);

      let L1 be LinearFunc of S;

      let L2 be Lipschitzian LinearOperator of S, T;

      (L2 * R1) is RestFunc of T & (R2 * (L1 + R1)) is RestFunc of T by A1, A2, Th4, Th3;

      hence thesis by NDIFF_3: 7;

    end;

    reconsider jj = 1 as Element of REAL by XREAL_0:def 1;

    theorem :: NDIFF_5:6

    

     Th6: for x0 be Real holds for g be PartFunc of REAL , the carrier of S st g is_differentiable_in x0 holds for f be PartFunc of the carrier of S, the carrier of T st f is_differentiable_in (g /. x0) holds (f * g) is_differentiable_in x0 & ( diff ((f * g),x0)) = (( diff (f,(g /. x0))) . ( diff (g,x0)))

    proof

      let x0 be Real;

      let g be PartFunc of REAL , the carrier of S such that

       A1: g is_differentiable_in x0;

      consider N1 be Neighbourhood of x0 such that

       A2: N1 c= ( dom g) and

       A3: ex L1 be LinearFunc of S, R1 be RestFunc of S st ( diff (g,x0)) = (L1 /. 1) & for x be Real st x in N1 holds ((g /. x) - (g /. x0)) = ((L1 /. (x - x0)) + (R1 /. (x - x0))) by A1, NDIFF_3:def 4;

      let f be PartFunc of the carrier of S, the carrier of T;

      assume f is_differentiable_in (g /. x0);

      then

      consider N2 be Neighbourhood of (g /. x0) such that

       A4: N2 c= ( dom f) and

       A5: ex R2 be RestFunc of S, T st (R2 /. ( 0. S)) = ( 0. T) & R2 is_continuous_in ( 0. S) & for y be Point of S st y in N2 holds ((f /. y) - (f /. (g /. x0))) = ((( diff (f,(g /. x0))) . (y - (g /. x0))) + (R2 /. (y - (g /. x0)))) by NDIFF_1: 47;

      consider R2 be RestFunc of S, T such that

       A6: (R2 /. ( 0. S)) = ( 0. T) and

       A7: for y be Point of S st y in N2 holds ((f /. y) - (f /. (g /. x0))) = ((( diff (f,(g /. x0))) . (y - (g /. x0))) + (R2 /. (y - (g /. x0)))) by A5;

      reconsider L2 = ( diff (f,(g /. x0))) as Lipschitzian LinearOperator of S, T by LOPBAN_1:def 9;

      consider L1 be LinearFunc of S, R1 be RestFunc of S such that

       A8: ( diff (g,x0)) = (L1 /. 1) & for x be Real st x in N1 holds ((g /. x) - (g /. x0)) = ((L1 /. (x - x0)) + (R1 /. (x - x0))) by A3;

      consider r be Point of S such that

       A9: for p be Real holds (L1 /. p) = (p * r) by NDIFF_3:def 2;

      reconsider p0 = 0 as Element of REAL by XREAL_0:def 1;

      ((g /. x0) - (g /. x0)) = ((L1 /. (x0 - x0)) + (R1 /. (x0 - x0))) by A8, RCOMP_1: 16;

      then ( 0. S) = ((L1 /. 0 ) + (R1 /. 0 )) by RLVECT_1: 15;

      then ( 0. S) = ((p0 * r) + (R1 /. 0 )) by A9;

      then ( 0. S) = (( 0. S) + (R1 /. 0 )) by RLVECT_1: 10;

      then (R1 /. 0 ) = ( 0. S) by RLVECT_1: 4;

      then

      reconsider R0 = ((L2 * R1) + (R2 * (L1 + R1))) as RestFunc of T by A6, Th5;

      

       A10: ( dom (L2 * L1)) = REAL by FUNCT_2:def 1;

      reconsider q = (L2 . r) as Point of T;

      now

        let pp be Real;

        reconsider p = pp as Element of REAL by XREAL_0:def 1;

        (L2 . (L1 /. p)) = (L2 . (p * r)) by A9;

        then (L2 . (L1 /. p)) = (p * q) by LOPBAN_1:def 5;

        then ((L2 * L1) . p) = (p * q) by A10, FUNCT_1: 12;

        hence ((L2 * L1) /. pp) = (pp * q) by A10, PARTFUN1:def 6;

      end;

      then

      reconsider L0 = (L2 * L1) as LinearFunc of T by NDIFF_3:def 2;

      g is_continuous_in x0 by A1, NDIFF_3: 22;

      then

      consider N3 be Neighbourhood of x0 such that

       A11: (g .: N3) c= N2 by NFCONT_3: 10;

      consider N be Neighbourhood of x0 such that

       A12: N c= N1 and

       A13: N c= N3 by RCOMP_1: 17;

      

       A14: ( dom L2) = the carrier of S by FUNCT_2:def 1;

      then

       A15: ( rng R1) c= ( dom L2);

      

       A16: ( rng L1) c= ( dom L2) by A14;

      now

        let x be object;

        assume

         A17: x in N;

        then

        reconsider x9 = x as Real;

        

         A18: x in N1 by A12, A17;

        then (g . x9) in (g .: N3) by A2, A13, A17, FUNCT_1:def 6;

        then (g . x9) in N2 by A11;

        hence x in ( dom (f * g)) by A2, A4, A18, FUNCT_1: 11;

      end;

      then

       A19: N c= ( dom (f * g));

       A20:

      now

        let x be Real such that

         A21: x in N;

        

         A22: ((g /. x) - (g /. x0)) = ((L1 /. (x - x0)) + (R1 /. (x - x0))) by A8, A12, A21;

        x in N1 by A12, A21;

        then (g . x) in (g .: N3) by A2, A13, A21, FUNCT_1:def 6;

        then (g . x) in N2 by A11;

        then

         A24: (g /. x) in N2 by A2, A12, A21, PARTFUN1:def 6;

        

         A25: x0 in N by RCOMP_1: 16;

        

         A26: R1 is total by NDIFF_3:def 1;

        then

         A27: ( dom R1) = REAL by PARTFUN1:def 2;

        

         A28: ( dom (L2 * R1)) = REAL by A26, PARTFUN1:def 2;

        (L1 + R1) is total by A26, VFUNCT_1: 32;

        then

         A29: ( dom (L1 + R1)) = REAL by PARTFUN1:def 2;

        R2 is total by NDIFF_1:def 5;

        then ( dom R2) = the carrier of S by PARTFUN1:def 2;

        then

         A30: ( rng (L1 + R1)) c= ( dom R2);

        then ( dom (R2 * (L1 + R1))) = ( dom (L1 + R1)) by RELAT_1: 27;

        then

         A31: ( dom ((L2 * R1) + (R2 * (L1 + R1)))) = ( REAL /\ REAL ) by A28, A29, VFUNCT_1:def 1;

        reconsider dxx0 = (x - x0) as Element of REAL by XREAL_0:def 1;

        (L2 . (R1 /. (x - x0))) = (L2 /. (R1 /. (x - x0)));

        then

         A32: (L2 . (R1 /. (x - x0))) = ((L2 * R1) /. dxx0) by A27, A15, PARTFUN2: 5;

        

         A33: (R2 /. ((L1 /. (x - x0)) + (R1 /. (x - x0)))) = (R2 /. ((L1 + R1) /. dxx0)) by A29, VFUNCT_1:def 1

        .= ((R2 * (L1 + R1)) /. dxx0) by A29, A30, PARTFUN2: 5;

        

         A34: ( dom L1) = REAL by FUNCT_2:def 1;

        

         A35: ((L2 * L1) /. (x - x0)) = (L2 /. (L1 /. dxx0)) by PARTFUN2: 5, A34, A16;

        

        thus (((f * g) /. x) - ((f * g) /. x0)) = ((f /. (g /. x)) - ((f * g) /. x0)) by A19, A21, PARTFUN2: 3

        .= ((f /. (g /. x)) - (f /. (g /. x0))) by A19, A25, PARTFUN2: 3

        .= ((( diff (f,(g /. x0))) . ((g /. x) - (g /. x0))) + (R2 /. ((g /. x) - (g /. x0)))) by A7, A24

        .= (((L2 . (L1 /. (x - x0))) + (L2 . (R1 /. (x - x0)))) + ((R2 * (L1 + R1)) /. (x - x0))) by A22, A33, VECTSP_1:def 20

        .= ((L2 . (L1 /. (x - x0))) + (((L2 * R1) /. (x - x0)) + ((R2 * (L1 + R1)) /. (x - x0)))) by A32, RLVECT_1:def 3

        .= ((L0 /. (x - x0)) + (R0 /. (x - x0))) by A35, A31, VFUNCT_1:def 1;

      end;

      hence

       A36: (f * g) is_differentiable_in x0 by A19, NDIFF_3:def 3;

      ( dom L1) = REAL by FUNCT_2:def 1;

      

      then ((L2 * L1) /. 1) = (L2 /. (L1 /. jj)) by PARTFUN2: 5, A16

      .= (( diff (f,(g /. x0))) . ( diff (g,x0))) by A8;

      hence thesis by A36, A19, A20, NDIFF_3:def 4;

    end;

    theorem :: NDIFF_5:7

    

     Th7: for S be RealNormSpace, xseq be FinSequence of S, yseq be FinSequence of REAL st ( len xseq) = ( len yseq) & (for i be Element of NAT st i in ( dom xseq) holds (yseq . i) = ||.(xseq /. i).||) holds ||.( Sum xseq).|| <= ( Sum yseq)

    proof

      let S be RealNormSpace, xseq be FinSequence of S, yseq be FinSequence of REAL ;

      assume that

       A1: ( len xseq) = ( len yseq) and

       A2: for i be Element of NAT st i in ( dom xseq) holds (yseq . i) = ||.(xseq /. i).||;

      defpred P[ Nat] means for xseq be FinSequence of S, yseq be FinSequence of REAL st $1 = ( len xseq) & ( len xseq) = ( len yseq) & (for i be Element of NAT st i in ( dom xseq) holds (yseq . i) = ||.(xseq /. i).||) holds ||.( Sum xseq).|| <= ( Sum yseq);

      

       A3: P[ 0 ]

      proof

        let xseq be FinSequence of S, yseq be FinSequence of REAL ;

        assume

         A4: 0 = ( len xseq) & ( len xseq) = ( len yseq) & (for i be Element of NAT st i in ( dom xseq) holds (yseq . i) = ||.(xseq /. i).||);

        consider Sx be sequence of the carrier of S such that

         A5: ( Sum xseq) = (Sx . ( len xseq)) & (Sx . 0 ) = ( 0. S) & for j be Nat, v be Element of S st j < ( len xseq) & v = (xseq . (j + 1)) holds (Sx . (j + 1)) = ((Sx . j) + v) by RLVECT_1:def 12;

        yseq = {} by A4;

        hence thesis by A4, A5, RVSUM_1: 72;

      end;

       A6:

      now

        let i be Nat;

        assume

         A7: P[i];

        now

          let xseq be FinSequence of S, yseq be FinSequence of REAL ;

          set xseq0 = (xseq | i), yseq0 = (yseq | i);

          assume

           A8: (i + 1) = ( len xseq) & ( len xseq) = ( len yseq) & (for i be Element of NAT st i in ( dom xseq) holds (yseq . i) = ||.(xseq /. i).||);

          

           A9: for k be Element of NAT st k in ( dom xseq0) holds (yseq0 . k) = ||.(xseq0 /. k).||

          proof

            let k be Element of NAT ;

            assume

             A10: k in ( dom xseq0);

            then

             A11: k in ( Seg i) & k in ( dom xseq) by RELAT_1: 57;

            then

             A12: (yseq . k) = ||.(xseq /. k).|| by A8;

            (xseq /. k) = (xseq . k) by A11, PARTFUN1:def 6;

            then (xseq /. k) = (xseq0 . k) by A11, FUNCT_1: 49;

            then (xseq /. k) = (xseq0 /. k) by A10, PARTFUN1:def 6;

            hence thesis by A11, A12, FUNCT_1: 49;

          end;

          

           A13: ( dom xseq) = ( Seg (i + 1)) by A8, FINSEQ_1:def 3;

          then

           A14: (yseq . (i + 1)) = ||.(xseq /. (i + 1)).|| by A8, FINSEQ_1: 4;

          

           A15: 1 <= (i + 1) by NAT_1: 11;

          yseq = ((yseq | i) ^ <*(yseq /. (i + 1))*>) by A8, FINSEQ_5: 21;

          then yseq = (yseq0 ^ <*(yseq . (i + 1))*>) by A8, A15, FINSEQ_4: 15;

          then

           A16: ( Sum yseq) = (( Sum yseq0) + (yseq . (i + 1))) by RVSUM_1: 74;

          reconsider v = (xseq . ( len xseq)) as Element of S by A13, A8, FINSEQ_1: 4, PARTFUN1: 4;

          

           A18: v = (xseq /. (i + 1)) by A8, A13, FINSEQ_1: 4, PARTFUN1:def 6;

          

           A19: i = ( len xseq0) by A8, FINSEQ_1: 59, NAT_1: 11;

          then xseq0 = (xseq | ( dom xseq0)) by FINSEQ_1:def 3;

          then

           A20: ( Sum xseq) = (( Sum xseq0) + v) by A8, A19, RLVECT_1: 38;

          

           A21: ||.(( Sum xseq0) + v).|| <= ( ||.( Sum xseq0).|| + ||.v.||) by NORMSP_1:def 1;

          ( len xseq0) = ( len yseq0) by A8, A19, FINSEQ_1: 59, NAT_1: 11;

          then ( ||.( Sum xseq0).|| + ||.v.||) <= (( Sum yseq0) + (yseq . (i + 1))) by A7, A9, A19, A14, A18, XREAL_1: 6;

          hence ||.( Sum xseq).|| <= ( Sum yseq) by A16, A20, A21, XXREAL_0: 2;

        end;

        hence P[(i + 1)];

      end;

      for i be Nat holds P[i] from NAT_1:sch 2( A3, A6);

      hence thesis by A1, A2;

    end;

    theorem :: NDIFF_5:8

    

     Th8: for S be RealNormSpace, x be Point of S, N1,N2 be Neighbourhood of x holds (N1 /\ N2) is Neighbourhood of x

    proof

      let S be RealNormSpace, x be Point of S, N1,N2 be Neighbourhood of x;

      consider N be Neighbourhood of x such that

       A1: N c= N1 & N c= N2 by NDIFF_1: 1;

      

       A2: N c= (N1 /\ N2) by A1, XBOOLE_1: 19;

      consider g be Real such that

       A3: 0 < g and

       A4: { y where y be Point of S : ||.(y - x).|| < g } c= N by NFCONT_1:def 1;

      { y where y be Point of S : ||.(y - x).|| < g } c= (N1 /\ N2) by A2, A4;

      hence thesis by A3, NFCONT_1:def 1;

    end;

    theorem :: NDIFF_5:9

    

     Th9: for X be non-empty FinSequence, x be set st x in ( product X) holds x is FinSequence

    proof

      let X be non-empty FinSequence, x be set;

      assume x in ( product X);

      then

      consider g be Function such that

       A1: x = g & ( dom g) = ( dom X) & for i be object st i in ( dom X) holds (g . i) in (X . i) by CARD_3:def 5;

      ( dom g) = ( Seg ( len X)) by A1, FINSEQ_1:def 3;

      hence x is FinSequence by A1, FINSEQ_1:def 2;

    end;

    registration

      let G be RealNormSpace-Sequence;

      cluster ( product G) -> constituted-FinSeqs;

      coherence

      proof

        let a be Element of ( product G);

        ( product G) = NORMSTR (# ( product ( carr G)), ( zeros G), [:( addop G):], [:( multop G):], ( productnorm G) #) by PRVECT_2: 6;

        hence thesis by Th9;

      end;

    end

     Lm1:

    now

      let G be RealLinearSpace-Sequence;

      ( len ( carr G)) = ( len G) by PRVECT_1:def 11;

      

      hence ( dom ( carr G)) = ( Seg ( len G)) by FINSEQ_1:def 3

      .= ( dom G) by FINSEQ_1:def 3;

    end;

    definition

      let G be RealLinearSpace-Sequence;

      let z be Element of ( product ( carr G));

      let j be Element of ( dom G);

      :: original: .

      redefine

      func z . j -> Element of (G . j) ;

      correctness

      proof

        reconsider zz = z as FinSequence by Th9;

        ( dom ( carr G)) = ( dom G) by Lm1;

        then (zz . j) in (( carr G) . j) by CARD_3: 9;

        hence thesis by PRVECT_1:def 11;

      end;

    end

    theorem :: NDIFF_5:10

    

     Th10: the carrier of ( product G) = ( product ( carr G))

    proof

      ( product G) = NORMSTR (# ( product ( carr G)), ( zeros G), [:( addop G):], [:( multop G):], ( productnorm G) #) by PRVECT_2: 6;

      hence thesis;

    end;

    theorem :: NDIFF_5:11

    

     Th11: for i be Element of ( dom G), r be set, x be Function st r in the carrier of (G . i) & x in ( product ( carr G)) holds (x +* (i,r)) in the carrier of ( product G)

    proof

      let i be Element of ( dom G), r be set, x be Function;

      assume

       A1: r in the carrier of (G . i) & x in ( product ( carr G));

      then

      consider g be Function such that

       A2: x = g & ( dom g) = ( dom ( carr G)) & for i be object st i in ( dom ( carr G)) holds (g . i) in (( carr G) . i) by CARD_3:def 5;

      set h = (x +* (i,r));

      set s = (i .--> r);

      s = ( {i} --> r) by FUNCOP_1:def 9;

      then

       A3: ( dom s) = {i};

      

       A4: ( dom h) = ( dom ( carr G)) by A2, FUNCT_7: 30;

      for j be object st j in ( dom ( carr G)) holds (h . j) in (( carr G) . j)

      proof

        let j be object;

        assume

         A5: j in ( dom ( carr G));

        per cases ;

          suppose not j in ( dom s);

          then j <> i by A3, TARSKI:def 1;

          then (h . j) = (x . j) by FUNCT_7: 32;

          hence (h . j) in (( carr G) . j) by A2, A5;

        end;

          suppose j in ( dom s);

          then

           A6: j = i by TARSKI:def 1;

          then (h . j) = r by A5, A2, FUNCT_7: 31;

          hence (h . j) in (( carr G) . j) by A1, A6, PRVECT_1:def 11;

        end;

      end;

      then (x +* (i,r)) in ( product ( carr G)) by A4, CARD_3:def 5;

      hence thesis by Th10;

    end;

    definition

      let G be RealNormSpace-Sequence;

      :: NDIFF_5:def1

      attr G is non-trivial means

      : Def1: for j be Element of ( dom G) holds (G . j) is non trivial;

    end

    registration

      cluster non-trivial for RealNormSpace-Sequence;

      correctness

      proof

        take G = <* the non trivial RealNormSpace*>;

        let j be Element of ( dom G);

        ( dom G) = ( Seg 1) by FINSEQ_1: 38;

        then j = 1 by FINSEQ_1: 2, TARSKI:def 1;

        hence thesis by FINSEQ_1: 40;

      end;

    end

    registration

      let G be non-trivial RealNormSpace-Sequence;

      let i be Element of ( dom G);

      cluster (G . i) -> non trivial;

      correctness by Def1;

    end

    registration

      let G be non-trivial RealNormSpace-Sequence;

      cluster ( product G) -> non trivial;

      correctness

      proof

        

         A1: the carrier of ( product G) = ( product ( carr G)) by Th10;

         not for x,y be set st x in ( product ( carr G)) & y in ( product ( carr G)) holds x = y

        proof

          assume

           A2: for x,y be set st x in ( product ( carr G)) & y in ( product ( carr G)) holds x = y;

          consider z be object such that

           A3: z in ( product ( carr G)) by XBOOLE_0:def 1;

          consider g be Function such that

           A4: z = g & ( dom g) = ( dom ( carr G)) & for i be object st i in ( dom ( carr G)) holds (g . i) in (( carr G) . i) by A3, CARD_3:def 5;

          set i = the Element of ( dom G);

          now

            let r,s be object;

            assume

             A5: r in the carrier of (G . i) & s in the carrier of (G . i);

            (g +* (i,r)) in the carrier of ( product G) & (g +* (i,s)) in the carrier of ( product G) by Th11, A3, A4, A5;

            then (g +* (i,r)) in ( product ( carr G)) & (g +* (i,s)) in ( product ( carr G)) by Th10;

            then

             A6: (g +* (i,r)) = (g +* (i,s)) by A2;

            i in ( dom G);

            then

             A7: i in ( dom g) by A4, Lm1;

            then ((g +* (i,r)) . i) = r by FUNCT_7: 31;

            hence r = s by A6, A7, FUNCT_7: 31;

          end;

          hence contradiction by ZFMISC_1:def 10;

        end;

        hence thesis by A1;

      end;

    end

    theorem :: NDIFF_5:12

    

     Th12: for G be RealNormSpace-Sequence, p,q be Point of ( product G), r0,p0,q0 be Element of ( product ( carr G)) st p = p0 & q = q0 holds (p + q) = r0 iff for i be Element of ( dom G) holds (r0 . i) = ((p0 . i) + (q0 . i))

    proof

      let G be RealNormSpace-Sequence, p,q be Point of ( product G), r0,p0,q0 be Element of ( product ( carr G));

      assume

       A1: p = p0 & q = q0;

      ( len ( carr G)) = ( len G) by PRVECT_1:def 11;

      

      then

       A2: ( dom ( carr G)) = ( Seg ( len G)) by FINSEQ_1:def 3

      .= ( dom G) by FINSEQ_1:def 3;

      

       A3: ( product G) = NORMSTR (# ( product ( carr G)), ( zeros G), [:( addop G):], [:( multop G):], ( productnorm G) #) by PRVECT_2: 6;

      hereby

        assume

         A4: (p + q) = r0;

        hereby

          let i be Element of ( dom G);

          reconsider i0 = i as Element of ( dom ( carr G)) by A2;

          (( addop G) . i0) = the addF of (G . i0) by PRVECT_1:def 12;

          hence (r0 . i) = ((p0 . i) + (q0 . i)) by A1, A4, A3, PRVECT_1:def 8;

        end;

      end;

      assume

       A5: for i be Element of ( dom G) holds (r0 . i) = ((p0 . i) + (q0 . i));

      reconsider pq = (p + q) as Element of ( product ( carr G)) by Th10;

      

       A6: ex g be Function st pq = g & ( dom g) = ( dom ( carr G)) & for i be object st i in ( dom ( carr G)) holds (g . i) in (( carr G) . i) by CARD_3:def 5;

      

       A7: ex g be Function st r0 = g & ( dom g) = ( dom ( carr G)) & for i be object st i in ( dom ( carr G)) holds (g . i) in (( carr G) . i) by CARD_3:def 5;

      now

        let i0 be object;

        assume

         A8: i0 in ( dom pq);

        then

        reconsider i1 = i0 as Element of ( dom G) by A2, A6;

        reconsider i = i0 as Element of ( dom ( carr G)) by A8, A6;

        (( addop G) . i) = the addF of (G . i) by PRVECT_1:def 12;

        then (pq . i0) = ((p0 . i1) + (q0 . i1)) by A1, A3, PRVECT_1:def 8;

        hence (pq . i0) = (r0 . i0) by A5;

      end;

      hence (p + q) = r0 by A6, A7, FUNCT_1: 2;

    end;

    theorem :: NDIFF_5:13

    

     Th13: for G be RealNormSpace-Sequence, p be Point of ( product G), r be Real, r0,p0 be Element of ( product ( carr G)) st p = p0 holds (r * p) = r0 iff for i be Element of ( dom G) holds (r0 . i) = (r * (p0 . i))

    proof

      let G be RealNormSpace-Sequence, p be Point of ( product G), r be Real, r0,p0 be Element of ( product ( carr G));

      assume

       A1: p = p0;

      hereby

        assume

         A2: (r * p) = r0;

        hereby

          let i be Element of ( dom G);

          reconsider i0 = i as Element of ( dom ( carr G)) by Lm1;

          

           A3: (( multop G) . i0) = the Mult of (G . i0) by PRVECT_2:def 8;

          reconsider rr = r as Element of REAL by XREAL_0:def 1;

          ( product G) = NORMSTR (# ( product ( carr G)), ( zeros G), [:( addop G):], [:( multop G):], ( productnorm G) #) by PRVECT_2: 6;

          

          hence (r0 . i) = (rr * (p0 . i)) by A1, A2, A3, PRVECT_2:def 2

          .= (r * (p0 . i));

        end;

      end;

      assume

       A4: for i be Element of ( dom G) holds (r0 . i) = (r * (p0 . i));

      reconsider rp = (r * p) as Element of ( product ( carr G)) by Th10;

      

       A5: ex g be Function st rp = g & ( dom g) = ( dom ( carr G)) & for i be object st i in ( dom ( carr G)) holds (g . i) in (( carr G) . i) by CARD_3:def 5;

      

       A6: ex g be Function st r0 = g & ( dom g) = ( dom ( carr G)) & for i be object st i in ( dom ( carr G)) holds (g . i) in (( carr G) . i) by CARD_3:def 5;

      now

        let i0 be object;

        assume

         A7: i0 in ( dom rp);

        then

        reconsider i1 = i0 as Element of ( dom G) by Lm1, A5;

        reconsider i = i0 as Element of ( dom ( carr G)) by A7, A5;

        

         A8: ( product G) = NORMSTR (# ( product ( carr G)), ( zeros G), [:( addop G):], [:( multop G):], ( productnorm G) #) by PRVECT_2: 6;

        reconsider r as Element of REAL by XREAL_0:def 1;

        (( multop G) . i) = the Mult of (G . i) by PRVECT_2:def 8;

        then (rp . i0) = (r * (p0 . i1)) by A1, A8, PRVECT_2:def 2;

        hence (rp . i0) = (r0 . i0) by A4;

      end;

      hence (r * p) = r0 by A5, A6, FUNCT_1: 2;

    end;

    theorem :: NDIFF_5:14

    

     Th14: for G be RealNormSpace-Sequence, p0 be Element of ( product ( carr G)) holds ( 0. ( product G)) = p0 iff for i be Element of ( dom G) holds (p0 . i) = ( 0. (G . i))

    proof

      let G be RealNormSpace-Sequence, p0 be Element of ( product ( carr G));

      

       A1: ( dom ( carr G)) = ( dom G) by Lm1;

      

       A2: ( product G) = NORMSTR (# ( product ( carr G)), ( zeros G), [:( addop G):], [:( multop G):], ( productnorm G) #) by PRVECT_2: 6;

      hence ( 0. ( product G)) = p0 implies for i be Element of ( dom G) holds (p0 . i) = ( 0. (G . i)) by A1, PRVECT_1:def 14;

      assume

       A3: for i be Element of ( dom G) holds (p0 . i) = ( 0. (G . i));

      now

        let i0 be Element of ( dom ( carr G));

        reconsider i = i0 as Element of ( dom G) by Lm1;

        (p0 . i) = ( 0. (G . i)) by A3;

        hence (p0 . i0) = ( 0. (G . i0));

      end;

      hence ( 0. ( product G)) = p0 by A2, PRVECT_1:def 14;

    end;

    theorem :: NDIFF_5:15

    

     Th15: for G be RealNormSpace-Sequence, p,q be Point of ( product G), r0,p0,q0 be Element of ( product ( carr G)) st p = p0 & q = q0 holds (p - q) = r0 iff for i be Element of ( dom G) holds (r0 . i) = ((p0 . i) - (q0 . i))

    proof

      let G be RealNormSpace-Sequence, p,q be Point of ( product G), r0,p0,q0 be Element of ( product ( carr G));

      assume

       A1: p = p0 & q = q0;

      reconsider qq0 = (( - 1) * q) as Element of ( product ( carr G)) by Th10;

      

       A2: (p - q) = (p + (( - 1) * q)) by RLVECT_1: 16;

      hereby

        assume

         A3: (p - q) = r0;

        thus for i be Element of ( dom G) holds (r0 . i) = ((p0 . i) - (q0 . i))

        proof

          let i be Element of ( dom G);

          

           A4: (r0 . i) = ((p0 . i) + (qq0 . i)) by Th12, A3, A1, A2;

          (qq0 . i) = (( - 1) * (q0 . i)) by A1, Th13;

          hence thesis by A4, RLVECT_1: 16;

        end;

      end;

      assume

       A5: for i be Element of ( dom G) holds (r0 . i) = ((p0 . i) - (q0 . i));

      now

        let i be Element of ( dom G);

        

         A6: (qq0 . i) = (( - 1) * (q0 . i)) by A1, Th13;

        (r0 . i) = ((p0 . i) - (q0 . i)) by A5;

        hence (r0 . i) = ((p0 . i) + (qq0 . i)) by A6, RLVECT_1: 16;

      end;

      hence (p - q) = r0 by A2, Th12, A1;

    end;

    begin

     Lm2:

    now

      let S be RealLinearSpace;

      let p,q be Point of S;

      let z1 be Real;

      

      thus (p + (z1 * (q - p))) = (p + ((z1 * q) + (z1 * ( - p)))) by RLVECT_1:def 5

      .= (p + ((z1 * q) + ( - (z1 * p)))) by RLVECT_1: 25

      .= ((p + ( - (z1 * p))) + (z1 * q)) by RLVECT_1:def 3

      .= (((1 * p) - (z1 * p)) + (z1 * q)) by RLVECT_1:def 8

      .= (((1 - z1) * p) + (z1 * q)) by RLVECT_1: 35;

    end;

    notation

      let S be RealLinearSpace;

      let p,q be Point of S;

      synonym [.p,q.] for LSeg (p,q);

    end

    definition

      let S be RealLinearSpace;

      let p,q be Point of S;

      :: NDIFF_5:def2

      func ].p,q.[ -> Subset of S equals ( [.p, q.] \ {p, q});

      correctness ;

    end

    theorem :: NDIFF_5:16

    

     LMOPN: for S be RealLinearSpace, p,q be Point of S st p <> q holds ].p, q.[ = { (p + (t * (q - p))) where t be Real : 0 < t & t < 1 }

    proof

      let S be RealLinearSpace, p,q be Point of S;

      assume

       AS1: p <> q;

      set A = { (p + (t * (q - p))) where t be Real : 0 < t & t < 1 };

      for x be object holds (x in ].p, q.[ iff x in A)

      proof

        let x be object;

        hereby

          assume x in ].p, q.[;

          then

           P1: x in [.p, q.] & not x in {p, q} by XBOOLE_0:def 5;

          then x in { (((1 - r) * p) + (r * q)) where r be Real : 0 <= r & r <= 1 } by RLTOPSP1:def 2;

          then

          consider t be Real such that

           P2: x = (((1 - t) * p) + (t * q)) & 0 <= t & t <= 1;

          

           P3: x = (p + (t * (q - p))) by P2, Lm2;

          

           P4: 0 <> t

          proof

            assume t = 0 ;

            

            then x = (p + ( 0. S)) by P3, RLVECT_1: 10

            .= p by RLVECT_1: 4;

            hence contradiction by P1, TARSKI:def 2;

          end;

          1 <> t

          proof

            assume t = 1;

            

            then x = (p + (q - p)) by P3, RLVECT_1:def 8

            .= (q - (p - p)) by RLVECT_1: 29

            .= (q - ( 0. S)) by RLVECT_1: 15

            .= q by RLVECT_1: 13;

            hence contradiction by P1, TARSKI:def 2;

          end;

          then 0 < t & t < 1 by P2, P4, XXREAL_0: 1;

          hence x in A by P3;

        end;

        assume x in A;

        then

        consider t be Real such that

         P4: x = (p + (t * (q - p))) & 0 < t & t < 1;

        x = (((1 - t) * p) + (t * q)) by Lm2, P4;

        then x in { (((1 - r) * p) + (r * q)) where r be Real : 0 <= r & r <= 1 } by P4;

        then

         P5: x in [.p, q.] by RLTOPSP1:def 2;

        

         P6: x <> p

        proof

          assume x = p;

          

          then

           P7: ( 0. S) = (((t * (q - p)) + p) - p) by P4, RLVECT_1: 15

          .= ((t * (q - p)) + (p - p)) by RLVECT_1: 28

          .= ((t * (q - p)) + ( 0. S)) by RLVECT_1: 15

          .= (t * (q - p)) by RLVECT_1: 4;

          (q - p) <> ( 0. S) by AS1, RLVECT_1: 21;

          hence contradiction by P4, P7, RLVECT_1: 11;

        end;

        x <> q

        proof

          assume x = q;

          

          then (q - p) = ((t * (q - p)) + (p - p)) by P4, RLVECT_1: 28

          .= ((t * (q - p)) + ( 0. S)) by RLVECT_1: 15

          .= (t * (q - p)) by RLVECT_1: 4;

          then (1 * (q - p)) = (t * (q - p)) by RLVECT_1:def 8;

          then ((1 * (q - p)) - (t * (q - p))) = ( 0. S) by RLVECT_1: 15;

          then

           P7: ((1 - t) * (q - p)) = ( 0. S) by RLVECT_1: 35;

          (q - p) <> ( 0. S) by AS1, RLVECT_1: 21;

          then (1 - t) = 0 by RLVECT_1: 11, P7;

          hence contradiction by P4;

        end;

        then not x in {p, q} by P6, TARSKI:def 2;

        hence x in ].p, q.[ by P5, XBOOLE_0:def 5;

      end;

      hence thesis by TARSKI: 2;

    end;

    

     Lm3: for x be Real st for e be Real st 0 < e holds x <= e holds x <= 0

    proof

      let x be Real;

      assume

       A1: for e be Real st 0 < e holds x <= e;

      assume

       A2: not x <= 0 ;

      then x <= (x / 2) by A1;

      then (x - (x / 2)) <= ((x / 2) - (x / 2)) by XREAL_1: 9;

      hence contradiction by A2;

    end;

    theorem :: NDIFF_5:17

    

     Th17: for T be RealNormSpace, R be PartFunc of REAL , T st R is total holds R is RestFunc-like iff for r be Real st r > 0 holds ex d be Real st d > 0 & for z be Real st z <> 0 & |.z.| < d holds ( ||.(R /. z).|| / |.z.|) < r

    proof

      let T be RealNormSpace, R be PartFunc of REAL , T;

      assume

       A1: R is total;

       A2:

      now

        assume

         A3: R is RestFunc-like;

        assume not (for r be Real st r > 0 holds ex d be Real st d > 0 & for z be Real st z <> 0 & |.z.| < d holds ( ||.(R /. z).|| / |.z.|) < r);

        then

        consider r be Real such that

         A4: r > 0 and

         A5: for d be Real st d > 0 holds ex z be Real st z <> 0 & |.z.| < d & not ( ||.(R /. z).|| / |.z.|) < r;

        defpred P[ Nat, Element of REAL ] means $2 <> 0 & |.$2.| < (1 / ($1 + 1)) & not (( ||.(R /. $2).|| / |.$2.|) < r);

         A6:

        now

          let n be Element of NAT ;

          consider z be Real such that

           A7: z <> 0 & |.z.| < (1 / (n + 1)) & not ( ||.(R /. z).|| / |.z.|) < r by A5;

          reconsider z as Element of REAL by XREAL_0:def 1;

          take z;

          thus P[n, z] by A7;

        end;

        consider s be Real_Sequence such that

         A8: for n be Element of NAT holds P[n, (s . n)] from FUNCT_2:sch 3( A6);

        

         A9: for n be Nat holds P[n, (s . n)]

        proof

          let n be Nat;

          n in NAT by ORDINAL1:def 12;

          hence thesis by A8;

        end;

         A10:

        now

          let p be Real;

          assume

           A11: 0 < p;

          consider n be Nat such that

           A12: (p " ) < n by SEQ_4: 3;

          ((p " ) + 0 qua Real) < (n + 1) by A12, XREAL_1: 8;

          then

           A13: (1 / (n + 1)) < (1 / (p " )) by A11, XREAL_1: 76;

          take n;

          let m be Nat;

          assume n <= m;

          then (n + 1) <= (m + 1) by XREAL_1: 6;

          then (1 / (m + 1)) <= (1 / (n + 1)) by XREAL_1: 118;

          then |.((s . m) - 0 ).| < (1 / (n + 1)) by A9, XXREAL_0: 2;

          hence |.((s . m) - 0 ).| < p by A13, XXREAL_0: 2;

        end;

        then s is convergent by SEQ_2:def 6;

        then ( lim s) = 0 by A10, SEQ_2:def 7;

        then

        reconsider s as 0 -convergent non-zero Real_Sequence by A9, A10, SEQ_1: 5, SEQ_2:def 6, FDIFF_1:def 1;

        ((s " ) (#) (R /* s)) is convergent & ( lim ((s " ) (#) (R /* s))) = ( 0. T) by A3, NDIFF_3:def 1;

        then

        consider n be Nat such that

         A16: for m be Nat st n <= m holds ||.((((s " ) (#) (R /* s)) . m) - ( 0. T)).|| < r by A4, NORMSP_1:def 7;

        

         A17: n in NAT by ORDINAL1:def 12;

        

         A19: ||.(((s . n) " ) * (R /. (s . n))).|| = ( |.((s . n) " ).| * ||.(R /. (s . n)).||) by NORMSP_1:def 1

        .= ( ||.(R /. (s . n)).|| / |.(s . n).|) by COMPLEX1: 66;

        ( dom R) = REAL by A1, PARTFUN1:def 2;

        then

         A20: ( rng s) c= ( dom R);

         ||.((((s " ) (#) (R /* s)) . n) - ( 0. T)).|| = ||.(((s " ) (#) (R /* s)) . n).|| by RLVECT_1: 13

        .= ||.(((s " ) . n) * ((R /* s) . n)).|| by NDIFF_1:def 2

        .= ||.(((s . n) " ) * ((R /* s) . n)).|| by VALUED_1: 10

        .= ||.(((s . n) " ) * (R /. (s . n))).|| by A20, FUNCT_2: 109, A17;

        hence for r be Real st r > 0 holds ex d be Real st d > 0 & for z be Real st z <> 0 & |.z.| < d holds ( ||.(R /. z).|| / |.z.|) < r by A9, A16, A19;

      end;

      now

        assume

         A21: for r be Real st r > 0 holds ex d be Real st d > 0 & for z be Real st z <> 0 & |.z.| < d holds ( ||.(R /. z).|| / |.z.|) < r;

        now

          let s be 0 -convergent non-zero Real_Sequence;

          

           A22: s is convergent & ( lim s) = 0 ;

           A23:

          now

            let r be Real;

            assume r > 0 ;

            then

            consider d be Real such that

             A24: d > 0 and

             A25: for z be Real st z <> 0 & |.z.| < d holds ( ||.(R /. z).|| / |.z.|) < r by A21;

            consider n be Nat such that

             A26: for m be Nat st n <= m holds |.((s . m) - 0 ).| < d by A22, A24, SEQ_2:def 7;

            take n;

            thus for m be Nat st n <= m holds ||.((((s " ) (#) (R /* s)) . m) - ( 0. T)).|| < r

            proof

              ( dom R) = REAL by A1, PARTFUN1:def 2;

              then

               A27: ( rng s) c= ( dom R);

              let m be Nat;

              

               A28: m in NAT by ORDINAL1:def 12;

              assume n <= m;

              then

               A29: |.((s . m) - 0 ).| < d by A26;

              ( ||.(R /. (s . m)).|| / |.(s . m).|) = ( |.((s . m) " ).| * ||.(R /. (s . m)).||) by COMPLEX1: 66

              .= ||.(((s . m) " ) * (R /. (s . m))).|| by NORMSP_1:def 1

              .= ||.(((s . m) " ) * ((R /* s) . m)).|| by A27, FUNCT_2: 109, A28

              .= ||.(((s " ) . m) * ((R /* s) . m)).|| by VALUED_1: 10

              .= ||.(((s " ) (#) (R /* s)) . m).|| by NDIFF_1:def 2

              .= ||.((((s " ) (#) (R /* s)) . m) - ( 0. T)).|| by RLVECT_1: 13;

              hence thesis by A25, A29, SEQ_1: 5;

            end;

          end;

          hence ((s " ) (#) (R /* s)) is convergent by NORMSP_1:def 6;

          hence ( lim ((s " ) (#) (R /* s))) = ( 0. T) by A23, NORMSP_1:def 7;

        end;

        hence R is RestFunc-like by A1, NDIFF_3:def 1;

      end;

      hence thesis by A2;

    end;

    theorem :: NDIFF_5:18

    

     Th18: for R be Function of REAL , REAL holds R is RestFunc-like iff for r be Real st r > 0 holds ex d be Real st d > 0 & for z be Real st z <> 0 & |.z.| < d holds ( |.(R . z).| / |.z.|) < r

    proof

      let R be Function of REAL , REAL ;

       A1:

      now

        assume

         A2: R is RestFunc-like;

        assume not (for r be Real st r > 0 holds ex d be Real st d > 0 & for z be Real st z <> 0 & |.z.| < d holds ( |.(R . z).| / |.z.|) < r);

        then

        consider r be Real such that

         A3: r > 0 and

         A4: for d be Real st d > 0 holds ex z be Real st z <> 0 & |.z.| < d & not ( |.(R . z).| / |.z.|) < r;

        defpred P[ Nat, Element of REAL ] means $2 <> 0 & |.$2.| < (1 / ($1 + 1)) & not ( |.(R . $2).| / |.$2.|) < r;

         A5:

        now

          let n be Element of NAT ;

          consider z be Real such that

           A6: z <> 0 & |.z.| < (1 / (n + 1)) & not ( |.(R . z).| / |.z.|) < r by A4;

          reconsider z as Element of REAL by XREAL_0:def 1;

          take z;

          thus P[n, z] by A6;

        end;

        consider s be Real_Sequence such that

         A7: for n be Element of NAT holds P[n, (s . n)] from FUNCT_2:sch 3( A5);

        

         A8: for n be Nat holds P[n, (s . n)]

        proof

          let n be Nat;

          n in NAT by ORDINAL1:def 12;

          hence thesis by A7;

        end;

         A9:

        now

          let p be Real;

          assume

           A10: 0 < p;

          consider n be Nat such that

           A11: (p " ) < n by SEQ_4: 3;

          ((p " ) + 0 qua Real) < (n + 1) by A11, XREAL_1: 8;

          then

           A12: (1 / (n + 1)) < (1 / (p " )) by A10, XREAL_1: 76;

          take n;

          let m be Nat;

          assume n <= m;

          then (n + 1) <= (m + 1) by XREAL_1: 6;

          then (1 / (m + 1)) <= (1 / (n + 1)) by XREAL_1: 118;

          then |.((s . m) - 0 ).| < (1 / (n + 1)) by A8, XXREAL_0: 2;

          hence |.((s . m) - 0 ).| < p by A12, XXREAL_0: 2;

        end;

        then s is convergent by SEQ_2:def 6;

        then ( lim s) = 0 by A9, SEQ_2:def 7;

        then

        reconsider s as 0 -convergent non-zero Real_Sequence by A9, A8, SEQ_1: 5, SEQ_2:def 6, FDIFF_1:def 1;

        ((s " ) (#) (R /* s)) is convergent & ( lim ((s " ) (#) (R /* s))) = 0 by A2, FDIFF_1:def 2;

        then

        consider n be Nat such that

         A15: for m be Nat st n <= m holds |.((((s " ) (#) (R /* s)) . m) - 0 ).| < r by A3, SEQ_2:def 7;

        

         A16: n in NAT by ORDINAL1:def 12;

        

         A18: |.(((s . n) " ) * (R . (s . n))).| = ( |.((s . n) " ).| * |.(R . (s . n)).|) by COMPLEX1: 65

        .= ( |.(R . (s . n)).| / |.(s . n).|) by COMPLEX1: 66;

         |.((((s " ) (#) (R /* s)) . n) - 0 ).| = |.(((s " ) . n) * ((R /* s) . n)).| by SEQ_1: 8

        .= |.(((s . n) " ) * ((R /* s) . n)).| by VALUED_1: 10

        .= |.(((s . n) " ) * (R . (s . n))).| by FUNCT_2: 115, A16;

        hence for r be Real st r > 0 holds ex d be Real st d > 0 & for z be Real st z <> 0 & |.z.| < d holds ( |.(R . z).| / |.z.|) < r by A8, A15, A18;

      end;

      now

        assume

         A19: for r be Real st r > 0 holds ex d be Real st d > 0 & for z be Real st z <> 0 & |.z.| < d holds ( |.(R . z).| / |.z.|) < r;

        now

          let s be 0 -convergent non-zero Real_Sequence;

          

           A20: s is convergent & ( lim s) = 0 ;

           A21:

          now

            let r be Real;

            assume

             A22: r > 0 ;

            consider d be Real such that

             A23: d > 0 and

             A24: for z be Real st z <> 0 & |.z.| < d holds ( |.(R . z).| / |.z.|) < r by A22, A19;

            consider n be Nat such that

             A25: for m be Nat st n <= m holds |.((s . m) - 0 ).| < d by A20, A23, SEQ_2:def 7;

            take n;

            hereby

              let m be Nat;

              

               A26: m in NAT by ORDINAL1:def 12;

              assume n <= m;

              then

               A27: |.((s . m) - 0 ).| < d by A25;

              ( |.(R . (s . m)).| / |.(s . m).|) = ( |.((s . m) " ).| * |.(R . (s . m)).|) by COMPLEX1: 66

              .= |.(((s . m) " ) * (R . (s . m))).| by COMPLEX1: 65

              .= |.(((s . m) " ) * ((R /* s) . m)).| by FUNCT_2: 115, A26

              .= |.(((s " ) . m) * ((R /* s) . m)).| by VALUED_1: 10

              .= |.((((s " ) (#) (R /* s)) . m) - 0 ).| by SEQ_1: 8;

              hence |.((((s " ) (#) (R /* s)) . m) - 0 ).| < r by A24, A27, SEQ_1: 5;

            end;

          end;

          hence ((s " ) (#) (R /* s)) is convergent by SEQ_2:def 6;

          hence ( lim ((s " ) (#) (R /* s))) = 0 by A21, SEQ_2:def 7;

        end;

        hence R is RestFunc-like by FDIFF_1:def 2;

      end;

      hence thesis by A1;

    end;

    reconsider jj = 1 as Real;

    

     Lm4: for T be RealNormSpace, f be PartFunc of REAL , T, g be PartFunc of REAL , REAL st ( dom f) = [. 0 , 1.] & ( dom g) = [. 0 , 1.] & (f | [. 0 , 1.]) is continuous & (g | [. 0 , 1.]) is continuous & f is_differentiable_on ]. 0 , 1.[ & g is_differentiable_on ]. 0 , 1.[ & (for x be Real st x in ]. 0 , 1.[ holds ||.( diff (f,x)).|| <= ( diff (g,x))) holds ||.((f /. 1) - (f /. 0 )).|| <= ((g /. 1) - (g /. 0 ))

    proof

      let T be RealNormSpace, f be PartFunc of REAL , T, g be PartFunc of REAL , REAL ;

      assume

       A1: ( dom f) = [. 0 , 1.] & ( dom g) = [. 0 , 1.] & (f | [. 0 , 1.]) is continuous & (g | [. 0 , 1.]) is continuous & f is_differentiable_on ]. 0 , 1.[ & g is_differentiable_on ]. 0 , 1.[ & (for x be Real st x in ]. 0 , 1.[ holds ||.( diff (f,x)).|| <= ( diff (g,x)));

      now

        let e be Real;

        assume

         A2: 0 < e;

        set e1 = (e / 2);

        set B = { x where x be Real : x in [. 0 , 1.] & ((( ||.((f /. x) - (f /. 0 )).|| - ((g . x) - (g . 0 ))) - (e1 * x)) - e1) <= 0 };

        now

          let z be object;

          assume z in B;

          then ex x be Real st z = x & x in [. 0 , 1.] & ((( ||.((f /. x) - (f /. 0 )).|| - ((g . x) - (g . 0 ))) - (e1 * x)) - e1) <= 0 ;

          hence z in REAL ;

        end;

        then

        reconsider B as Subset of REAL by TARSKI:def 3;

        now

          let r be Real;

          assume r in B;

          then ex x be Real st r = x & x in [. 0 , 1.] & ((( ||.((f /. x) - (f /. 0 )).|| - ((g . x) - (g . 0 ))) - (e1 * x)) - e1) <= 0 ;

          then

           A3: ex t be Real st r = t & 0 <= t & t <= 1;

          then |.r.| = r by ABSVALUE:def 1;

          hence |.r.| < 2 by A3, XXREAL_0: 2;

        end;

        then

         A4: B is real-bounded by SEQ_4: 4;

        set s = ( upper_bound B);

        

         A5: ex d be Real st 0 < d & d in B

        proof

           0 in [. 0 , 1.];

          then

          consider d1 be Real such that

           A6: 0 < d1 & for x1 be Real st x1 in [. 0 , 1.] & |.(x1 - 0 ).| < d1 holds ||.((f /. x1) - (f /. 0 )).|| < e1 by A2, A1, NFCONT_3: 17;

          set d2 = (d1 / 2);

          

           A7: d2 < d1 by A6, XREAL_1: 216;

          take d = ( min (d2,1));

          thus

           A8: 0 < d by A6, XXREAL_0: 21;

          

           A9: d <= 1 by XXREAL_0: 17;

          then

           A10: d in [. 0 , 1.] by A8;

          

           A11: d <= d2 by XXREAL_0: 17;

           |.(d - 0 ).| = d by A8, ABSVALUE:def 1;

          then |.(d - 0 ).| < d1 by A11, A7, XXREAL_0: 2;

          then

           A12: ||.((f /. d) - (f /. 0 )).|| < e1 by A6, A10;

          

           A13: [. 0 , d.] c= ( dom g) by A1, A9, XXREAL_1: 34;

          

           A14: (g | [. 0 , d.]) is continuous by A1, A9, FCONT_1: 16, XXREAL_1: 34;

          

           A15: ]. 0 , d.[ c= ]. 0 , 1.[ by A9, XXREAL_1: 46;

          then

          consider x0 be Real such that

           A16: x0 in ]. 0 , d.[ & ( diff (g,x0)) = (((g . d) - (g . 0 )) / (d - 0 )) by A1, A8, A13, A14, FDIFF_1: 26, ROLLE: 3;

           ||.( diff (f,x0)).|| <= ( diff (g,x0)) by A1, A16, A15;

          then 0 <= ((g . d) - (g . 0 )) by A8, A16;

          then ( 0 qua Real + ||.((f /. d) - (f /. 0 )).||) <= (((g . d) - (g . 0 )) + e1) by A12, XREAL_1: 7;

          then ( 0 qua Real + ||.((f /. d) - (f /. 0 )).||) <= ((((g . d) - (g . 0 )) + e1) + (e1 * d)) by A8, A2, XREAL_1: 7;

          then ( ||.((f /. d) - (f /. 0 )).|| - ((((g . d) - (g . 0 )) + e1) + (e1 * d))) <= 0 by XREAL_1: 47;

          then ((( ||.((f /. d) - (f /. 0 )).|| - ((g . d) - (g . 0 ))) - (e1 * d)) - e1) <= 0 ;

          hence d in B by A10;

        end;

        then

         A17: 0 < s by A4, SEQ_4:def 1;

        now

          let r be Real;

          assume r in B;

          then ex x be Real st r = x & x in [. 0 , 1.] & ((( ||.((f /. x) - (f /. 0 )).|| - ((g . x) - (g . 0 ))) - (e1 * x)) - e1) <= 0 ;

          then ex t be Real st r = t & 0 <= t & t <= 1;

          hence r <= 1;

        end;

        then

         A18: s <= 1 by A5, SEQ_4: 45;

        defpred P[ Nat, Element of REAL ] means $2 in B & |.(s - $2).| <= (1 / ($1 + 1));

         A19:

        now

          let x be Element of NAT ;

          reconsider t = (1 / (1 + x)) as Real;

          consider r be Real such that

           A20: r in B & (s - t) < r by A4, A5, SEQ_4:def 1;

          reconsider r as Element of REAL by XREAL_0:def 1;

          take r;

          ((s - t) + t) < (r + t) by A20, XREAL_1: 8;

          then

           A21: (s - r) < ((t + r) - r) by XREAL_1: 14;

          r <= s by A4, A20, SEQ_4:def 1;

          then 0 <= (s - r) by XREAL_1: 48;

          hence P[x, r] by A20, A21, SEQ_2: 1;

        end;

        consider sq be sequence of REAL such that

         A22: for x be Element of NAT holds P[x, (sq . x)] from FUNCT_2:sch 3( A19);

        

         A23: for x be Nat holds P[x, (sq . x)]

        proof

          let x be Nat;

          x in NAT by ORDINAL1:def 12;

          hence thesis by A22;

        end;

        reconsider sq as Real_Sequence;

         A24:

        now

          let p1 be Real;

          assume

           A25: 0 < p1;

          set p = (p1 / 2);

          consider n be Nat such that

           A26: (1 / p) < n by SEQ_4: 3;

          ((1 / p) + 0 qua Real) < (n + 1) by A26, XREAL_1: 8;

          then

           A27: (1 / (n + 1)) <= (1 / (1 / p)) by A25, XREAL_1: 118;

          take n;

          thus for m be Nat st n <= m holds |.((sq . m) - s).| < p1

          proof

            let m be Nat;

            assume n <= m;

            then 0 < (n + 1) & (n + 1) <= (m + 1) by XREAL_1: 6;

            then (1 / (m + 1)) <= (1 / (n + 1)) by XREAL_1: 118;

            then

             A28: (1 / (m + 1)) <= p by A27, XXREAL_0: 2;

            (sq . m) in B & |.(s - (sq . m)).| <= (1 / (m + 1)) by A23;

            then |.((sq . m) - s).| <= (1 / (1 + m)) by COMPLEX1: 60;

            then

             A29: |.((sq . m) - s).| <= p by A28, XXREAL_0: 2;

            p < p1 by A25, XREAL_1: 216;

            hence thesis by A29, XXREAL_0: 2;

          end;

        end;

        then

         A30: sq is convergent by SEQ_2:def 6;

        then

         A31: ( lim sq) = s by A24, SEQ_2:def 7;

        deffunc F( Real) = ((( ||.((f /. $1) - (f /. 0 )).|| - ((g . $1) - (g . 0 ))) - (e1 * $1)) - e1);

        

         A32: for x be Element of REAL holds F(x) in REAL by XREAL_0:def 1;

        consider Lg0 be Function of REAL , REAL such that

         A33: for x be Element of REAL holds (Lg0 . x) = F(x) from FUNCT_2:sch 8( A32);

        

         A34: for x be Real holds (Lg0 . x) = F(x)

        proof

          let x be Real;

          reconsider x as Element of REAL by XREAL_0:def 1;

          (Lg0 . x) = F(x) by A33;

          hence thesis;

        end;

        set Lg = (Lg0 | [. 0 , 1.]);

        

         A35: ( dom Lg0) = REAL by FUNCT_2:def 1;

        then

         A36: ( dom Lg) = [. 0 , 1.] by RELAT_1: 62;

        now

          let y be object;

          assume y in ( rng sq);

          then ex x be object st x in NAT & (sq . x) = y by FUNCT_2: 11;

          then y in B by A23;

          then ex x be Real st y = x & x in [. 0 , 1.] & ((( ||.((f /. x) - (f /. 0 )).|| - ((g . x) - (g . 0 ))) - (e1 * x)) - e1) <= 0 ;

          hence y in [. 0 , 1.];

        end;

        then

         A37: ( rng sq) c= ( dom Lg) by A36;

        

         A38: s in [. 0 , 1.] by A18, A17;

        now

          let r be Real;

          set r3 = (r / 3);

          assume

           A39: 0 < r;

          then

          consider t1 be Real such that

           A40: 0 < t1 & for x1 be Real st x1 in [. 0 , 1.] & |.(x1 - s).| < t1 holds ||.((f /. x1) - (f /. s)).|| < r3 by A1, A38, NFCONT_3: 17;

          consider t2 be Real such that

           A41: 0 < t2 & for x1 be Real st x1 in [. 0 , 1.] & |.(x1 - s).| < t2 holds |.((g . x1) - (g . s)).| < r3 by A39, A38, A1, FCONT_1: 14;

          set t30 = (r3 / e1);

          set t3 = (t30 / 2);

           0 < t3 & t3 < t30 by A2, A39, XREAL_1: 216;

          then (e1 * t3) < ((r3 / e1) * e1) by A2, XREAL_1: 97;

          then

           A42: (e1 * t3) < r3 by A2, XCMPLX_1: 87;

          take t = ( min (( min (t1,t2)),t3));

          

           A43: ( min (t1,t2)) <= t1 & ( min (t1,t2)) <= t2 & 0 < ( min (t1,t2)) by A40, A41, XXREAL_0: 17, XXREAL_0: 21;

          hence 0 < t by A2, A39, XXREAL_0: 21;

          

           A44: t <= t3 by XXREAL_0: 17;

          

           A45: t <= ( min (t1,t2)) by XXREAL_0: 17;

          then

           A46: t <= t1 by A43, XXREAL_0: 2;

          

           A47: t <= t2 by A43, A45, XXREAL_0: 2;

          thus for x1 be Real st x1 in ( dom Lg) & |.(x1 - s).| < t holds |.((Lg . x1) - (Lg . s)).| < r

          proof

            let x1 be Real;

            assume that

             A48: x1 in ( dom Lg) and

             A49: |.(x1 - s).| < t;

            x1 in [. 0 , 1.] by A35, A48, RELAT_1: 62;

            

            then

             A50: (Lg . x1) = (Lg0 . x1) by FUNCT_1: 49

            .= ((( ||.((f /. x1) - (f /. 0 )).|| - ((g . x1) - (g . 0 ))) - (e1 * x1)) - e1) by A34;

            (Lg . s) = (Lg0 . s) by A38, FUNCT_1: 49;

            then (Lg . s) = ((( ||.((f /. s) - (f /. 0 )).|| - ((g . s) - (g . 0 ))) - (e1 * s)) - e1) by A34;

            then ((Lg . x1) - (Lg . s)) = ((( ||.((f /. x1) - (f /. 0 )).|| - ||.((f /. s) - (f /. 0 )).||) - ((g . x1) - (g . s))) - (e1 * (x1 - s))) by A50;

            then

             A51: |.((Lg . x1) - (Lg . s)).| <= ( |.(( ||.((f /. x1) - (f /. 0 )).|| - ||.((f /. s) - (f /. 0 )).||) - ((g . x1) - (g . s))).| + |.(e1 * (x1 - s)).|) by COMPLEX1: 57;

            ( |.(( ||.((f /. x1) - (f /. 0 )).|| - ||.((f /. s) - (f /. 0 )).||) - ((g . x1) - (g . s))).| + |.(e1 * (x1 - s)).|) <= (( |.( ||.((f /. x1) - (f /. 0 )).|| - ||.((f /. s) - (f /. 0 )).||).| + |.((g . x1) - (g . s)).|) + |.(e1 * (x1 - s)).|) by COMPLEX1: 57, XREAL_1: 6;

            then

             A52: |.((Lg . x1) - (Lg . s)).| <= (( |.( ||.((f /. x1) - (f /. 0 )).|| - ||.((f /. s) - (f /. 0 )).||).| + |.((g . x1) - (g . s)).|) + |.(e1 * (x1 - s)).|) by A51, XXREAL_0: 2;

            (((f /. x1) - (f /. 0 )) - ((f /. s) - (f /. 0 ))) = ((f /. x1) - ((f /. 0 ) - ( - ((f /. s) - (f /. 0 ))))) by RLVECT_1: 29

            .= ((f /. x1) - ((f /. 0 ) + ((f /. s) - (f /. 0 )))) by RLVECT_1: 17

            .= ((f /. x1) - ((f /. s) - ((f /. 0 ) - (f /. 0 )))) by RLVECT_1: 29

            .= ((f /. x1) - ((f /. s) - ( 0. T))) by RLVECT_1: 5

            .= ((f /. x1) - (f /. s)) by RLVECT_1: 13;

            then ( |.( ||.((f /. x1) - (f /. 0 )).|| - ||.((f /. s) - (f /. 0 )).||).| + |.((g . x1) - (g . s)).|) <= ( ||.((f /. x1) - (f /. s)).|| + |.((g . x1) - (g . s)).|) by NORMSP_1: 9, XREAL_1: 6;

            then (( |.( ||.((f /. x1) - (f /. 0 )).|| - ||.((f /. s) - (f /. 0 )).||).| + |.((g . x1) - (g . s)).|) + |.(e1 * (x1 - s)).|) <= (( ||.((f /. x1) - (f /. s)).|| + |.((g . x1) - (g . s)).|) + |.(e1 * (x1 - s)).|) by XREAL_1: 6;

            then

             A54: |.((Lg . x1) - (Lg . s)).| <= (( ||.((f /. x1) - (f /. s)).|| + |.((g . x1) - (g . s)).|) + |.(e1 * (x1 - s)).|) by A52, XXREAL_0: 2;

             |.(x1 - s).| < t2 by A49, A47, XXREAL_0: 2;

            then |.((g . x1) - (g . s)).| < r3 by A48, A36, A41;

            then

             A55: ( ||.((f /. x1) - (f /. s)).|| + |.((g . x1) - (g . s)).|) < ( ||.((f /. x1) - (f /. s)).|| + r3) by XREAL_1: 8;

             |.(x1 - s).| < t3 by A49, A44, XXREAL_0: 2;

            then ( |.(x1 - s).| * e1) <= (t3 * e1) by A2, XREAL_1: 64;

            then ( |.(x1 - s).| * |.e1.|) <= (t3 * e1) by A2, ABSVALUE:def 1;

            then |.(e1 * (x1 - s)).| <= (t3 * e1) by COMPLEX1: 65;

            then

             A56: |.(e1 * (x1 - s)).| < r3 by A42, XXREAL_0: 2;

             |.(x1 - s).| < t1 by A49, A46, XXREAL_0: 2;

            then ||.((f /. x1) - (f /. s)).|| < r3 by A48, A36, A40;

            then ( ||.((f /. x1) - (f /. s)).|| + r3) < (r3 + r3) by XREAL_1: 8;

            then ( ||.((f /. x1) - (f /. s)).|| + |.((g . x1) - (g . s)).|) < (r3 + r3) by A55, XXREAL_0: 2;

            then (( ||.((f /. x1) - (f /. s)).|| + |.((g . x1) - (g . s)).|) + |.(e1 * (x1 - s)).|) < ((r3 + r3) + r3) by A56, XREAL_1: 8;

            hence |.((Lg . x1) - (Lg . s)).| < r by A54, XXREAL_0: 2;

          end;

        end;

        then

         A57: Lg is_continuous_in s by FCONT_1: 3;

        then

         A57a: (Lg /* sq) is convergent & (Lg . s) = ( lim (Lg /* sq)) by A30, A31, A37, FCONT_1:def 1;

        

         A58: for n be Nat holds 0 <= (( - (Lg /* sq)) . n)

        proof

          let n be Nat;

          

           A59: n in NAT by ORDINAL1:def 12;

          (( - (Lg /* sq)) . n) = ( - ((Lg /* sq) . n)) by SEQ_1: 10;

          then

           A60: (( - (Lg /* sq)) . n) = ( - (Lg . (sq . n))) by A37, FUNCT_2: 108, A59;

           P[n, (sq . n)] by A23;

          then

           A61: ex x be Real st (sq . n) = x & x in [. 0 , 1.] & ((( ||.((f /. x) - (f /. 0 )).|| - ((g . x) - (g . 0 ))) - (e1 * x)) - e1) <= 0 ;

          then (Lg0 . (sq . n)) <= 0 by A34;

          then (Lg . (sq . n)) <= 0 by A61, FUNCT_1: 49;

          hence 0 <= (( - (Lg /* sq)) . n) by A60;

        end;

        ( - (Lg /* sq)) is convergent by A57, A30, A31, A37, FCONT_1:def 1, SEQ_2: 9;

        then 0 <= ( lim ( - (Lg /* sq))) by A58, SEQ_2: 17;

        then 0 <= ( - ( lim (Lg /* sq))) by A57a, SEQ_2: 10;

        then (Lg . s) <= 0 by A57a;

        then (Lg0 . s) <= 0 by A38, FUNCT_1: 49;

        then

         A62: ((( ||.((f /. s) - (f /. 0 )).|| - ((g . s) - (g . 0 ))) - (e1 * s)) - e1) <= 0 by A34;

        

         A63: s = 1

        proof

          assume s <> 1;

          then s < 1 by A18, XXREAL_0: 1;

          then

           A64: s in ]. 0 , 1.[ by A17;

          then f is_differentiable_in s by A1, NDIFF_3: 10;

          then

          consider N1 be Neighbourhood of s such that

           A65: N1 c= ( dom f) & ex L1 be LinearFunc of T, R1 be RestFunc of T st ( diff (f,s)) = (L1 /. 1) & for x be Real st x in N1 holds ((f /. x) - (f /. s)) = ((L1 /. (x - s)) + (R1 /. (x - s))) by NDIFF_3:def 4;

          consider L1 be LinearFunc of T, R1 be RestFunc of T such that

           A66: ( diff (f,s)) = (L1 /. 1) & for x be Real st x in N1 holds ((f /. x) - (f /. s)) = ((L1 /. (x - s)) + (R1 /. (x - s))) by A65;

          g is_differentiable_in s by A1, A64, FDIFF_1: 9;

          then

          consider N2 be Neighbourhood of s such that

           A67: N2 c= ( dom g) & ex L2 be LinearFunc, R2 be RestFunc st ( diff (g,s)) = (L2 . 1) & for x be Real st x in N2 holds ((g . x) - (g . s)) = ((L2 . (x - s)) + (R2 . (x - s))) by FDIFF_1:def 5;

          consider L2 be LinearFunc, R2 be RestFunc such that

           A68: ( diff (g,s)) = (L2 . 1) & for x be Real st x in N2 holds ((g . x) - (g . s)) = ((L2 . (x - s)) + (R2 . (x - s))) by A67;

          consider NN3 be Neighbourhood of s such that

           A69: NN3 c= N1 & NN3 c= N2 by RCOMP_1: 17;

          consider g0 be Real such that

           A70: 0 < g0 & ].(s - g0), (s + g0).[ c= ]. 0 , 1.[ by A64, RCOMP_1: 19;

          reconsider NN4 = ].(s - g0), (s + g0).[ as Neighbourhood of s by A70, RCOMP_1:def 6;

          consider N3 be Neighbourhood of s such that

           A71: N3 c= NN3 & N3 c= NN4 by RCOMP_1: 17;

          consider d1 be Real such that

           A73: 0 < d1 & N3 = ].(s - d1), (s + d1).[ by RCOMP_1:def 6;

          set e2 = (e1 / 2);

          R1 is total & R1 is RestFunc-like by NDIFF_3:def 1;

          then

          consider d2 be Real such that

           A74: 0 < d2 & for t be Real st t <> 0 & |.t.| < d2 holds ( ||.(R1 /. t).|| / |.t.|) < e2 by A2, Th17;

          R2 is total & R2 is RestFunc-like by FDIFF_1:def 2;

          then

          consider d3 be Real such that

           A75: 0 < d3 & for t be Real st t <> 0 & |.t.| < d3 holds ( |.(R2 . t).| / |.t.|) < e2 by A2, Th18;

          

           A76: ( min (d1,d2)) <= d1 & ( min (d1,d2)) <= d2 & 0 < ( min (d1,d2)) by A73, A74, XXREAL_0: 17, XXREAL_0: 21;

          set d40 = ( min (( min (d1,d2)),d3));

          

           A77: d40 <= ( min (d1,d2)) & d40 <= d3 & 0 < d40 by A75, A76, XXREAL_0: 17, XXREAL_0: 21;

          set d4 = (d40 / 2);

          

           A78: d40 <= d1 & d40 <= d2 by A76, A77, XXREAL_0: 2;

          d4 < d40 by A77, XREAL_1: 216;

          then

           A79a: 0 < d4 & d4 < d1 & d4 < d2 & d4 < d3 by A77, A78, XXREAL_0: 2;

          then (s - d1) < (s + d4) & (s + d4) < (s + d1) by XREAL_1: 8;

          then

           A80: (s + d4) in N3 by A73;

          then

           A81: ((f /. (s + d4)) - (f /. s)) = ((L1 /. ((s + d4) - s)) + (R1 /. ((s + d4) - s))) by A66, A71, A69;

          

           A82: ((g . (s + d4)) - (g . s)) = ((L2 . ((s + d4) - s)) + (R2 . ((s + d4) - s))) by A71, A69, A80, A68;

          consider df1 be Point of T such that

           A83: for p be Real holds (L1 /. p) = (p * df1) by NDIFF_3:def 2;

          (L1 /. 1) = (1 * df1) by A83;

          then (L1 /. 1) = df1 by RLVECT_1:def 8;

          then

           A84: (L1 /. d4) = (d4 * ( diff (f,s))) by A66, A83;

          consider df2 be Real such that

           A85: for p be Real holds (L2 . p) = (df2 * p) by FDIFF_1:def 3;

          (L2 . 1) = (df2 * 1) by A85;

          then

           A86: (L2 . d4) = (d4 * ( diff (g,s))) by A68, A85;

          

           A87: ||.((f /. (s + d4)) - (f /. s)).|| <= ( ||.(L1 /. d4).|| + ||.(R1 /. d4).||) by A81, NORMSP_1:def 1;

          

           A88: ||.(L1 /. d4).|| = ( |.d4.| * ||.( diff (f,s)).||) by A84, NORMSP_1:def 1

          .= ( ||.( diff (f,s)).|| * d4) by A77, ABSVALUE:def 1;

          

           A89: 0 < |.d4.| by A77, ABSVALUE:def 1;

           |.d4.| < d2 by A79a, ABSVALUE:def 1;

          then ( ||.(R1 /. d4).|| / |.d4.|) < e2 by A74, A77;

          then ||.(R1 /. d4).|| <= (e2 * |.d4.|) by A89, XREAL_1: 81;

          then ||.(R1 /. d4).|| <= (e2 * d4) by A77, ABSVALUE:def 1;

          then ( ||.(L1 /. d4).|| + ||.(R1 /. d4).||) <= (( ||.( diff (f,s)).|| * d4) + (e2 * d4)) by A88, XREAL_1: 6;

          then

           A90: ||.((f /. (s + d4)) - (f /. s)).|| <= (( ||.( diff (f,s)).|| * d4) + (e2 * d4)) by A87, XXREAL_0: 2;

          ( ||.( diff (f,s)).|| * d4) <= (( diff (g,s)) * d4) by A64, A1, A77, XREAL_1: 64;

          then (( ||.( diff (f,s)).|| * d4) + (e2 * d4)) <= ((( diff (g,s)) * d4) + (e2 * d4)) by XREAL_1: 6;

          then

           A91: ||.((f /. (s + d4)) - (f /. s)).|| <= ((( diff (g,s)) * d4) + (e2 * d4)) by A90, XXREAL_0: 2;

           |.d4.| < d3 by A79a, ABSVALUE:def 1;

          then ( |.(R2 . d4).| / |.d4.|) < e2 by A75, A77;

          then |.(R2 . d4).| <= (e2 * |.d4.|) by A89, XREAL_1: 81;

          then |.(R2 . d4).| <= (e2 * d4) by A77, ABSVALUE:def 1;

          then ( - (e2 * d4)) <= (R2 . d4) by ABSVALUE: 5;

          then ((d4 * ( diff (g,s))) - (e2 * d4)) <= ((g . (s + d4)) - (g . s)) by A82, A86, XREAL_1: 6;

          then (d4 * ( diff (g,s))) <= (((g . (s + d4)) - (g . s)) + (e2 * d4)) by XREAL_1: 20;

          then ((( diff (g,s)) * d4) + (e2 * d4)) <= ((((g . (s + d4)) - (g . s)) + (e2 * d4)) + (e2 * d4)) by XREAL_1: 6;

          then ||.((f /. (s + d4)) - (f /. s)).|| <= (((g . (s + d4)) - (g . s)) + (e1 * d4)) by A91, XXREAL_0: 2;

          then ( ||.((f /. (s + d4)) - (f /. s)).|| - (((g . (s + d4)) - (g . s)) + (e1 * d4))) <= 0 by XREAL_1: 47;

          then

           A92: (((( ||.((f /. (s + d4)) - (f /. s)).|| - (g . (s + d4))) + (g . s)) - (e1 * d4)) + ((( ||.((f /. s) - (f /. 0 )).|| - ((g . s) - (g . 0 ))) - (e1 * s)) - e1)) <= ( 0 qua Real + 0 qua Real) by A62;

          ( ||.((f /. (s + d4)) - (f /. 0 )).|| - ((((g . (s + d4)) - (g . 0 )) + (e1 * (d4 + s))) + e1)) <= (( ||.((f /. (s + d4)) - (f /. s)).|| + ||.((f /. s) - (f /. 0 )).||) - ((((g . (s + d4)) - (g . 0 )) + (e1 * (d4 + s))) + e1)) by NORMSP_1: 10, XREAL_1: 9;

          then

           A93: ((( ||.((f /. (s + d4)) - (f /. 0 )).|| - ((g . (s + d4)) - (g . 0 ))) - (e1 * (s + d4))) - e1) <= 0 by A92;

           |.(( 0 qua Real + 1) - (2 * (s + d4))).| < (1 - 0 qua Real) by A80, A71, A70, RCOMP_1: 3;

          then (s + d4) in [. 0 , 1.] by RCOMP_1: 2;

          then

           A94: (s + d4) in B by A93;

          (s + 0 qua Real) < (s + d4) by A77, XREAL_1: 8;

          hence contradiction by A94, A4, SEQ_4:def 1;

        end;

         0 in ( dom g) & 1 in ( dom g) by A1;

        then (g /. 1) = (g . 1) & (g /. 0 ) = (g . 0 ) by PARTFUN1:def 6;

        then ((( ||.((f /. 1) - (f /. 0 )).|| - ((g /. 1) - (g /. 0 ))) - e) + e) <= ( 0 qua Real + e) by A63, A62, XREAL_1: 6;

        hence ( ||.((f /. 1) - (f /. 0 )).|| - ((g /. 1) - (g /. 0 ))) <= e;

      end;

      then (( ||.((f /. 1) - (f /. 0 )).|| - ((g /. 1) - (g /. 0 ))) + ((g /. 1) - (g /. 0 ))) <= ( 0 qua Real + ((g /. 1) - (g /. 0 ))) by Lm3, XREAL_1: 6;

      hence thesis;

    end;

    theorem :: NDIFF_5:19

    

     Th19: for S,T be RealNormSpace, f be PartFunc of S, T, p,q be Point of S, M be Real st [.p, q.] c= ( dom f) & (for x be Point of S st x in [.p, q.] holds f is_continuous_in x) & (for x be Point of S st x in ].p, q.[ holds f is_differentiable_in x) & (for x be Point of S st x in ].p, q.[ holds ||.( diff (f,x)).|| <= M) holds ||.((f /. q) - (f /. p)).|| <= (M * ||.(q - p).||)

    proof

      let S,T be RealNormSpace, f be PartFunc of S, T, p,q be Point of S, M be Real;

      assume

       A1: [.p, q.] c= ( dom f) & (for x be Point of S st x in [.p, q.] holds f is_continuous_in x) & (for x be Point of S st x in ].p, q.[ holds f is_differentiable_in x) & (for x be Point of S st x in ].p, q.[ holds ||.( diff (f,x)).|| <= M);

      per cases ;

        suppose

         B2: p = q;

        

         B3: ||.((f /. q) - (f /. p)).|| = ||.( 0. T).|| by B2, RLVECT_1: 15

        .= 0 ;

        (M * ||.(q - p).||) = (M * ||.( 0. S).||) by B2, RLVECT_1: 15

        .= 0 ;

        hence thesis by B3;

      end;

        suppose p <> q;

        then

         X1: ].p, q.[ = { (p + (t * (q - p))) where t be Real : 0 < t & t < 1 } by LMOPN;

        deffunc PP( Real) = (($1 * (q - p)) + p);

        consider pt0 be Function of REAL , the carrier of S such that

         A2: for t be Element of REAL holds (pt0 . t) = PP(t) from FUNCT_2:sch 4;

        

         A3: for t be Real holds (pt0 . t) = PP(t)

        proof

          let t be Real;

          reconsider t as Element of REAL by XREAL_0:def 1;

          (pt0 . t) = PP(t) by A2;

          hence thesis;

        end;

        set pt = (pt0 | [. 0 , 1.]);

        

         A4: ( dom pt0) = REAL by FUNCT_2:def 1;

        then

         A5: ( dom pt) = [. 0 , 1.] by RELAT_1: 62;

         A6:

        now

          let t be Real;

          assume t in [. 0 , 1.];

          (pt0 /. t) = (pt0 . t) by A4, PARTFUN1:def 6, XREAL_0:def 1;

          hence (pt0 /. t) = ((t * (q - p)) + p) by A3;

        end;

        

         A7: ]. 0 , 1.[ c= [. 0 , 1.] by XXREAL_1: 25;

         A8:

        now

          let t be Real;

          assume t in ]. 0 , 1.[;

          

          hence (pt /. t) = (pt0 /. t) by A5, A7, PARTFUN2: 15

          .= (pt0 . t) by A4, PARTFUN1:def 6, XREAL_0:def 1

          .= ((t * (q - p)) + p) by A3;

        end;

        then

         A10: pt is_differentiable_on ]. 0 , 1.[ & for t be Real st t in ]. 0 , 1.[ holds ((pt `| ]. 0 , 1.[) . t) = (q - p) by A5, A7, NDIFF_3: 21;

        reconsider phi = (f * pt) as PartFunc of REAL , T;

        

         A11: ( rng pt) c= [.p, q.]

        proof

          let y be object;

          assume y in ( rng pt);

          then

          consider x be object such that

           A12: x in ( dom pt) & y = (pt . x) by FUNCT_1:def 3;

          

           A13: y = (pt0 . x) by A12, FUNCT_1: 47;

          reconsider x as Element of REAL by A12;

          consider r be Real such that

           A14: x = r & 0 <= r & r <= 1 by A12, A5;

          y = (p + (x * (q - p))) by A3, A13

          .= (((1 - x) * p) + (x * q)) by Lm2;

          then y in { (((1 - r1) * p) + (r1 * q)) where r1 be Real : 0 <= r1 & r1 <= 1 } by A14;

          hence y in [.p, q.] by RLTOPSP1:def 2;

        end;

        then ( rng pt) c= ( dom f) by A1;

        then

         A15: ( dom phi) = [. 0 , 1.] by A5, RELAT_1: 27;

        

         A16: for t be Real st t in [. 0 , 1.] holds (phi /. t) = (f /. (p + (t * (q - p))))

        proof

          let t be Real;

          assume

           A17: t in [. 0 , 1.];

          

          then

           A18: (phi /. t) = (phi . t) by A15, PARTFUN1:def 6

          .= (f . (pt . t)) by A17, A15, FUNCT_1: 12;

          

           A19: (pt . t) in ( rng pt) by A17, A5, FUNCT_1:def 3;

          (pt . t) = (pt0 . t) by A17, A5, FUNCT_1: 47

          .= (p + (t * (q - p))) by A3;

          hence thesis by A18, A11, A19, A1, PARTFUN1:def 6;

        end;

        now

          let x0 be Real;

          assume

           A20: x0 in ( dom phi);

          then

           A21: pt is_continuous_in x0 by A5, A6, A15, NFCONT_3: 33, NFCONT_3:def 2;

          (pt . x0) in ( rng pt) by A5, A20, A15, FUNCT_1:def 3;

          then (pt . x0) in [.p, q.] by A11;

          then (pt /. x0) in [.p, q.] by A20, A15, A5, PARTFUN1:def 6;

          hence phi is_continuous_in x0 by A1, A20, A21, NFCONT_3: 15;

        end;

        then phi is continuous by NFCONT_3:def 2;

        then

         A22: (phi | [. 0 , 1.]) is continuous;

         A23:

        now

          let x be Real;

          assume

           A24: x in ]. 0 , 1.[;

          then

           A25: pt is_differentiable_in x by A10, NDIFF_3: 10;

          ((pt `| ]. 0 , 1.[) . x) = (q - p) by A24, A8, A5, A7, NDIFF_3: 21;

          then

           A26: ( diff (pt,x)) = (q - p) by A10, A24, NDIFF_3:def 6;

          

           A27: (pt . x) = (pt /. x) by A24, A7, A5, PARTFUN1:def 6;

          

           A28: ex r be Real st x = r & 0 < r & r < 1 by A24;

          

           A29: (pt . x) = (pt0 . x) by A24, A7, A5, FUNCT_1: 47;

          

           A30: (pt0 . x) = (p + (x * (q - p))) by A3;

          (pt . x) in ].p, q.[ by X1, A28, A29, A30;

          then

           A31: f is_differentiable_in (pt /. x) by A27, A1;

          hence phi is_differentiable_in x by A25, Th6;

          thus ( diff (phi,x)) = (( diff (f,(p + (x * (q - p))))) . (q - p)) by A26, A27, A29, A30, A31, A25, Th6;

        end;

        then ]. 0 , 1.[ c= ( dom phi) & for x be Real st x in ]. 0 , 1.[ holds phi is_differentiable_in x by A15, XXREAL_1: 25;

        then

         A32: phi is_differentiable_on ]. 0 , 1.[ by NDIFF_3: 10;

        deffunc GG( Real) = ( In (((M * ||.(q - p).||) * $1), REAL ));

        consider g0 be Function of REAL , REAL such that

         A33: for t be Element of REAL holds (g0 . t) = GG(t) from FUNCT_2:sch 4;

        

         A34: for t be Real holds (g0 . t) = GG(t)

        proof

          let t be Real;

          reconsider t as Element of REAL by XREAL_0:def 1;

          (g0 . t) = GG(t) by A33;

          hence thesis;

        end;

        set g = (g0 | [. 0 , 1.]);

        

         A35: for t be Real st t in [. 0 , 1.] holds (g0 . t) = (((M * ||.(q - p).||) * t) + 0 qua Real)

        proof

          let t be Real;

          assume t in [. 0 , 1.];

          

          thus (g0 . t) = GG(t) by A34

          .= (((M * ||.(q - p).||) * t) + 0 qua Real);

        end;

        ( dom g0) = REAL by FUNCT_2:def 1;

        then

         A36: ( dom g) = [. 0 , 1.] by RELAT_1: 62;

        

         A37: (g | [. 0 , 1.]) is continuous by A35, FCONT_1: 41;

         A38:

        now

          let t be Real;

          assume t in ]. 0 , 1.[;

          

          hence (g . t) = (g0 . t) by A36, A7, FUNCT_1: 47

          .= GG(t) by A34

          .= (((M * ||.(q - p).||) * t) + 0 qua Real);

        end;

        then

         A39: g is_differentiable_on ]. 0 , 1.[ & for t be Real st t in ]. 0 , 1.[ holds ((g `| ]. 0 , 1.[) . t) = (M * ||.(q - p).||) by A36, A7, FDIFF_1: 23;

        for t be Real st t in ]. 0 , 1.[ holds ||.( diff (phi,t)).|| <= ( diff (g,t))

        proof

          let t be Real;

          assume

           A40: t in ]. 0 , 1.[;

          then

           A41: ||.( diff (phi,t)).|| = ||.(( diff (f,(p + (t * (q - p))))) . (q - p)).|| by A23;

          reconsider L = ( diff (f,(p + (t * (q - p))))) as Lipschitzian LinearOperator of S, T by LOPBAN_1:def 9;

          

           A42: ||.(L . (q - p)).|| <= ( ||.( diff (f,(p + (t * (q - p))))).|| * ||.(q - p).||) by LOPBAN_1: 32;

          

           A43: ex r be Real st t = r & 0 < r & r < 1 by A40;

          (p + (t * (q - p))) in ].p, q.[ by A43, X1;

          then

           A44: ( ||.( diff (f,(p + (t * (q - p))))).|| * ||.(q - p).||) <= (M * ||.(q - p).||) by A1, XREAL_1: 64;

          ( diff (g,t)) = ((g `| ]. 0 , 1.[) . t) by A40, A39, FDIFF_1:def 7;

          then ( diff (g,t)) = (M * ||.(q - p).||) by A40, A38, A36, A7, FDIFF_1: 23;

          hence thesis by A44, A42, A41, XXREAL_0: 2;

        end;

        then

         A45: ||.((phi /. 1) - (phi /. 0 )).|| <= ((g /. 1) - (g /. 0 )) by Lm4, A15, A22, A32, A36, A37, A38, A7, FDIFF_1: 23;

        

         A46: 1 in [. 0 , 1.] & 0 in [. 0 , 1.];

        

        then

         A47: (g /. 1) = (g . 1) by A36, PARTFUN1:def 6

        .= (g0 . 1) by A36, A46, FUNCT_1: 47

        .= GG() by A34

        .= ((M * ||.(q - p).||) * 1);

        

         A48: (g /. 0 ) = (g . 0 ) by A36, A46, PARTFUN1:def 6

        .= (g0 . 0 ) by A36, A46, FUNCT_1: 47

        .= GG(0) by A34

        .= ((M * ||.(q - p).||) * 0 qua Real);

        

         A49: (phi /. 1) = (f /. (p + (1 * (q - p)))) by A16, A46

        .= (f /. (p + (q - p))) by RLVECT_1:def 8

        .= (f /. (q - (p - p))) by RLVECT_1: 29

        .= (f /. (q - ( 0. S))) by RLVECT_1: 15

        .= (f /. q) by RLVECT_1: 13;

        (phi /. 0 ) = (f /. (p + ( 0 qua Real * (q - p)))) by A16, A46

        .= (f /. (p + ( 0. S))) by RLVECT_1: 10

        .= (f /. p) by RLVECT_1: 4;

        hence thesis by A45, A47, A48, A49;

      end;

    end;

    theorem :: NDIFF_5:20

    

     Th20: for S,T be RealNormSpace, f be PartFunc of S, T, p,q be Point of S, M be Real, L be Point of ( R_NormSpace_of_BoundedLinearOperators (S,T)) st [.p, q.] c= ( dom f) & (for x be Point of S st x in [.p, q.] holds f is_continuous_in x) & (for x be Point of S st x in ].p, q.[ holds f is_differentiable_in x) & (for x be Point of S st x in ].p, q.[ holds ||.(( diff (f,x)) - L).|| <= M) holds ||.(((f /. q) - (f /. p)) - (L . (q - p))).|| <= (M * ||.(q - p).||)

    proof

      let S,T be RealNormSpace, f be PartFunc of S, T, p,q be Point of S, M be Real, L be Point of ( R_NormSpace_of_BoundedLinearOperators (S,T));

      assume that

       A1: [.p, q.] c= ( dom f) and

       A2: (for x be Point of S st x in [.p, q.] holds f is_continuous_in x) & (for x be Point of S st x in ].p, q.[ holds f is_differentiable_in x) & (for x be Point of S st x in ].p, q.[ holds ||.(( diff (f,x)) - L).|| <= M);

      reconsider LP = L as Lipschitzian LinearOperator of S, T by LOPBAN_1:def 9;

      deffunc LL( Point of S) = (L . ($1 - p));

      consider L0 be Function of the carrier of S, the carrier of T such that

       A3: for t be Element of the carrier of S holds (L0 . t) = LL(t) from FUNCT_2:sch 4;

      

       A4: ( dom L0) = the carrier of S by FUNCT_2:def 1;

      now

        let x1,x2 be Point of S;

        assume x1 in ( dom L0) & x2 in ( dom L0);

        (L0 /. x1) = (L . (x1 - p)) & (L0 /. x2) = (L . (x2 - p)) by A3;

        

        then ||.((L0 /. x1) - (L0 /. x2)).|| = ||.((LP . (x1 - p)) + (( - 1) * (LP . (x2 - p)))).|| by RLVECT_1: 16

        .= ||.((LP . (x1 - p)) + (LP . (( - 1) * (x2 - p)))).|| by LOPBAN_1:def 5

        .= ||.(LP . ((x1 - p) + (( - 1) * (x2 - p)))).|| by VECTSP_1:def 20

        .= ||.(LP . ((x1 - p) - (x2 - p))).|| by RLVECT_1: 16

        .= ||.(LP . (x1 - ((x2 - p) + p))).|| by RLVECT_1: 27

        .= ||.(LP . (x1 - (x2 - (p - p)))).|| by RLVECT_1: 29

        .= ||.(LP . (x1 - (x2 - ( 0. S)))).|| by RLVECT_1: 15

        .= ||.(LP . (x1 - x2)).|| by RLVECT_1: 13;

        then

         A5: ||.((L0 /. x1) - (L0 /. x2)).|| <= ( ||.L.|| * ||.(x1 - x2).||) by LOPBAN_1: 32;

        ( 0 qua Real + ||.L.||) < (1 + ||.L.||) by XREAL_1: 8;

        then ( ||.L.|| * ||.(x1 - x2).||) <= (( ||.L.|| + 1) * ||.(x1 - x2).||) by XREAL_1: 64;

        hence ||.((L0 /. x1) - (L0 /. x2)).|| <= (( ||.L.|| + 1) * ||.(x1 - x2).||) by A5, XXREAL_0: 2;

      end;

      then

       A6: L0 is_continuous_on ( dom L0) by NFCONT_1: 45, NFCONT_1:def 9;

      reconsider R = (the carrier of S --> ( 0. T)) as PartFunc of S, T;

      

       A7: ( dom R) = the carrier of S;

      now

        let h be ( 0. S) -convergent sequence of S;

        assume h is non-zero;

         A8:

        now

          let n be Nat;

          

           A9: (R /. (h . n)) = (R . (h . n)) by A7, PARTFUN1:def 6

          .= ( 0. T);

          

           A10: ( rng h) c= ( dom R);

          

           A11: n in NAT by ORDINAL1:def 12;

          

          thus ((( ||.h.|| " ) (#) (R /* h)) . n) = ((( ||.h.|| " ) . n) * ((R /* h) . n)) by NDIFF_1:def 2

          .= ((( ||.h.|| " ) . n) * (R /. (h . n))) by A11, A10, FUNCT_2: 109

          .= ( 0. T) by A9, RLVECT_1: 10;

        end;

        then

         A12: (( ||.h.|| " ) (#) (R /* h)) is constant by VALUED_0:def 18;

        hence (( ||.h.|| " ) (#) (R /* h)) is convergent by NDIFF_1: 18;

        ((( ||.h.|| " ) (#) (R /* h)) . 0 ) = ( 0. T) by A8;

        hence ( lim (( ||.h.|| " ) (#) (R /* h))) = ( 0. T) by A12, NDIFF_1: 18;

      end;

      then

      reconsider R as RestFunc of S, T by NDIFF_1:def 5;

       A13:

      now

        let x0 be Point of S;

        set N = the Neighbourhood of x0;

        

         A14: for x be Point of S st x in N holds ((L0 /. x) - (L0 /. x0)) = ((L . (x - x0)) + (R /. (x - x0)))

        proof

          let x be Point of S;

          

           A15: (R /. (x - x0)) = (R . (x - x0)) by A7, PARTFUN1:def 6

          .= ( 0. T);

          assume x in N;

          

          thus ((L0 /. x) - (L0 /. x0)) = ((L . (x - p)) - (L0 /. x0)) by A3

          .= ((L . (x - p)) - (L . (x0 - p))) by A3

          .= ((LP . (x - p)) + (( - 1) * (LP . (x0 - p)))) by RLVECT_1: 16

          .= ((LP . (x - p)) + (LP . (( - 1) * (x0 - p)))) by LOPBAN_1:def 5

          .= (LP . ((x - p) + (( - 1) * (x0 - p)))) by VECTSP_1:def 20

          .= (LP . ((x - p) - (x0 - p))) by RLVECT_1: 16

          .= (LP . (x - ((x0 - p) + p))) by RLVECT_1: 27

          .= (LP . (x - (x0 - (p - p)))) by RLVECT_1: 29

          .= (LP . (x - (x0 - ( 0. S)))) by RLVECT_1: 15

          .= (LP . (x - x0)) by RLVECT_1: 13

          .= ((L . (x - x0)) + (R /. (x - x0))) by A15, RLVECT_1: 4;

        end;

        hence L0 is_differentiable_in x0 by A4, NDIFF_1:def 6;

        hence ( diff (L0,x0)) = L by A4, A14, NDIFF_1:def 7;

      end;

      set g = (f - L0);

      

       A16: ( dom g) = (( dom f) /\ ( dom L0)) by VFUNCT_1:def 2

      .= ( dom f) by A4, XBOOLE_1: 28;

      

       A17: for x be Point of S st x in ( dom g) holds (g /. x) = ((f /. x) - (L . (x - p)))

      proof

        let x be Point of S;

        assume x in ( dom g);

        

        hence (g /. x) = ((f /. x) - (L0 /. x)) by VFUNCT_1:def 2

        .= ((f /. x) - (L . (x - p))) by A3;

      end;

      

       A18: for x be Point of S st x in [.p, q.] holds g is_continuous_in x

      proof

        let x be Point of S;

        assume x in [.p, q.];

        then

         A19: f is_continuous_in x by A2;

        (L0 | ( dom L0)) is_continuous_in x by A4, A6, NFCONT_1:def 7;

        hence thesis by A19, NFCONT_1: 15;

      end;

      

       A20: for x be Point of S st x in ].p, q.[ holds g is_differentiable_in x

      proof

        let x be Point of S;

        assume x in ].p, q.[;

        then f is_differentiable_in x & L0 is_differentiable_in x by A2, A13;

        hence g is_differentiable_in x by NDIFF_1: 36;

      end;

      for x be Point of S st x in ].p, q.[ holds ||.( diff (g,x)).|| <= M

      proof

        let x be Point of S;

        assume

         A21: x in ].p, q.[;

        then

         A22: f is_differentiable_in x by A2;

        L0 is_differentiable_in x & ( diff (L0,x)) = L by A13;

        then ( diff (g,x)) = (( diff (f,x)) - L) by A22, NDIFF_1: 36;

        hence ||.( diff (g,x)).|| <= M by A2, A21;

      end;

      then

       A23: ||.((g /. q) - (g /. p)).|| <= (M * ||.(q - p).||) by Th19, A1, A16, A18, A20;

      p in [.p, q.] by RLTOPSP1: 68;

      then (g /. p) = ((f /. p) - (L . (p - p))) by A1, A16, A17;

      

      then

       A24: (g /. p) = ((f /. p) - (LP . ( 0. S))) by RLVECT_1: 15

      .= ((f /. p) - (LP . ( 0 * p))) by RLVECT_1: 10

      .= ((f /. p) - ( 0 * (LP . p))) by LOPBAN_1:def 5

      .= ((f /. p) - ( 0. T)) by RLVECT_1: 10

      .= (f /. p) by RLVECT_1: 13;

      q in [.p, q.] by RLTOPSP1: 68;

      then (g /. q) = ((f /. q) - (L . (q - p))) by A1, A16, A17;

      then ((f /. q) - ((L . (q - p)) + (f /. p))) = ((g /. q) - (g /. p)) by A24, RLVECT_1: 27;

      hence thesis by A23, RLVECT_1: 27;

    end;

    begin

    definition

      let G be RealNormSpace-Sequence;

      let i be Element of ( dom G);

      :: NDIFF_5:def3

      func proj i -> Function of ( product G), (G . i) means

      : Def3: for x be Element of ( product ( carr G)) holds (it . x) = (x . i);

      existence

      proof

        deffunc F( Element of ( product ( carr G))) = ($1 . i);

        consider f be Function of ( product ( carr G)), (G . i) such that

         A1: for x be Element of ( product ( carr G)) holds (f . x) = F(x) from FUNCT_2:sch 4;

        ( product G) = NORMSTR (# ( product ( carr G)), ( zeros G), [:( addop G):], [:( multop G):], ( productnorm G) #) by PRVECT_2: 6;

        then

        reconsider f as Function of ( product G), (G . i);

        take f;

        thus thesis by A1;

      end;

      uniqueness

      proof

        let f,g be Function of the carrier of ( product G), the carrier of (G . i);

        assume that

         A2: for x be Element of ( product ( carr G)) holds (f . x) = (x . i) and

         A3: for x be Element of ( product ( carr G)) holds (g . x) = (x . i);

        

         A4: ( product G) = NORMSTR (# ( product ( carr G)), ( zeros G), [:( addop G):], [:( multop G):], ( productnorm G) #) by PRVECT_2: 6;

        now

          let x1 be Element of the carrier of ( product G);

          reconsider x = x1 as Element of ( product ( carr G)) by A4;

          (f . x1) = (x . i) by A2;

          hence (f . x1) = (g . x1) by A3;

        end;

        hence thesis;

      end;

    end

    definition

      let G be RealNormSpace-Sequence;

      let i be Element of ( dom G);

      let x be Element of ( product G);

      :: NDIFF_5:def4

      func reproj (i,x) -> Function of (G . i), ( product G) means

      : Def4: for r be Element of (G . i) holds (it . r) = (x +* (i,r));

      existence

      proof

        reconsider x1 = x as Element of ( product ( carr G)) by Th10;

        defpred P[ Element of (G . i), Element of the carrier of ( product G)] means $2 = (x1 +* (i,$1));

        

         A1: for r be Element of (G . i) holds ex y be Element of the carrier of ( product G) st P[r, y]

        proof

          let r be Element of (G . i);

          (x1 +* (i,r)) is Element of the carrier of ( product G) by Th11;

          hence thesis;

        end;

        ex f be Function of the carrier of (G . i), the carrier of ( product G) st for r be Element of (G . i) holds P[r, (f . r)] from FUNCT_2:sch 3( A1);

        hence thesis;

      end;

      uniqueness

      proof

        let f,g be Function of the carrier of (G . i), the carrier of ( product G);

        assume that

         A2: for r be Element of (G . i) holds (f . r) = (x +* (i,r)) and

         A3: for r be Element of (G . i) holds (g . r) = (x +* (i,r));

        let r be Element of (G . i);

        (f . r) = (x +* (i,r)) by A2;

        hence (f . r) = (g . r) by A3;

      end;

    end

    definition

      ::$Canceled

      let G be RealNormSpace-Sequence;

      let F be RealNormSpace;

      let i be set;

      let f be PartFunc of ( product G), F;

      let x be Element of ( product G);

      :: NDIFF_5:def6

      pred f is_partial_differentiable_in x,i means (f * ( reproj (( In (i,( dom G))),x))) is_differentiable_in (( proj ( In (i,( dom G)))) . x);

    end

    definition

      let G be RealNormSpace-Sequence;

      let F be RealNormSpace;

      let i be set;

      let f be PartFunc of ( product G), F;

      let x be Point of ( product G);

      :: NDIFF_5:def7

      func partdiff (f,x,i) -> Point of ( R_NormSpace_of_BoundedLinearOperators ((G . ( In (i,( dom G)))),F)) equals ( diff ((f * ( reproj (( In (i,( dom G))),x))),(( proj ( In (i,( dom G)))) . x)));

      coherence ;

    end

    begin

    reserve G for RealNormSpace-Sequence;

    reserve F for RealNormSpace;

    reserve i for Element of ( dom G);

    reserve f,f1,f2 for PartFunc of ( product G), F;

    reserve x for Point of ( product G);

    reserve X for set;

    definition

      let G be RealNormSpace-Sequence;

      let F be RealNormSpace;

      let i be set;

      let f be PartFunc of ( product G), F;

      let X be set;

      :: NDIFF_5:def8

      pred f is_partial_differentiable_on X,i means X c= ( dom f) & for x be Point of ( product G) st x in X holds (f | X) is_partial_differentiable_in (x,i);

    end

    theorem :: NDIFF_5:21

    

     Th21: for xi be Element of (G . i) holds ||.(( reproj (i,( 0. ( product G)))) . xi).|| = ||.xi.||

    proof

      let xi be Element of (G . i);

      set j = ( len G);

      reconsider i0 = i as Element of NAT ;

      ( Seg ( len G)) = ( dom G) by FINSEQ_1:def 3;

      then

       A1: 1 <= i0 & i0 <= j by FINSEQ_1: 1;

      set z = ( 0. ( product G));

      

       A3: the carrier of ( product G) = ( product ( carr G)) by Th10;

      then

      reconsider w = (z +* (i0,xi)) as Element of ( product ( carr G)) by Th11;

      

       A4: ||.(( reproj (i,z)) . xi).|| = |.( normsequence (G,w)).| by Def4, PRVECT_2: 7;

      reconsider q = ||.xi.|| as Element of REAL ;

      set q1 = <*q*>;

      set y = ( 0* j);

      

       A5: ( len ( normsequence (G,w))) = j by PRVECT_2:def 11;

      

       A6: ( len y) = j by CARD_1:def 7;

      then

       A7: (((y | (i0 -' 1)) ^ <*q*>) ^ (y /^ i0)) = ( Replace (y,i0,q)) by A1, FINSEQ_7:def 1;

      then

       A8: ( len (((y | (i0 -' 1)) ^ <*q*>) ^ (y /^ i0))) = ( len y) by FINSEQ_7: 5;

      

       A9: ( len y) = ( len ( Replace (y,i0,q))) by FINSEQ_7: 5;

      for k be Nat st 1 <= k & k <= ( len ( normsequence (G,w))) holds (( normsequence (G,w)) . k) = ((((y | (i0 -' 1)) ^ <*q*>) ^ (y /^ i0)) . k)

      proof

        let k be Nat;

        assume

         A10: 1 <= k & k <= ( len ( normsequence (G,w)));

        then

        reconsider k1 = k as Element of ( dom G) by A5, FINSEQ_3: 25;

        

         A11: k1 in ( dom G);

        z in the carrier of ( product G);

        then z in ( product ( carr G)) by Th10;

        then

        consider g be Function such that

         A12: z = g & ( dom g) = ( dom ( carr G)) & for y be object st y in ( dom ( carr G)) holds (g . y) in (( carr G) . y) by CARD_3:def 5;

        

         A13: k in ( dom z) by A11, A12, Lm1;

        

         A14: (( normsequence (G,w)) . k) = (the normF of (G . k1) . (w . k1)) by PRVECT_2:def 11;

        per cases ;

          suppose

           A15: k = i0;

          then

           A16: (( normsequence (G,w)) . k) = ||.xi.|| by A14, A13, FUNCT_7: 31;

          (( Replace (y,i0,q)) /. k) = q by A15, A10, A5, A6, FINSEQ_7: 8;

          hence (( normsequence (G,w)) . k) = ((((y | (i0 -' 1)) ^ <*q*>) ^ (y /^ i0)) . k) by A16, A7, A10, A5, A6, A9, FINSEQ_4: 15;

        end;

          suppose

           A17: k <> i0;

          then (w . k1) = (z . k1) by FUNCT_7: 32;

          then

           A18: (( normsequence (G,w)) . k) = ||.( 0. (G . k1)).|| by A14, Th14, A3;

          (( Replace (y,i0,q)) /. k) = (y /. k) by A17, A10, A5, A6, FINSEQ_7: 10;

          then (( Replace (y,i0,q)) . k) = (y /. k) by A10, A5, A6, A9, FINSEQ_4: 15;

          then (( Replace (y,i0,q)) . k) = (y . k) by A10, A5, A6, FINSEQ_4: 15;

          hence (( normsequence (G,w)) . k) = ((((y | (i0 -' 1)) ^ <*q*>) ^ (y /^ i0)) . k) by A18, A6, A1, FINSEQ_7:def 1;

        end;

      end;

      then

       A19: ( normsequence (G,w)) = (((y | (i0 -' 1)) ^ <*q*>) ^ (y /^ i0)) by A6, A8, PRVECT_2:def 11;

      ( sqrt ( Sum ( sqr (y | (i0 -' 1))))) = |.( 0* (i0 -' 1)).| by A1, PDIFF_7: 2;

      then ( sqrt ( Sum ( sqr (y | (i0 -' 1))))) = 0 by EUCLID: 7;

      then

       A20: ( Sum ( sqr (y | (i0 -' 1)))) = 0 by RVSUM_1: 86, SQUARE_1: 24;

      ( sqrt ( Sum ( sqr (y /^ i0)))) = |.( 0* (j -' i0)).| by PDIFF_7: 3;

      then

       A21: ( sqrt ( Sum ( sqr (y /^ i0)))) = 0 by EUCLID: 7;

      reconsider q2 = (q ^2 ) as Element of REAL by XREAL_0:def 1;

      ( sqr (((y | (i0 -' 1)) ^ <*q*>) ^ (y /^ i0))) = (( sqr ((y | (i0 -' 1)) ^ <*q*>)) ^ ( sqr (y /^ i0))) by RVSUM_1: 144

      .= ((( sqr (y | (i0 -' 1))) ^ ( sqr <*q*>)) ^ ( sqr (y /^ i0))) by RVSUM_1: 144

      .= ((( sqr (y | (i0 -' 1))) ^ <*(q ^2 )*>) ^ ( sqr (y /^ i0))) by RVSUM_1: 55;

      

      then ( Sum ( sqr (((y | (i0 -' 1)) ^ <*q*>) ^ (y /^ i0)))) = (( Sum (( sqr (y | (i0 -' 1))) ^ <*q2*>)) + ( Sum ( sqr (y /^ i0)))) by RVSUM_1: 75

      .= ((( Sum ( sqr (y | (i0 -' 1)))) + (q ^2 )) + ( Sum ( sqr (y /^ i0)))) by RVSUM_1: 74

      .= (q ^2 ) by A20, A21, RVSUM_1: 86, SQUARE_1: 24;

      then ||.(( reproj (i,z)) . xi).|| = |.q.| by A19, A4, COMPLEX1: 72;

      hence thesis by COMPLEX1: 43;

    end;

    theorem :: NDIFF_5:22

    

     Th22: for G be RealNormSpace-Sequence, i be Element of ( dom G), x be Point of ( product G), r be Point of (G . i) holds ((( reproj (i,x)) . r) - x) = (( reproj (i,( 0. ( product G)))) . (r - (( proj i) . x))) & (x - (( reproj (i,x)) . r)) = (( reproj (i,( 0. ( product G)))) . ((( proj i) . x) - r))

    proof

      let G be RealNormSpace-Sequence, i be Element of ( dom G), x be Point of ( product G), r be Point of (G . i);

      set m = ( len G);

      reconsider xf = x as Element of ( product ( carr G)) by Th10;

      

       A1: ( dom ( carr G)) = ( dom G) by Lm1;

      reconsider Zr = ( 0. ( product G)) as Element of ( product ( carr G)) by Th10;

      reconsider ixr = (( reproj (i,x)) . r) as Element of ( product ( carr G)) by Th10;

      reconsider p = ((( reproj (i,x)) . r) - x) as Element of ( product ( carr G)) by Th10;

      reconsider q = (( reproj (i,( 0. ( product G)))) . (r - (( proj i) . x))) as Element of ( product ( carr G)) by Th10;

      

       A3: ( dom q) = ( dom ( carr G)) by CARD_3: 9;

      reconsider s = (x - (( reproj (i,x)) . r)) as Element of ( product ( carr G)) by Th10;

      reconsider t = (( reproj (i,( 0. ( product G)))) . ((( proj i) . x) - r)) as Element of ( product ( carr G)) by Th10;

      

       A5: ( dom t) = ( dom ( carr G)) by CARD_3: 9;

      

       A6: (( reproj (i,x)) . r) = (x +* (i,r)) by Def4;

      reconsider xfi = (xf . i) as Point of (G . i);

      

       A7: (( reproj (i,( 0. ( product G)))) . (r - (( proj i) . x))) = (( 0. ( product G)) +* (i,(r - (( proj i) . x)))) by Def4;

      then

       A7a: q = (Zr +* (i,(r - xfi))) by Def3;

      

       A8: (( reproj (i,( 0. ( product G)))) . ((( proj i) . x) - r)) = (( 0. ( product G)) +* (i,((( proj i) . x) - r))) by Def4;

      then

       A8a: t = (Zr +* (i,(xfi - r))) by Def3;

      set ir = (i .--> r);

      set irx1 = (i .--> (r - xfi));

      set irx2 = (i .--> (xfi - r));

      x in the carrier of ( product G);

      then

       A9: x in ( product ( carr G)) by Th10;

      consider g1 be Function such that

       A10: x = g1 & ( dom g1) = ( dom ( carr G)) & for i be object st i in ( dom ( carr G)) holds (g1 . i) in (( carr G) . i) by A9, CARD_3:def 5;

      for k be object st k in ( dom p) holds (p . k) = (q . k)

      proof

        let k be object;

        assume

         A11: k in ( dom p);

        then

        reconsider k0 = k as Element of ( dom G) by A1, CARD_3: 9;

        consider g be Function such that

         A12: Zr = g & ( dom g) = ( dom ( carr G)) & for i be object st i in ( dom ( carr G)) holds (g . i) in (( carr G) . i) by CARD_3:def 5;

        

         A13: k in ( dom Zr) by A12, A11, CARD_3: 9;

        

         A14: k in ( dom x) by A10, A11, CARD_3: 9;

        per cases ;

          suppose not k in {i};

          then

           A15: k <> i by TARSKI:def 1;

          then

           A16: (q . k0) = (Zr . k0) by A7, FUNCT_7: 32;

          (p . k) = ((ixr . k0) - (xf . k0)) by Th15

          .= ((xf . k0) - (xf . k0)) by A15, A6, FUNCT_7: 32;

          then (p . k) = ( 0. (G . k0)) by RLVECT_1: 15;

          hence (p . k) = (q . k) by A16, Th14;

        end;

          suppose k in {i};

          then

           A17: k = i by TARSKI:def 1;

          then

           A18: (q . k0) = (r - xfi) by A7a, A13, FUNCT_7: 31;

          (p . k) = ((ixr . k0) - (xf . k0)) by Th15;

          hence (p . k) = (q . k) by A18, A6, A17, A14, FUNCT_7: 31;

        end;

      end;

      hence ((( reproj (i,x)) . r) - x) = (( reproj (i,( 0. ( product G)))) . (r - (( proj i) . x))) by A3, FUNCT_1: 2, CARD_3: 9;

      for k be object st k in ( dom s) holds (s . k) = (t . k)

      proof

        let k be object;

        assume

         A19: k in ( dom s);

        then

        reconsider k0 = k as Element of ( dom G) by A1, CARD_3: 9;

        consider g be Function such that

         A20: Zr = g & ( dom g) = ( dom ( carr G)) & for i be object st i in ( dom ( carr G)) holds (g . i) in (( carr G) . i) by CARD_3:def 5;

        

         A21: k in ( dom Zr) by A20, A19, CARD_3: 9;

        

         A22: k in ( dom x) by A10, A19, CARD_3: 9;

        per cases ;

          suppose not k in {i};

          then

           A23: k <> i by TARSKI:def 1;

          then

           A24: (t . k0) = (Zr . k0) by A8, FUNCT_7: 32;

          (s . k) = ((xf . k0) - (ixr . k0)) by Th15

          .= ((xf . k0) - (xf . k0)) by A6, A23, FUNCT_7: 32;

          then (s . k) = ( 0. (G . k0)) by RLVECT_1: 15;

          hence (s . k) = (t . k) by A24, Th14;

        end;

          suppose k in {i};

          then

           A25: k = i by TARSKI:def 1;

          then

           A26: (t . k0) = (xfi - r) by A8a, A21, FUNCT_7: 31;

          (s . k) = ((xf . k0) - (ixr . k0)) by Th15;

          hence (s . k) = (t . k) by A26, A6, A25, A22, FUNCT_7: 31;

        end;

      end;

      hence thesis by A5, FUNCT_1: 2, CARD_3: 9;

    end;

    theorem :: NDIFF_5:23

    

     Th23: for G be RealNormSpace-Sequence, i be Element of ( dom G), x be Point of ( product G), Z be Subset of ( product G) st Z is open & x in Z holds ex N be Neighbourhood of (( proj i) . x) st for z be Point of (G . i) st z in N holds (( reproj (i,x)) . z) in Z

    proof

      let G be RealNormSpace-Sequence, i be Element of ( dom G), x be Point of ( product G), Z be Subset of ( product G);

      assume Z is open & x in Z;

      then

      consider r be Real such that

       A1: 0 < r & { y where y be Point of ( product G) : ||.(y - x).|| < r } c= Z by NDIFF_1: 3;

      set N = { y where y be Point of (G . i) : ||.(y - (( proj i) . x)).|| < r };

      reconsider N as Neighbourhood of (( proj i) . x) by A1, NFCONT_1: 3;

      take N;

      thus for z be Point of (G . i) st z in N holds (( reproj (i,x)) . z) in Z

      proof

        let z be Point of (G . i);

        assume z in N;

        then

         A2: ex y be Point of (G . i) st y = z & ||.(y - (( proj i) . x)).|| < r;

         ||.((( reproj (i,x)) . z) - x).|| = ||.(( reproj (i,( 0. ( product G)))) . (z - (( proj i) . x))).|| by Th22

        .= ||.(z - (( proj i) . x)).|| by Th21;

        then (( reproj (i,x)) . z) in { y where y be Point of ( product G) : ||.(y - x).|| < r } by A2;

        hence thesis by A1;

      end;

    end;

    theorem :: NDIFF_5:24

    

     Th24: for G be RealNormSpace-Sequence, T be RealNormSpace, i be set, f be PartFunc of ( product G), T, Z be Subset of ( product G) st Z is open holds f is_partial_differentiable_on (Z,i) iff Z c= ( dom f) & for x be Point of ( product G) st x in Z holds f is_partial_differentiable_in (x,i)

    proof

      let G be RealNormSpace-Sequence, T be RealNormSpace, i be set, f be PartFunc of ( product G), T, Z be Subset of ( product G);

      assume

       A1: Z is open;

      set i0 = ( In (i,( dom G)));

      set S = (G . i0);

      set RNS = ( R_NormSpace_of_BoundedLinearOperators (S,T));

      thus f is_partial_differentiable_on (Z,i) implies Z c= ( dom f) & for x be Point of ( product G) st x in Z holds f is_partial_differentiable_in (x,i)

      proof

        assume

         A2: f is_partial_differentiable_on (Z,i);

        hence Z c= ( dom f);

        let nx0 be Point of ( product G);

        reconsider x0 = (( proj i0) . nx0) as Point of S;

        assume

         A4: nx0 in Z;

        then (f | Z) is_partial_differentiable_in (nx0,i) by A2;

        then

        consider N0 be Neighbourhood of x0 such that

         A5: N0 c= ( dom ((f | Z) * ( reproj (i0,nx0)))) and

         A6: ex L be Point of RNS, R be RestFunc of S, T st for x be Point of S st x in N0 holds ((((f | Z) * ( reproj (i0,nx0))) /. x) - (((f | Z) * ( reproj (i0,nx0))) /. x0)) = ((L . (x - x0)) + (R /. (x - x0))) by NDIFF_1:def 6;

        consider L be Point of RNS, R be RestFunc of S, T such that

         A7: for x be Point of S st x in N0 holds ((((f | Z) * ( reproj (i0,nx0))) /. x) - (((f | Z) * ( reproj (i0,nx0))) /. x0)) = ((L . (x - x0)) + (R /. (x - x0))) by A6;

        consider N1 be Neighbourhood of x0 such that

         A8: for x be Point of S st x in N1 holds (( reproj (i0,nx0)) . x) in Z by A1, A4, Th23;

         A9:

        now

          let x be Point of S;

          assume x in N1;

          then (( reproj (i0,nx0)) . x) in Z by A8;

          then (( reproj (i0,nx0)) . x) in (( dom f) /\ Z) by A2, XBOOLE_0:def 4;

          hence (( reproj (i0,nx0)) . x) in ( dom (f | Z)) by RELAT_1: 61;

        end;

        reconsider N = (N0 /\ N1) as Neighbourhood of x0 by Th8;

        ((f | Z) * ( reproj (i0,nx0))) c= (f * ( reproj (i0,nx0))) by RELAT_1: 29, RELAT_1: 59;

        then

         A10: ( dom ((f | Z) * ( reproj (i0,nx0)))) c= ( dom (f * ( reproj (i0,nx0)))) by RELAT_1: 11;

        N c= N0 by XBOOLE_1: 17;

        then

         A11: N c= ( dom (f * ( reproj (i0,nx0)))) by A5, A10;

        now

          let x be Point of S;

          assume

           A12: x in N;

          then

           A13: x in N0 by XBOOLE_0:def 4;

          

           A14: ( dom ( reproj (i0,nx0))) = the carrier of (G . i0) by FUNCT_2:def 1;

          x in N1 by A12, XBOOLE_0:def 4;

          then

           A15: (( reproj (i0,nx0)) . x) in ( dom (f | Z)) by A9;

          then

           A16: (( reproj (i0,nx0)) . x) in ( dom f) & (( reproj (i0,nx0)) . x) in Z by RELAT_1: 57;

          

           A17: (( reproj (i0,nx0)) . x0) in ( dom (f | Z)) by A9, NFCONT_1: 4;

          then

           A18: (( reproj (i0,nx0)) . x0) in ( dom f) & (( reproj (i0,nx0)) . x0) in Z by RELAT_1: 57;

          

           A19: (((f | Z) * ( reproj (i0,nx0))) /. x) = ((f | Z) /. (( reproj (i0,nx0)) /. x)) by A15, A14, PARTFUN2: 4

          .= (f /. (( reproj (i0,nx0)) /. x)) by A16, PARTFUN2: 17

          .= ((f * ( reproj (i0,nx0))) /. x) by A14, A16, PARTFUN2: 4;

          (((f | Z) * ( reproj (i0,nx0))) /. x0) = ((f | Z) /. (( reproj (i0,nx0)) /. x0)) by A14, A17, PARTFUN2: 4

          .= (f /. (( reproj (i0,nx0)) /. x0)) by A18, PARTFUN2: 17

          .= ((f * ( reproj (i0,nx0))) /. x0) by A14, A18, PARTFUN2: 4;

          hence (((f * ( reproj (i0,nx0))) /. x) - ((f * ( reproj (i0,nx0))) /. x0)) = ((L . (x - x0)) + (R /. (x - x0))) by A7, A13, A19;

        end;

        hence f is_partial_differentiable_in (nx0,i) by A11, NDIFF_1:def 6;

      end;

      assume that

       A20: Z c= ( dom f) and

       A21: for nx be Point of ( product G) st nx in Z holds f is_partial_differentiable_in (nx,i);

      now

        let nx0 be Point of ( product G);

        assume

         A22: nx0 in Z;

        then

         A23: f is_partial_differentiable_in (nx0,i) by A21;

        reconsider x0 = (( proj i0) . nx0) as Point of S;

        consider N0 be Neighbourhood of x0 such that N0 c= ( dom (f * ( reproj (i0,nx0)))) and

         A24: ex L be Point of RNS, R be RestFunc of S, T st for x be Point of S st x in N0 holds (((f * ( reproj (i0,nx0))) /. x) - ((f * ( reproj (i0,nx0))) /. x0)) = ((L . (x - x0)) + (R /. (x - x0))) by A23, NDIFF_1:def 6;

        consider N1 be Neighbourhood of x0 such that

         A25: for x be Point of S st x in N1 holds (( reproj (i0,nx0)) . x) in Z by A1, A22, Th23;

         A26:

        now

          let x be Point of S;

          assume x in N1;

          then (( reproj (i0,nx0)) . x) in Z by A25;

          then (( reproj (i0,nx0)) . x) in (( dom f) /\ Z) by A20, XBOOLE_0:def 4;

          hence (( reproj (i0,nx0)) . x) in ( dom (f | Z)) by RELAT_1: 61;

        end;

        

         A27: N1 c= ( dom ((f | Z) * ( reproj (i0,nx0))))

        proof

          let z be object;

          assume

           A28: z in N1;

          then

           A29: z in the carrier of S;

          reconsider x = z as Point of S by A28;

          

           A30: (( reproj (i0,nx0)) . x) in ( dom (f | Z)) by A28, A26;

          z in ( dom ( reproj (i0,nx0))) by A29, FUNCT_2:def 1;

          hence z in ( dom ((f | Z) * ( reproj (i0,nx0)))) by A30, FUNCT_1: 11;

        end;

        reconsider N = (N0 /\ N1) as Neighbourhood of x0 by Th8;

        N c= N1 by XBOOLE_1: 17;

        then

         A31: N c= ( dom ((f | Z) * ( reproj (i0,nx0)))) by A27;

        consider L be Point of RNS, R be RestFunc of S, T such that

         A32: for x be Point of S st x in N0 holds (((f * ( reproj (i0,nx0))) /. x) - ((f * ( reproj (i0,nx0))) /. x0)) = ((L . (x - x0)) + (R /. (x - x0))) by A24;

        now

          let x be Point of S;

          assume

           A33: x in N;

          then

           A34: x in N0 by XBOOLE_0:def 4;

          

           A35: ( dom ( reproj (i0,nx0))) = the carrier of (G . i0) by FUNCT_2:def 1;

          x in N1 by A33, XBOOLE_0:def 4;

          then

           A36: (( reproj (i0,nx0)) . x) in ( dom (f | Z)) by A26;

          then

           A37: (( reproj (i0,nx0)) . x) in (( dom f) /\ Z) by RELAT_1: 61;

          then

           A38: (( reproj (i0,nx0)) . x) in ( dom f) by XBOOLE_0:def 4;

          

           A39: (( reproj (i0,nx0)) . x0) in ( dom (f | Z)) by A26, NFCONT_1: 4;

          then

           A40: (( reproj (i0,nx0)) . x0) in (( dom f) /\ Z) by RELAT_1: 61;

          then

           A41: (( reproj (i0,nx0)) . x0) in ( dom f) by XBOOLE_0:def 4;

          

           A42: (((f | Z) * ( reproj (i0,nx0))) /. x) = ((f | Z) /. (( reproj (i0,nx0)) /. x)) by A36, A35, PARTFUN2: 4

          .= (f /. (( reproj (i0,nx0)) /. x)) by A37, PARTFUN2: 16

          .= ((f * ( reproj (i0,nx0))) /. x) by A35, A38, PARTFUN2: 4;

          (((f | Z) * ( reproj (i0,nx0))) /. x0) = ((f | Z) /. (( reproj (i0,nx0)) /. x0)) by A35, A39, PARTFUN2: 4

          .= (f /. (( reproj (i0,nx0)) /. x0)) by A40, PARTFUN2: 16

          .= ((f * ( reproj (i0,nx0))) /. x0) by A35, A41, PARTFUN2: 4;

          hence ((((f | Z) * ( reproj (i0,nx0))) /. x) - (((f | Z) * ( reproj (i0,nx0))) /. x0)) = ((L . (x - x0)) + (R /. (x - x0))) by A42, A34, A32;

        end;

        hence (f | Z) is_partial_differentiable_in (nx0,i) by A31, NDIFF_1:def 6;

      end;

      hence thesis by A20;

    end;

    theorem :: NDIFF_5:25

    for i be set st i in ( dom G) & f is_partial_differentiable_on (X,i) holds X is Subset of ( product G) by XBOOLE_1: 1;

    definition

      let G be RealNormSpace-Sequence;

      let S be RealNormSpace;

      let i be set;

      let f be PartFunc of ( product G), S;

      let X be set;

      assume

       A2: f is_partial_differentiable_on (X,i);

      :: NDIFF_5:def9

      func f `partial| (X,i) -> PartFunc of ( product G), ( R_NormSpace_of_BoundedLinearOperators ((G . ( In (i,( dom G)))),S)) means

      : Def9: ( dom it ) = X & for x be Point of ( product G) st x in X holds (it /. x) = ( partdiff (f,x,i));

      existence

      proof

        deffunc F( Element of ( product G)) = ( partdiff (f,$1,i));

        defpred P[ Element of ( product G)] means $1 in X;

        consider F be PartFunc of ( product G), ( R_NormSpace_of_BoundedLinearOperators ((G . ( In (i,( dom G)))),S)) such that

         A3: (for x be Point of ( product G) holds x in ( dom F) iff P[x]) & for x be Point of ( product G) st x in ( dom F) holds (F . x) = F(x) from SEQ_1:sch 3;

        take F;

        now

          

           A4: X is Subset of ( product G) by A2, XBOOLE_1: 1;

          let y be object;

          assume y in X;

          hence y in ( dom F) by A3, A4;

        end;

        then

         A5: X c= ( dom F);

        ( dom F) c= X by A3;

        hence ( dom F) = X by A5, XBOOLE_0:def 10;

        hereby

          let x be Point of ( product G);

          assume

           A6: x in X;

          then (F . x) = ( partdiff (f,x,i)) by A3;

          hence (F /. x) = ( partdiff (f,x,i)) by A3, A6, PARTFUN1:def 6;

        end;

      end;

      uniqueness

      proof

        let F,H be PartFunc of ( product G), ( R_NormSpace_of_BoundedLinearOperators ((G . ( In (i,( dom G)))),S));

        assume that

         A7: ( dom F) = X and

         A8: for x be Point of ( product G) st x in X holds (F /. x) = ( partdiff (f,x,i)) and

         A9: ( dom H) = X and

         A10: for x be Point of ( product G) st x in X holds (H /. x) = ( partdiff (f,x,i));

        now

          let x be Point of ( product G);

          assume

           A11: x in ( dom F);

          then (F /. x) = ( partdiff (f,x,i)) by A7, A8;

          hence (F /. x) = (H /. x) by A7, A10, A11;

        end;

        hence thesis by A7, A9, PARTFUN2: 1;

      end;

    end

    theorem :: NDIFF_5:26

    

     Th26: for i be set st i in ( dom G) holds ((f1 + f2) * ( reproj (( In (i,( dom G))),x))) = ((f1 * ( reproj (( In (i,( dom G))),x))) + (f2 * ( reproj (( In (i,( dom G))),x)))) & ((f1 - f2) * ( reproj (( In (i,( dom G))),x))) = ((f1 * ( reproj (( In (i,( dom G))),x))) - (f2 * ( reproj (( In (i,( dom G))),x))))

    proof

      let i0 be set;

      assume i0 in ( dom G);

      set i = ( In (i0,( dom G)));

      

       A1: ( dom ( reproj (i,x))) = the carrier of (G . i) by FUNCT_2:def 1;

      

       A2: ( dom (f1 + f2)) = (( dom f1) /\ ( dom f2)) by VFUNCT_1:def 1;

      

       A3b: for s be Element of (G . i) holds s in ( dom ((f1 + f2) * ( reproj (i,x)))) iff s in ( dom ((f1 * ( reproj (i,x))) + (f2 * ( reproj (i,x)))))

      proof

        let s be Element of (G . i);

        s in ( dom ((f1 + f2) * ( reproj (i,x)))) iff (( reproj (i,x)) . s) in (( dom f1) /\ ( dom f2)) by A2, A1, FUNCT_1: 11;

        then s in ( dom ((f1 + f2) * ( reproj (i,x)))) iff (( reproj (i,x)) . s) in ( dom f1) & (( reproj (i,x)) . s) in ( dom f2) by XBOOLE_0:def 4;

        then s in ( dom ((f1 + f2) * ( reproj (i,x)))) iff s in ( dom (f1 * ( reproj (i,x)))) & s in ( dom (f2 * ( reproj (i,x)))) by A1, FUNCT_1: 11;

        then s in ( dom ((f1 + f2) * ( reproj (i,x)))) iff s in (( dom (f1 * ( reproj (i,x)))) /\ ( dom (f2 * ( reproj (i,x))))) by XBOOLE_0:def 4;

        hence thesis by VFUNCT_1:def 1;

      end;

      then

       A3: for s be object holds s in ( dom ((f1 + f2) * ( reproj (i,x)))) iff s in ( dom ((f1 * ( reproj (i,x))) + (f2 * ( reproj (i,x)))));

      then

       A3a: ( dom ((f1 + f2) * ( reproj (i,x)))) = ( dom ((f1 * ( reproj (i,x))) + (f2 * ( reproj (i,x))))) by TARSKI: 2;

      

       A4: for z be Element of (G . i) st z in ( dom ((f1 + f2) * ( reproj (i,x)))) holds (((f1 + f2) * ( reproj (i,x))) . z) = (((f1 * ( reproj (i,x))) + (f2 * ( reproj (i,x)))) . z)

      proof

        let z be Element of (G . i);

        assume

         A5: z in ( dom ((f1 + f2) * ( reproj (i,x))));

        then

         A6: (( reproj (i,x)) . z) in ( dom (f1 + f2)) by FUNCT_1: 11;

        z in (( dom (f1 * ( reproj (i,x)))) /\ ( dom (f2 * ( reproj (i,x))))) by A3a, A5, VFUNCT_1:def 1;

        then

         A7: z in ( dom (f1 * ( reproj (i,x)))) & z in ( dom (f2 * ( reproj (i,x)))) by XBOOLE_0:def 4;

        

         A8: (( reproj (i,x)) . z) in (( dom f1) /\ ( dom f2)) by A2, A5, FUNCT_1: 11;

        then (( reproj (i,x)) . z) in ( dom f1) by XBOOLE_0:def 4;

        

        then

         A9: (f1 /. (( reproj (i,x)) . z)) = (f1 . (( reproj (i,x)) . z)) by PARTFUN1:def 6

        .= ((f1 * ( reproj (i,x))) . z) by A7, FUNCT_1: 12

        .= ((f1 * ( reproj (i,x))) /. z) by A7, PARTFUN1:def 6;

        (( reproj (i,x)) . z) in ( dom f2) by A8, XBOOLE_0:def 4;

        

        then

         A10: (f2 /. (( reproj (i,x)) . z)) = (f2 . (( reproj (i,x)) . z)) by PARTFUN1:def 6

        .= ((f2 * ( reproj (i,x))) . z) by A7, FUNCT_1: 12

        .= ((f2 * ( reproj (i,x))) /. z) by A7, PARTFUN1:def 6;

        (((f1 + f2) * ( reproj (i,x))) . z) = ((f1 + f2) . (( reproj (i,x)) . z)) by A5, FUNCT_1: 12

        .= ((f1 + f2) /. (( reproj (i,x)) . z)) by A6, PARTFUN1:def 6

        .= ((f1 /. (( reproj (i,x)) . z)) + (f2 /. (( reproj (i,x)) . z))) by A6, VFUNCT_1:def 1

        .= (((f1 * ( reproj (i,x))) + (f2 * ( reproj (i,x)))) /. z) by A3b, A5, A9, A10, VFUNCT_1:def 1;

        hence thesis by A3b, A5, PARTFUN1:def 6;

      end;

      

       A11: ( dom (f1 - f2)) = (( dom f1) /\ ( dom f2)) by VFUNCT_1:def 2;

      

       A12b: for s be Element of (G . i) holds s in ( dom ((f1 - f2) * ( reproj (i,x)))) iff s in ( dom ((f1 * ( reproj (i,x))) - (f2 * ( reproj (i,x)))))

      proof

        let s be Element of (G . i);

        s in ( dom ((f1 - f2) * ( reproj (i,x)))) iff (( reproj (i,x)) . s) in (( dom f1) /\ ( dom f2)) by A11, A1, FUNCT_1: 11;

        then s in ( dom ((f1 - f2) * ( reproj (i,x)))) iff (( reproj (i,x)) . s) in ( dom f1) & (( reproj (i,x)) . s) in ( dom f2) by XBOOLE_0:def 4;

        then s in ( dom ((f1 - f2) * ( reproj (i,x)))) iff s in ( dom (f1 * ( reproj (i,x)))) & s in ( dom (f2 * ( reproj (i,x)))) by A1, FUNCT_1: 11;

        then s in ( dom ((f1 - f2) * ( reproj (i,x)))) iff s in (( dom (f1 * ( reproj (i,x)))) /\ ( dom (f2 * ( reproj (i,x))))) by XBOOLE_0:def 4;

        hence thesis by VFUNCT_1:def 2;

      end;

      then

       A12: for s be object holds s in ( dom ((f1 - f2) * ( reproj (i,x)))) iff s in ( dom ((f1 * ( reproj (i,x))) - (f2 * ( reproj (i,x)))));

      then

       A12a: ( dom ((f1 - f2) * ( reproj (i,x)))) = ( dom ((f1 * ( reproj (i,x))) - (f2 * ( reproj (i,x))))) by TARSKI: 2;

      for z be Element of (G . i) st z in ( dom ((f1 - f2) * ( reproj (i,x)))) holds (((f1 - f2) * ( reproj (i,x))) . z) = (((f1 * ( reproj (i,x))) - (f2 * ( reproj (i,x)))) . z)

      proof

        let z be Element of (G . i);

        assume

         A13: z in ( dom ((f1 - f2) * ( reproj (i,x))));

        then

         A14: (( reproj (i,x)) . z) in ( dom (f1 - f2)) by FUNCT_1: 11;

        z in (( dom (f1 * ( reproj (i,x)))) /\ ( dom (f2 * ( reproj (i,x))))) by A12a, A13, VFUNCT_1:def 2;

        then

         A15: z in ( dom (f1 * ( reproj (i,x)))) & z in ( dom (f2 * ( reproj (i,x)))) by XBOOLE_0:def 4;

        

         A16: (( reproj (i,x)) . z) in (( dom f1) /\ ( dom f2)) by A11, A13, FUNCT_1: 11;

        then (( reproj (i,x)) . z) in ( dom f1) by XBOOLE_0:def 4;

        

        then

         A17: (f1 /. (( reproj (i,x)) . z)) = (f1 . (( reproj (i,x)) . z)) by PARTFUN1:def 6

        .= ((f1 * ( reproj (i,x))) . z) by A15, FUNCT_1: 12

        .= ((f1 * ( reproj (i,x))) /. z) by A15, PARTFUN1:def 6;

        (( reproj (i,x)) . z) in ( dom f2) by A16, XBOOLE_0:def 4;

        

        then

         A18: (f2 /. (( reproj (i,x)) . z)) = (f2 . (( reproj (i,x)) . z)) by PARTFUN1:def 6

        .= ((f2 * ( reproj (i,x))) . z) by A15, FUNCT_1: 12

        .= ((f2 * ( reproj (i,x))) /. z) by A15, PARTFUN1:def 6;

        

        thus (((f1 - f2) * ( reproj (i,x))) . z) = ((f1 - f2) . (( reproj (i,x)) . z)) by A13, FUNCT_1: 12

        .= ((f1 - f2) /. (( reproj (i,x)) . z)) by A14, PARTFUN1:def 6

        .= ((f1 /. (( reproj (i,x)) . z)) - (f2 /. (( reproj (i,x)) . z))) by A14, VFUNCT_1:def 2

        .= (((f1 * ( reproj (i,x))) - (f2 * ( reproj (i,x)))) /. z) by A12b, A13, A17, A18, VFUNCT_1:def 2

        .= (((f1 * ( reproj (i,x))) - (f2 * ( reproj (i,x)))) . z) by A12b, A13, PARTFUN1:def 6;

      end;

      hence thesis by A3, A12, A4, TARSKI: 2, PARTFUN1: 5;

    end;

    theorem :: NDIFF_5:27

    

     Th27: for i be set st i in ( dom G) holds (r (#) (f * ( reproj (( In (i,( dom G))),x)))) = ((r (#) f) * ( reproj (( In (i,( dom G))),x)))

    proof

      let i0 be set;

      assume i0 in ( dom G);

      set i = ( In (i0,( dom G)));

      

       A1: ( dom (r (#) f)) = ( dom f) by VFUNCT_1:def 4;

      

       A2: ( dom (r (#) (f * ( reproj (i,x))))) = ( dom (f * ( reproj (i,x)))) by VFUNCT_1:def 4;

      

       A3: ( dom ( reproj (i,x))) = the carrier of (G . i) by FUNCT_2:def 1;

      

       A4b: for s be Element of (G . i) holds s in ( dom ((r (#) f) * ( reproj (i,x)))) iff s in ( dom (f * ( reproj (i,x))))

      proof

        let s be Element of (G . i);

        s in ( dom ((r (#) f) * ( reproj (i,x)))) iff (( reproj (i,x)) . s) in ( dom (r (#) f)) by A3, FUNCT_1: 11;

        hence thesis by A1, A3, FUNCT_1: 11;

      end;

      then

       A4: for s be object holds s in ( dom (r (#) (f * ( reproj (i,x))))) iff s in ( dom ((r (#) f) * ( reproj (i,x)))) by A2;

      then

       A4a: ( dom (r (#) (f * ( reproj (i,x))))) = ( dom ((r (#) f) * ( reproj (i,x)))) by TARSKI: 2;

      

       A5: for s be Element of (G . i) holds s in ( dom ((r (#) f) * ( reproj (i,x)))) iff (( reproj (i,x)) . s) in ( dom (r (#) f))

      proof

        let s be Element of (G . i);

        ( dom ( reproj (i,x))) = the carrier of (G . i) by FUNCT_2:def 1;

        hence thesis by FUNCT_1: 11;

      end;

      for z be Element of (G . i) st z in ( dom (r (#) (f * ( reproj (i,x))))) holds ((r (#) (f * ( reproj (i,x)))) . z) = (((r (#) f) * ( reproj (i,x))) . z)

      proof

        let z be Element of (G . i);

        assume

         A6: z in ( dom (r (#) (f * ( reproj (i,x)))));

        then

         A7: z in ( dom (f * ( reproj (i,x)))) by VFUNCT_1:def 4;

        

         A9: (f /. (( reproj (i,x)) . z)) = (f . (( reproj (i,x)) . z)) by A1, A5, A4a, A6, PARTFUN1:def 6

        .= ((f * ( reproj (i,x))) . z) by A7, FUNCT_1: 12

        .= ((f * ( reproj (i,x))) /. z) by A7, PARTFUN1:def 6;

        

         A10: ((r (#) (f * ( reproj (i,x)))) . z) = ((r (#) (f * ( reproj (i,x)))) /. z) by A6, PARTFUN1:def 6

        .= (r * (f /. (( reproj (i,x)) . z))) by A6, A9, VFUNCT_1:def 4;

        (((r (#) f) * ( reproj (i,x))) . z) = ((r (#) f) . (( reproj (i,x)) . z)) by A2, A4b, A6, FUNCT_1: 12

        .= ((r (#) f) /. (( reproj (i,x)) . z)) by A5, A4a, A6, PARTFUN1:def 6

        .= (r * (f /. (( reproj (i,x)) . z))) by A5, A4a, A6, VFUNCT_1:def 4;

        hence thesis by A10;

      end;

      hence thesis by A4, TARSKI: 2, PARTFUN1: 5;

    end;

    theorem :: NDIFF_5:28

    for i be set st i in ( dom G) & f1 is_partial_differentiable_in (x,i) & f2 is_partial_differentiable_in (x,i) holds (f1 + f2) is_partial_differentiable_in (x,i) & ( partdiff ((f1 + f2),x,i)) = (( partdiff (f1,x,i)) + ( partdiff (f2,x,i)))

    proof

      let i0 be set;

      set i = ( In (i0,( dom G)));

      assume

       A1: i0 in ( dom G);

      then

       A2: ((f1 + f2) * ( reproj (i,x))) = ((f1 * ( reproj (i,x))) + (f2 * ( reproj (i,x)))) by Th26;

      assume

       A3: f1 is_partial_differentiable_in (x,i0) & f2 is_partial_differentiable_in (x,i0);

      hence (f1 + f2) is_partial_differentiable_in (x,i0) by A2, NDIFF_1: 35;

      

      thus (( partdiff (f1,x,i0)) + ( partdiff (f2,x,i0))) = ( diff (((f1 * ( reproj (i,x))) + (f2 * ( reproj (i,x)))),(( proj i) . x))) by A3, NDIFF_1: 35

      .= ( partdiff ((f1 + f2),x,i0)) by A1, Th26;

    end;

    theorem :: NDIFF_5:29

    for i be set st i in ( dom G) & f1 is_partial_differentiable_in (x,i) & f2 is_partial_differentiable_in (x,i) holds (f1 - f2) is_partial_differentiable_in (x,i) & ( partdiff ((f1 - f2),x,i)) = (( partdiff (f1,x,i)) - ( partdiff (f2,x,i)))

    proof

      let i0 be set;

      assume

       A1: i0 in ( dom G);

      set i = ( In (i0,( dom G)));

      assume

       A2: f1 is_partial_differentiable_in (x,i0) & f2 is_partial_differentiable_in (x,i0);

      ((f1 - f2) * ( reproj (i,x))) = ((f1 * ( reproj (i,x))) - (f2 * ( reproj (i,x)))) by A1, Th26;

      hence (f1 - f2) is_partial_differentiable_in (x,i0) by A2, NDIFF_1: 36;

      

      thus (( partdiff (f1,x,i0)) - ( partdiff (f2,x,i0))) = ( diff (((f1 * ( reproj (i,x))) - (f2 * ( reproj (i,x)))),(( proj i) . x))) by A2, NDIFF_1: 36

      .= ( partdiff ((f1 - f2),x,i0)) by A1, Th26;

    end;

    theorem :: NDIFF_5:30

    for i be set st i in ( dom G) & f is_partial_differentiable_in (x,i) holds (r (#) f) is_partial_differentiable_in (x,i) & ( partdiff ((r (#) f),x,i)) = (r * ( partdiff (f,x,i)))

    proof

      let i0 be set;

      assume

       A1: i0 in ( dom G);

      set i = ( In (i0,( dom G)));

      assume

       A2: f is_partial_differentiable_in (x,i0);

      (r (#) (f * ( reproj (i,x)))) = ((r (#) f) * ( reproj (i,x))) by A1, Th27;

      hence (r (#) f) is_partial_differentiable_in (x,i0) by A2, NDIFF_1: 37;

      

      thus ( partdiff ((r (#) f),x,i0)) = ( diff ((r (#) (f * ( reproj (i,x)))),(( proj i) . x))) by A1, Th27

      .= (r * ( partdiff (f,x,i0))) by A2, NDIFF_1: 37;

    end;

    begin

    theorem :: NDIFF_5:31

    

     Th31: ||.(( proj i) . x).|| <= ||.x.||

    proof

      reconsider y = x as Element of ( product ( carr G)) by Th10;

      (( proj i) . x) = (y . i) by Def3;

      hence thesis by PRVECT_2: 10;

    end;

    registration

      let G be RealNormSpace-Sequence;

      cluster -> ( len G) -element for Point of ( product G);

      coherence

      proof

        let x be Point of ( product G);

        

         A1: the carrier of ( product G) = ( product ( carr G)) by Th10;

        

         A2: ( dom x) = ( dom ( carr G)) & for i be set st i in ( dom ( carr G)) holds (x . i) in (( carr G) . i) by A1, CARD_3: 9;

        ( len ( carr G)) = ( len G) by PRVECT_1:def 11;

        then ( dom x) = ( Seg ( len G)) by A2, FINSEQ_1:def 3;

        then ( len x) = ( len G) by FINSEQ_1:def 3;

        hence thesis by CARD_1:def 7;

      end;

    end

    theorem :: NDIFF_5:32

    

     Th32: for G be RealNormSpace-Sequence, T be RealNormSpace, i be set, Z be Subset of ( product G), f be PartFunc of ( product G), T st Z is open holds f is_partial_differentiable_on (Z,i) iff Z c= ( dom f) & for x be Point of ( product G) st x in Z holds f is_partial_differentiable_in (x,i)

    proof

      let G be RealNormSpace-Sequence, T be RealNormSpace, i0 be set, Z be Subset of ( product G), f be PartFunc of ( product G), T;

      assume

       A1: Z is open;

      set i = ( In (i0,( dom G)));

      set S = (G . i);

      set RNS = ( R_NormSpace_of_BoundedLinearOperators (S,T));

      hereby

        assume

         A2: f is_partial_differentiable_on (Z,i0);

        hence Z c= ( dom f);

        let nx0 be Point of ( product G);

        reconsider x0 = (( proj i) . nx0) as Point of S;

        assume

         A4: nx0 in Z;

        then (f | Z) is_partial_differentiable_in (nx0,i0) by A2;

        then

        consider N0 be Neighbourhood of x0 such that

         A5: N0 c= ( dom ((f | Z) * ( reproj (i,nx0)))) and

         A6: ex L be Point of RNS, R be RestFunc of S, T st for x be Point of S st x in N0 holds ((((f | Z) * ( reproj (i,nx0))) /. x) - (((f | Z) * ( reproj (i,nx0))) /. x0)) = ((L . (x - x0)) + (R /. (x - x0))) by NDIFF_1:def 6;

        consider L be Point of RNS, R be RestFunc of S, T such that

         A7: for x be Point of S st x in N0 holds ((((f | Z) * ( reproj (i,nx0))) /. x) - (((f | Z) * ( reproj (i,nx0))) /. x0)) = ((L . (x - x0)) + (R /. (x - x0))) by A6;

        consider N1 be Neighbourhood of x0 such that

         A8: for x be Point of S st x in N1 holds (( reproj (i,nx0)) . x) in Z by A1, A4, Th23;

         A9:

        now

          let x be Point of S;

          assume x in N1;

          then (( reproj (i,nx0)) . x) in Z by A8;

          then (( reproj (i,nx0)) . x) in (( dom f) /\ Z) by A2, XBOOLE_0:def 4;

          hence (( reproj (i,nx0)) . x) in ( dom (f | Z)) by RELAT_1: 61;

        end;

        reconsider N = (N0 /\ N1) as Neighbourhood of x0 by Th8;

        ((f | Z) * ( reproj (i,nx0))) c= (f * ( reproj (i,nx0))) by RELAT_1: 29, RELAT_1: 59;

        then

         A10: ( dom ((f | Z) * ( reproj (i,nx0)))) c= ( dom (f * ( reproj (i,nx0)))) by RELAT_1: 11;

        N c= N0 by XBOOLE_1: 17;

        then

         A11: N c= ( dom (f * ( reproj (i,nx0)))) by A5, A10;

        

         A12: ( dom ( reproj (i,nx0))) = the carrier of (G . i) by FUNCT_2:def 1;

        now

          let x be Point of S;

          assume x in N;

          then

           A13: x in N0 & x in N1 by XBOOLE_0:def 4;

          then

           A14: (( reproj (i,nx0)) . x) in ( dom (f | Z)) by A9;

          then

           A15: (( reproj (i,nx0)) . x) in ( dom f) & (( reproj (i,nx0)) . x) in Z by RELAT_1: 57;

          

           A16: (( reproj (i,nx0)) . x0) in ( dom (f | Z)) by A9, NFCONT_1: 4;

          then

           A17: (( reproj (i,nx0)) . x0) in ( dom f) & (( reproj (i,nx0)) . x0) in Z by RELAT_1: 57;

          

           A18: (((f | Z) * ( reproj (i,nx0))) /. x) = ((f | Z) /. (( reproj (i,nx0)) /. x)) by A14, A12, PARTFUN2: 4

          .= (f /. (( reproj (i,nx0)) /. x)) by A15, PARTFUN2: 17

          .= ((f * ( reproj (i,nx0))) /. x) by A12, A15, PARTFUN2: 4;

          (((f | Z) * ( reproj (i,nx0))) /. x0) = ((f | Z) /. (( reproj (i,nx0)) /. x0)) by A12, A16, PARTFUN2: 4

          .= (f /. (( reproj (i,nx0)) /. x0)) by A17, PARTFUN2: 17

          .= ((f * ( reproj (i,nx0))) /. x0) by A12, A17, PARTFUN2: 4;

          hence (((f * ( reproj (i,nx0))) /. x) - ((f * ( reproj (i,nx0))) /. x0)) = ((L . (x - x0)) + (R /. (x - x0))) by A7, A13, A18;

        end;

        hence f is_partial_differentiable_in (nx0,i0) by A11, NDIFF_1:def 6;

      end;

      assume that

       A19: Z c= ( dom f) and

       A20: for nx be Point of ( product G) st nx in Z holds f is_partial_differentiable_in (nx,i0);

      now

        let nx0 be Point of ( product G);

        assume

         A21: nx0 in Z;

        then

         A22: f is_partial_differentiable_in (nx0,i0) by A20;

        reconsider x0 = (( proj i) . nx0) as Point of S;

        consider N0 be Neighbourhood of x0 such that N0 c= ( dom (f * ( reproj (i,nx0)))) and

         A23: ex L be Point of RNS, R be RestFunc of S, T st for x be Point of S st x in N0 holds (((f * ( reproj (i,nx0))) /. x) - ((f * ( reproj (i,nx0))) /. x0)) = ((L . (x - x0)) + (R /. (x - x0))) by A22, NDIFF_1:def 6;

        consider N1 be Neighbourhood of x0 such that

         A24: for x be Point of S st x in N1 holds (( reproj (i,nx0)) . x) in Z by A1, A21, Th23;

         A25:

        now

          let x be Point of S;

          assume x in N1;

          then (( reproj (i,nx0)) . x) in Z by A24;

          then (( reproj (i,nx0)) . x) in (( dom f) /\ Z) by A19, XBOOLE_0:def 4;

          hence (( reproj (i,nx0)) . x) in ( dom (f | Z)) by RELAT_1: 61;

        end;

        

         A26: N1 c= ( dom ((f | Z) * ( reproj (i,nx0))))

        proof

          let z be object;

          assume

           A27: z in N1;

          then z in the carrier of S;

          then

           A28: z in ( dom ( reproj (i,nx0))) by FUNCT_2:def 1;

          reconsider x = z as Point of S by A27;

          (( reproj (i,nx0)) . x) in ( dom (f | Z)) by A27, A25;

          hence z in ( dom ((f | Z) * ( reproj (i,nx0)))) by A28, FUNCT_1: 11;

        end;

        reconsider N = (N0 /\ N1) as Neighbourhood of x0 by Th8;

        N c= N1 by XBOOLE_1: 17;

        then

         A29: N c= ( dom ((f | Z) * ( reproj (i,nx0)))) by A26;

        consider L be Point of RNS, R be RestFunc of S, T such that

         A30: for x be Point of S st x in N0 holds (((f * ( reproj (i,nx0))) /. x) - ((f * ( reproj (i,nx0))) /. x0)) = ((L . (x - x0)) + (R /. (x - x0))) by A23;

        now

          let x be Point of S;

          assume

           A31: x in N;

          then

           A32: x in N0 by XBOOLE_0:def 4;

          

           A33: ( dom ( reproj (i,nx0))) = the carrier of (G . i) by FUNCT_2:def 1;

          x in N1 by A31, XBOOLE_0:def 4;

          then

           A34: (( reproj (i,nx0)) . x) in ( dom (f | Z)) by A25;

          then

           A35: (( reproj (i,nx0)) . x) in (( dom f) /\ Z) by RELAT_1: 61;

          then

           A36: (( reproj (i,nx0)) . x) in ( dom f) by XBOOLE_0:def 4;

          

           A37: (( reproj (i,nx0)) . x0) in ( dom (f | Z)) by A25, NFCONT_1: 4;

          then

           A38: (( reproj (i,nx0)) . x0) in (( dom f) /\ Z) by RELAT_1: 61;

          then

           A39: (( reproj (i,nx0)) . x0) in ( dom f) by XBOOLE_0:def 4;

          

           A40: (((f | Z) * ( reproj (i,nx0))) /. x) = ((f | Z) /. (( reproj (i,nx0)) /. x)) by A34, A33, PARTFUN2: 4

          .= (f /. (( reproj (i,nx0)) /. x)) by A35, PARTFUN2: 16

          .= ((f * ( reproj (i,nx0))) /. x) by A33, A36, PARTFUN2: 4;

          (((f | Z) * ( reproj (i,nx0))) /. x0) = ((f | Z) /. (( reproj (i,nx0)) /. x0)) by A33, A37, PARTFUN2: 4

          .= (f /. (( reproj (i,nx0)) /. x0)) by A38, PARTFUN2: 16

          .= ((f * ( reproj (i,nx0))) /. x0) by A33, A39, PARTFUN2: 4;

          hence ((((f | Z) * ( reproj (i,nx0))) /. x) - (((f | Z) * ( reproj (i,nx0))) /. x0)) = ((L . (x - x0)) + (R /. (x - x0))) by A40, A32, A30;

        end;

        hence (f | Z) is_partial_differentiable_in (nx0,i0) by A29, NDIFF_1:def 6;

      end;

      hence thesis by A19;

    end;

    theorem :: NDIFF_5:33

    

     Th33: for i,j be Element of ( dom G), x be Point of (G . i), z be Element of ( product ( carr G)) st z = (( reproj (i,( 0. ( product G)))) . x) holds (i = j implies (z . j) = x) & (i <> j implies (z . j) = ( 0. (G . j)))

    proof

      let i,j be Element of ( dom G), x be Point of (G . i), z be Element of ( product ( carr G));

      assume

       A1: z = (( reproj (i,( 0. ( product G)))) . x);

      reconsider Zr = ( 0. ( product G)) as Element of ( product ( carr G)) by Th10;

      reconsider ixr = (( reproj (i,( 0. ( product G)))) . x) as Element of ( product ( carr G)) by Th10;

      

       A2: (( reproj (i,( 0. ( product G)))) . x) = (( 0. ( product G)) +* (i,x)) by Def4;

      set ix = (i .--> x);

      consider g be Function such that

       A3: Zr = g & ( dom g) = ( dom ( carr G)) & for i be object st i in ( dom ( carr G)) holds (g . i) in (( carr G) . i) by CARD_3:def 5;

      

       A4: ( dom Zr) = ( dom G) by A3, Lm1;

      now

        assume i <> j;

        then (z . j) = (Zr . j) by A1, A2, FUNCT_7: 32;

        hence (z . j) = ( 0. (G . j)) by Th14;

      end;

      hence thesis by A1, A2, A4, FUNCT_7: 31;

    end;

    theorem :: NDIFF_5:34

    

     Th34: for x,y be Point of (G . i) holds (( reproj (i,( 0. ( product G)))) . (x + y)) = ((( reproj (i,( 0. ( product G)))) . x) + (( reproj (i,( 0. ( product G)))) . y))

    proof

      let x,y be Point of (G . i);

      reconsider v = (( reproj (i,( 0. ( product G)))) . (x + y)) as Element of ( product ( carr G)) by Th10;

      reconsider s = (( reproj (i,( 0. ( product G)))) . x) as Element of ( product ( carr G)) by Th10;

      reconsider t = (( reproj (i,( 0. ( product G)))) . y) as Element of ( product ( carr G)) by Th10;

      for j be Element of ( dom G) holds (v . j) = ((s . j) + (t . j))

      proof

        let j be Element of ( dom G);

        per cases ;

          suppose

           A1: i = j;

          then

          reconsider yy = y as Point of (G . j);

          (v . j) = (x + y) by Th33, A1;

          then (v . j) = ((s . j) + yy) by Th33, A1;

          hence (v . j) = ((s . j) + (t . j)) by Th33, A1;

        end;

          suppose

           A2: i <> j;

          then (v . j) = ( 0. (G . j)) by Th33;

          then (v . j) = (( 0. (G . j)) + ( 0. (G . j))) by RLVECT_1:def 4;

          then (v . j) = ((s . j) + ( 0. (G . j))) by Th33, A2;

          hence (v . j) = ((s . j) + (t . j)) by Th33, A2;

        end;

      end;

      hence thesis by Th12;

    end;

    theorem :: NDIFF_5:35

    

     Th35: for x,y be Point of ( product G) holds (( proj i) . (x + y)) = ((( proj i) . x) + (( proj i) . y))

    proof

      let x,y be Point of ( product G);

      reconsider v = (x + y) as Element of ( product ( carr G)) by Th10;

      reconsider s = x as Element of ( product ( carr G)) by Th10;

      reconsider t = y as Element of ( product ( carr G)) by Th10;

      (( proj i) . (x + y)) = (v . i) & (( proj i) . x) = (s . i) & (( proj i) . y) = (t . i) by Def3;

      hence thesis by Th12;

    end;

    theorem :: NDIFF_5:36

    for x,y be Point of (G . i) holds (( reproj (i,( 0. ( product G)))) . (x - y)) = ((( reproj (i,( 0. ( product G)))) . x) - (( reproj (i,( 0. ( product G)))) . y))

    proof

      let x,y be Point of (G . i);

      reconsider v = (( reproj (i,( 0. ( product G)))) . (x - y)) as Element of ( product ( carr G)) by Th10;

      reconsider s = (( reproj (i,( 0. ( product G)))) . x) as Element of ( product ( carr G)) by Th10;

      reconsider t = (( reproj (i,( 0. ( product G)))) . y) as Element of ( product ( carr G)) by Th10;

      for j be Element of ( dom G) holds (v . j) = ((s . j) - (t . j))

      proof

        let j be Element of ( dom G);

        per cases ;

          suppose

           A1: i = j;

          then

          reconsider yy = y as Point of (G . j);

          (v . j) = (x - y) by Th33, A1;

          then (v . j) = ((s . j) - yy) by Th33, A1;

          hence (v . j) = ((s . j) - (t . j)) by Th33, A1;

        end;

          suppose

           A2: i <> j;

          then (v . j) = ( 0. (G . j)) by Th33;

          then (v . j) = (( 0. (G . j)) - ( 0. (G . j))) by RLVECT_1: 13;

          then (v . j) = ((s . j) - ( 0. (G . j))) by Th33, A2;

          hence (v . j) = ((s . j) - (t . j)) by Th33, A2;

        end;

      end;

      hence thesis by Th15;

    end;

    theorem :: NDIFF_5:37

    

     Th37: for x,y be Point of ( product G) holds (( proj i) . (x - y)) = ((( proj i) . x) - (( proj i) . y))

    proof

      let x,y be Point of ( product G);

      reconsider v = (x - y) as Element of ( product ( carr G)) by Th10;

      reconsider s = x as Element of ( product ( carr G)) by Th10;

      reconsider t = y as Element of ( product ( carr G)) by Th10;

      (( proj i) . (x - y)) = (v . i) & (( proj i) . x) = (s . i) & (( proj i) . y) = (t . i) by Def3;

      hence thesis by Th15;

    end;

    theorem :: NDIFF_5:38

    

     Th38: for x be Point of (G . i) st x <> ( 0. (G . i)) holds (( reproj (i,( 0. ( product G)))) . x) <> ( 0. ( product G))

    proof

      let x be Point of (G . i);

      assume

       A1: x <> ( 0. (G . i));

      assume

       A2: (( reproj (i,( 0. ( product G)))) . x) = ( 0. ( product G));

      reconsider v = (( reproj (i,( 0. ( product G)))) . x) as Element of ( product ( carr G)) by Th10;

      x = (v . i) by Th33;

      hence contradiction by A1, Th14, A2;

    end;

    theorem :: NDIFF_5:39

    

     Th39: for x be Point of (G . i), a be Real holds (( reproj (i,( 0. ( product G)))) . (a * x)) = (a * (( reproj (i,( 0. ( product G)))) . x))

    proof

      let x be Point of (G . i), a be Real;

      reconsider a as Real;

      reconsider v = (( reproj (i,( 0. ( product G)))) . (a * x)) as Element of ( product ( carr G)) by Th10;

      reconsider s = (( reproj (i,( 0. ( product G)))) . x) as Element of ( product ( carr G)) by Th10;

      for j be Element of ( dom G) holds (v . j) = (a * (s . j))

      proof

        let j be Element of ( dom G);

        per cases ;

          suppose

           A1: i = j;

          then (v . j) = (a * x) by Th33;

          hence (v . j) = (a * (s . j)) by Th33, A1;

        end;

          suppose

           A2: i <> j;

          then (v . j) = ( 0. (G . j)) by Th33;

          then (v . j) = (a * ( 0. (G . j))) by RLVECT_1: 10;

          hence (v . j) = (a * (s . j)) by Th33, A2;

        end;

      end;

      hence thesis by Th13;

    end;

    theorem :: NDIFF_5:40

    

     Th40: for x be Point of ( product G), a be Real holds (( proj i) . (a * x)) = (a * (( proj i) . x))

    proof

      let x be Point of ( product G), a be Real;

      reconsider a as Real;

      reconsider v = (a * x) as Element of ( product ( carr G)) by Th10;

      reconsider s = x as Element of ( product ( carr G)) by Th10;

      (( proj i) . (a * x)) = (v . i) & (( proj i) . x) = (s . i) by Def3;

      hence thesis by Th13;

    end;

    theorem :: NDIFF_5:41

    

     Th41: for G be RealNormSpace-Sequence, S be RealNormSpace, f be PartFunc of ( product G), S, x be Point of ( product G), i be set st f is_differentiable_in x holds f is_partial_differentiable_in (x,i) & ( partdiff (f,x,i)) = (( diff (f,x)) * ( reproj (( In (i,( dom G))),( 0. ( product G)))))

    proof

      let G be RealNormSpace-Sequence, S be RealNormSpace, f be PartFunc of ( product G), S, x be Point of ( product G), i0 be set;

      assume

       A1: f is_differentiable_in x;

      set i = ( In (i0,( dom G)));

      consider N be Neighbourhood of x such that

       A2: N c= ( dom f) & ex R be RestFunc of ( product G), S st for y be Point of ( product G) st y in N holds ((f /. y) - (f /. x)) = ((( diff (f,x)) . (y - x)) + (R /. (y - x))) by A1, NDIFF_1:def 7;

      consider R be RestFunc of ( product G), S such that

       A3: for y be Point of ( product G) st y in N holds ((f /. y) - (f /. x)) = ((( diff (f,x)) . (y - x)) + (R /. (y - x))) by A2;

      consider r0 be Real such that

       A4: 0 < r0 & { z where z be Point of ( product G) : ||.(z - x).|| < r0 } c= N by NFCONT_1:def 1;

      set u = (f * ( reproj (i,x)));

      reconsider x0 = (( proj i) . x) as Point of (G . i);

      set Z = ( 0. ( product G));

      set Nx0 = { z where z be Point of (G . i) : ||.(z - x0).|| < r0 };

      now

        let s be object;

        assume s in Nx0;

        then ex z be Point of (G . i) st s = z & ||.(z - x0).|| < r0;

        hence s in the carrier of (G . i);

      end;

      then Nx0 is Subset of (G . i) by TARSKI:def 3;

      then

      reconsider Nx0 as Neighbourhood of x0 by A4, NFCONT_1:def 1;

      

       A5: for xi be Element of (G . i) st xi in Nx0 holds (( reproj (i,x)) . xi) in N

      proof

        let xi be Element of (G . i);

        assume xi in Nx0;

        then

         A6: ex z be Point of (G . i) st xi = z & ||.(z - x0).|| < r0;

        ((( reproj (i,x)) . xi) - x) = (( reproj (i,Z)) . (xi - x0)) by Th22;

        then ||.((( reproj (i,x)) . xi) - x).|| < r0 by Th21, A6;

        then (( reproj (i,x)) . xi) in { z where z be Point of ( product G) : ||.(z - x).|| < r0 };

        hence thesis by A4;

      end;

      

       A7: R is total by NDIFF_1:def 5;

      then

       A8: ( dom R) = the carrier of ( product G) by PARTFUN1:def 2;

      reconsider R1 = (R * ( reproj (i,( 0. ( product G))))) as PartFunc of (G . i), S;

      

       A9: ( dom ( reproj (i,( 0. ( product G))))) = the carrier of (G . i) by FUNCT_2:def 1;

      

       A10: ( dom R1) = the carrier of (G . i) by A7, PARTFUN1:def 2;

      for r be Real st r > 0 holds ex d be Real st d > 0 & for z be Point of (G . i) st z <> ( 0. (G . i)) & ||.z.|| < d holds (( ||.z.|| " ) * ||.(R1 /. z).||) < r

      proof

        let r be Real;

        assume r > 0 ;

        then

        consider d be Real such that

         A11: d > 0 & for z be Point of ( product G) st z <> ( 0. ( product G)) & ||.z.|| < d holds (( ||.z.|| " ) * ||.(R /. z).||) < r by A7, NDIFF_1: 23;

        take d;

        now

          let z be Point of (G . i);

          assume

           A12: z <> ( 0. (G . i)) & ||.z.|| < d;

          

           A13: ||.(( reproj (i,Z)) . z).|| = ||.z.|| by Th21;

          (R /. (( reproj (i,Z)) . z)) = (R . (( reproj (i,Z)) . z)) by A8, PARTFUN1:def 6;

          then (R /. (( reproj (i,Z)) . z)) = (R1 . z) by A9, FUNCT_1: 13;

          then (R /. (( reproj (i,Z)) . z)) = (R1 /. z) by A10, PARTFUN1:def 6;

          hence (( ||.z.|| " ) * ||.(R1 /. z).||) < r by A11, A13, A12, Th38;

        end;

        hence thesis by A11;

      end;

      then

      reconsider R1 as RestFunc of (G . i), S by A7, NDIFF_1: 23;

      reconsider dfx = ( diff (f,x)) as Lipschitzian LinearOperator of ( product G), S by LOPBAN_1:def 9;

      reconsider LD1 = (dfx * ( reproj (i,( 0. ( product G))))) as Function of (G . i), S;

       A14:

      now

        let x,y be Element of (G . i);

        (LD1 . (x + y)) = (dfx . (( reproj (i,Z)) . (x + y))) by FUNCT_2: 15;

        then (LD1 . (x + y)) = (dfx . ((( reproj (i,Z)) . x) + (( reproj (i,Z)) . y))) by Th34;

        then (LD1 . (x + y)) = ((dfx . (( reproj (i,Z)) . x)) + (dfx . (( reproj (i,Z)) . y))) by VECTSP_1:def 20;

        then (LD1 . (x + y)) = ((LD1 . x) + (dfx . (( reproj (i,Z)) . y))) by FUNCT_2: 15;

        hence (LD1 . (x + y)) = ((LD1 . x) + (LD1 . y)) by FUNCT_2: 15;

      end;

      now

        let x be Element of (G . i), a be Real;

        (LD1 . (a * x)) = (dfx . (( reproj (i,Z)) . (a * x))) by FUNCT_2: 15;

        then (LD1 . (a * x)) = (dfx . (a * (( reproj (i,Z)) . x))) by Th39;

        then (LD1 . (a * x)) = (a * (dfx . (( reproj (i,Z)) . x))) by LOPBAN_1:def 5;

        hence (LD1 . (a * x)) = (a * (LD1 . x)) by FUNCT_2: 15;

      end;

      then

      reconsider LD1 as LinearOperator of (G . i), S by A14, LOPBAN_1:def 5, VECTSP_1:def 20;

      consider K0 be Real such that

       A15: 0 <= K0 & for x be VECTOR of ( product G) holds ||.(dfx . x).|| <= (K0 * ||.x.||) by LOPBAN_1:def 8;

      now

        let r be VECTOR of (G . i);

         ||.(dfx . (( reproj (i,Z)) . r)).|| <= (K0 * ||.(( reproj (i,Z)) . r).||) by A15;

        then ||.(dfx . (( reproj (i,Z)) . r)).|| <= (K0 * ||.r.||) by Th21;

        hence ||.(LD1 . r).|| <= (K0 * ||.r.||) by FUNCT_2: 15;

      end;

      then LD1 is Lipschitzian by A15;

      then

      reconsider LD1 as Point of ( R_NormSpace_of_BoundedLinearOperators ((G . i),S)) by LOPBAN_1:def 9;

      now

        let s be object;

        assume s in (( reproj (i,x)) .: Nx0);

        then ex t be Element of (G . i) st t in Nx0 & s = (( reproj (i,x)) . t) by FUNCT_2: 65;

        hence s in ( dom f) by A2, A5;

      end;

      then

       A16: (( reproj (i,x)) .: Nx0) c= ( dom f);

      ( dom ( reproj (i,x))) = the carrier of (G . i) by FUNCT_2:def 1;

      then

       A17: Nx0 c= ( dom u) by A16, FUNCT_3: 3;

      

       A18: for y be Point of (G . i) st y in Nx0 holds ((u /. y) - (u /. x0)) = ((LD1 . (y - x0)) + (R1 /. (y - x0)))

      proof

        let y be Point of (G . i);

        assume

         A19: y in Nx0;

        then

         A20: (( reproj (i,x)) . y) in N by A5;

        

         A21: (( reproj (i,x)) . x0) = (x +* (i,x0)) by Def4;

        

         A22: the carrier of ( product G) = ( product ( carr G)) by Th10;

        (x . i) = x0 by Def3, A22;

        then

         A23: x = (x +* (i,x0)) by FUNCT_7: 35;

        

         A24: (( reproj (i,x)) . x0) in N by A5, NFCONT_1: 4;

        (u /. y) = (u . y) by A19, A17, PARTFUN1:def 6;

        then (u /. y) = (f . (( reproj (i,x)) . y)) by FUNCT_2: 15;

        then

         A25: (u /. y) = (f /. (( reproj (i,x)) . y)) by A20, A2, PARTFUN1:def 6;

        (R /. (( reproj (i,Z)) . (y - x0))) = (R . (( reproj (i,Z)) . (y - x0))) by A8, PARTFUN1:def 6;

        then (R /. (( reproj (i,Z)) . (y - x0))) = (R1 . (y - x0)) by A9, FUNCT_1: 13;

        then

         A26: (R /. (( reproj (i,Z)) . (y - x0))) = (R1 /. (y - x0)) by A10, PARTFUN1:def 6;

        x0 in Nx0 by NFCONT_1: 4;

        then (u /. x0) = (u . x0) by A17, PARTFUN1:def 6;

        then (u /. x0) = (f . (( reproj (i,x)) . x0)) by FUNCT_2: 15;

        then ((u /. y) - (u /. x0)) = ((f /. (( reproj (i,x)) . y)) - (f /. x)) by A25, A23, A24, A2, A21, PARTFUN1:def 6;

        then ((u /. y) - (u /. x0)) = ((( diff (f,x)) . ((( reproj (i,x)) . y) - x)) + (R /. ((( reproj (i,x)) . y) - x))) by A3, A19, A5;

        then ((u /. y) - (u /. x0)) = ((dfx . (( reproj (i,Z)) . (y - x0))) + (R /. ((( reproj (i,x)) . y) - x))) by Th22;

        then ((u /. y) - (u /. x0)) = ((dfx . (( reproj (i,Z)) . (y - x0))) + (R /. (( reproj (i,Z)) . (y - x0)))) by Th22;

        hence ((u /. y) - (u /. x0)) = ((LD1 . (y - x0)) + (R1 /. (y - x0))) by A26, FUNCT_2: 15;

      end;

      hence f is_partial_differentiable_in (x,i0) by A17, NDIFF_1:def 6;

      u is_differentiable_in x0 by A17, A18, NDIFF_1:def 6;

      hence thesis by A17, A18, NDIFF_1:def 7;

    end;

    

     Lm5: for G be RealNormSpace-Sequence, S be RealNormSpace, f be PartFunc of ( product G), S, x be Point of ( product G) holds ex L be Lipschitzian LinearOperator of ( product G), S st for h be Element of ( product G) holds ex w be FinSequence of S st ( dom w) = ( Seg ( len G)) & (for i be Element of NAT st i in ( Seg ( len G)) holds (w . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . h))) & (L . h) = ( Sum w)

    proof

      let G be RealNormSpace-Sequence, S be RealNormSpace, f be PartFunc of ( product G), S, x be Point of ( product G);

      set m = ( len G);

      defpred LX[ set, set] means ex w be FinSequence of S st ( dom w) = ( Seg m) & (for i be Element of NAT st i in ( Seg m) holds (w . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . $1))) & $2 = ( Sum w);

      

       A1: for v be Element of ( product G) holds ex y be Element of S st LX[v, y]

      proof

        let v be Element of ( product G);

        defpred YX[ set, set] means ex i be Element of NAT st i = $1 & $2 = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . v));

        

         A2: for i be Nat st i in ( Seg m) holds ex r be Element of S st YX[i, r]

        proof

          let i be Nat;

          assume i in ( Seg m);

          reconsider i0 = i as Element of NAT by ORDINAL1:def 12;

          (( partdiff (f,x,i0)) . (( proj ( In (i0,( dom G)))) . v)) in the carrier of S;

          hence thesis;

        end;

        consider w be FinSequence of S such that

         A3: ( dom w) = ( Seg m) & for i be Nat st i in ( Seg m) holds YX[i, (w . i)] from FINSEQ_1:sch 5( A2);

         A4:

        now

          let i be Element of NAT ;

          assume i in ( Seg m);

          then YX[i, (w . i)] by A3;

          hence (w . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . v));

        end;

        reconsider w0 = ( Sum w) as Element of S;

        ex w be FinSequence of S st ( dom w) = ( Seg m) & (for i be Element of NAT st i in ( Seg m) holds (w . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . v))) & w0 = ( Sum w) by A4, A3;

        hence ex y0 be Element of S st LX[v, y0];

      end;

      consider L be Function of ( product G), S such that

       A5: for h be Element of ( product G) holds LX[h, (L . h)] from FUNCT_2:sch 3( A1);

      

       A6: for s,t be Element of ( product G) holds (L . (s + t)) = ((L . s) + (L . t))

      proof

        let s,t be Element of ( product G);

        consider w be FinSequence of S such that

         A7: ( dom w) = ( Seg m) & (for i be Element of NAT st i in ( Seg m) holds (w . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . s))) & (L . s) = ( Sum w) by A5;

        consider v be FinSequence of S such that

         A8: ( dom v) = ( Seg m) & (for i be Element of NAT st i in ( Seg m) holds (v . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . t))) & (L . t) = ( Sum v) by A5;

        consider u be FinSequence of S such that

         A9: ( dom u) = ( Seg m) & (for i be Element of NAT st i in ( Seg m) holds (u . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . (s + t)))) & (L . (s + t)) = ( Sum u) by A5;

        

         A10: ( len w) = m by A7, FINSEQ_1:def 3;

        

         A11: ( len v) = m by A8, FINSEQ_1:def 3;

        

         A12: ( len u) = m by A9, FINSEQ_1:def 3;

        now

          let i be Nat;

          assume

           A13: i in ( dom w);

          then

           A14: 1 <= i & i <= m by A7, FINSEQ_1: 1;

          then (w /. i) = (w . i) by A10, FINSEQ_4: 15;

          then

           A15: (w /. i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . s)) by A7, A13;

          (v /. i) = (v . i) by A14, A11, FINSEQ_4: 15;

          then

           A16: (v /. i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . t)) by A7, A8, A13;

          

           A17: ( partdiff (f,x,i)) is Lipschitzian LinearOperator of (G . ( In (i,( dom G)))), S by LOPBAN_1:def 9;

          (u . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . (s + t))) by A7, A9, A13;

          then (u . i) = (( partdiff (f,x,i)) . ((( proj ( In (i,( dom G)))) . s) + (( proj ( In (i,( dom G)))) . t))) by Th35;

          hence (u . i) = ((w /. i) + (v /. i)) by A15, A16, A17, VECTSP_1:def 20;

        end;

        hence (L . (s + t)) = ((L . s) + (L . t)) by A9, A7, A8, A10, A11, A12, RLVECT_2: 2;

      end;

      for s be Element of ( product G), r be Real holds (L . (r * s)) = (r * (L . s))

      proof

        let s be Element of ( product G), r be Real;

        consider w be FinSequence of S such that

         A18: ( dom w) = ( Seg m) & (for i be Element of NAT st i in ( Seg m) holds (w . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . s))) & (L . s) = ( Sum w) by A5;

        consider u be FinSequence of S such that

         A19: ( dom u) = ( Seg m) & (for i be Element of NAT st i in ( Seg m) holds (u . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . (r * s)))) & (L . (r * s)) = ( Sum u) by A5;

        

         A20: ( len w) = m & ( len u) = m by A18, A19, FINSEQ_1:def 3;

        now

          let i be Nat;

          assume

           A21: i in ( dom w);

          then 1 <= i & i <= m by A18, FINSEQ_1: 1;

          then (w /. i) = (w . i) by A20, FINSEQ_4: 15;

          then

           A22: (w /. i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . s)) by A18, A21;

          

           A23: ( partdiff (f,x,i)) is Lipschitzian LinearOperator of (G . ( In (i,( dom G)))), S by LOPBAN_1:def 9;

          (u . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . (r * s))) by A18, A19, A21;

          then (u . i) = (( partdiff (f,x,i)) . (r * (( proj ( In (i,( dom G)))) . s))) by Th40;

          hence (u . i) = (r * (w /. i)) by A22, A23, LOPBAN_1:def 5;

        end;

        hence (L . (r * s)) = (r * (L . s)) by A18, A19, A20, RLVECT_2: 3;

      end;

      then

      reconsider L as LinearOperator of ( product G), S by A6, LOPBAN_1:def 5, VECTSP_1:def 20;

      defpred YXL[ set, set] means ex i be Element of NAT st i = $1 & $2 = ||.( partdiff (f,x,i)).||;

      

       A24: for i be Nat st i in ( Seg m) holds ex r be Element of REAL st YXL[i, r]

      proof

        let i be Nat;

        assume i in ( Seg m);

        reconsider i0 = i as Element of NAT by ORDINAL1:def 12;

        reconsider r = ||.( partdiff (f,x,i0)).|| as Element of REAL ;

         YXL[i, r];

        hence thesis;

      end;

      consider Kw be FinSequence of REAL such that

       A25: ( dom Kw) = ( Seg m) & for i be Nat st i in ( Seg m) holds YXL[i, (Kw . i)] from FINSEQ_1:sch 5( A24);

       A26:

      now

        let i be Element of NAT ;

        assume i in ( Seg m);

        then YXL[i, (Kw . i)] by A25;

        hence (Kw . i) = ||.( partdiff (f,x,i)).||;

      end;

       A27:

      now

        let i be Nat;

        assume i in ( dom Kw);

        then (Kw . i) = ||.( partdiff (f,x,i)).|| by A26, A25;

        hence 0 <= (Kw . i);

      end;

      set LK = ( Sum Kw);

      for s be Element of ( product G) holds ||.(L . s).|| <= (LK * ||.s.||)

      proof

        let s be Element of ( product G);

        consider w be FinSequence of S such that

         A29: ( dom w) = ( Seg m) & (for i be Element of NAT st i in ( Seg m) holds (w . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . s))) & (L . s) = ( Sum w) by A5;

        defpred YXD[ set, set] means ex i be Element of NAT st i = $1 & $2 = ( ||.( partdiff (f,x,i)).|| * ||.s.||);

        

         A30: for i be Nat st i in ( Seg m) holds ex r be Element of REAL st YXD[i, r]

        proof

          let i be Nat;

          assume i in ( Seg m);

          reconsider i0 = i as Element of NAT by ORDINAL1:def 12;

          reconsider r = ( ||.( partdiff (f,x,i0)).|| * ||.s.||) as Element of REAL by XREAL_0:def 1;

           YXD[i, r];

          hence thesis;

        end;

        consider Dw be FinSequence of REAL such that

         A31: ( dom Dw) = ( Seg m) & for i be Nat st i in ( Seg m) holds YXD[i, (Dw . i)] from FINSEQ_1:sch 5( A30);

         A32:

        now

          let i be Element of NAT ;

          assume i in ( Seg m);

          then YXD[i, (Dw . i)] by A31;

          hence (Dw . i) = ( ||.( partdiff (f,x,i)).|| * ||.s.||);

        end;

        defpred YXH[ set, set] means ex i be Element of NAT st i = $1 & $2 = ||.(w /. i).||;

        

         A33: for i be Nat st i in ( Seg m) holds ex r be Element of REAL st YXH[i, r]

        proof

          let i be Nat;

          assume i in ( Seg m);

          reconsider i0 = i as Element of NAT by ORDINAL1:def 12;

          reconsider r = ||.(w /. i0).|| as Element of REAL ;

           YXH[i, r];

          hence thesis;

        end;

        consider yseq be FinSequence of REAL such that

         A34: ( dom yseq) = ( Seg m) & for i be Nat st i in ( Seg m) holds YXH[i, (yseq . i)] from FINSEQ_1:sch 5( A33);

         A35:

        now

          let i be Element of NAT ;

          assume i in ( Seg m);

          then YXH[i, (yseq . i)] by A34;

          hence (yseq . i) = ||.(w /. i).||;

        end;

        ( len w) = ( len yseq) by A29, A34, FINSEQ_3: 29;

        then

         A36: ||.(L . s).|| <= ( Sum yseq) by A29, A35, Th7;

        m = ( len yseq) by A34, FINSEQ_1:def 3;

        then

         A37: yseq is Element of (m -tuples_on REAL ) by FINSEQ_2: 92;

        ( len Dw) = m by A31, FINSEQ_1:def 3;

        then

         A38: Dw is Element of (m -tuples_on REAL ) by FINSEQ_2: 92;

        now

          let i be Nat;

          assume

           A39: i in ( Seg m);

          then

           A40: (yseq . i) = ||.(w /. i).|| by A35;

          (w /. i) = (w . i) by A39, A29, PARTFUN1:def 6;

          then

           A41: ||.(w /. i).|| = ||.(( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . s)).|| by A29, A39;

          reconsider DF1 = ( partdiff (f,x,i)) as Lipschitzian LinearOperator of (G . ( In (i,( dom G)))), S by LOPBAN_1:def 9;

          

           A42: ||.(DF1 . (( proj ( In (i,( dom G)))) . s)).|| <= ( ||.( partdiff (f,x,i)).|| * ||.(( proj ( In (i,( dom G)))) . s).||) by LOPBAN_1: 32;

          ( product G) = NORMSTR (# ( product ( carr G)), ( zeros G), [:( addop G):], [:( multop G):], ( productnorm G) #) by PRVECT_2: 6;

          then

          reconsider ss = s as Element of ( product ( carr G));

          reconsider xi = (( proj ( In (i,( dom G)))) . s) as Point of (G . ( In (i,( dom G))));

          xi = (ss . ( In (i,( dom G)))) by Def3;

          then ( ||.( partdiff (f,x,i)).|| * ||.(( proj ( In (i,( dom G)))) . s).||) <= ( ||.( partdiff (f,x,i)).|| * ||.s.||) by PRVECT_2: 10, XREAL_1: 64;

          then ||.(w /. i).|| <= ( ||.( partdiff (f,x,i)).|| * ||.s.||) by A41, A42, XXREAL_0: 2;

          hence (yseq . i) <= (Dw . i) by A32, A39, A40;

        end;

        then

         A43: ( Sum yseq) <= ( Sum Dw) by A37, A38, RVSUM_1: 82;

        ( len Kw) = m by A25, FINSEQ_1:def 3;

        then

        reconsider KKw = Kw as Element of (m -tuples_on REAL ) by FINSEQ_2: 92;

        ( ||.s.|| * KKw) in (m -tuples_on REAL );

        then ex t be Element of ( REAL * ) st t = ( ||.s.|| * KKw) & ( len t) = m;

        then

         A44: ( dom Dw) = ( dom ( ||.s.|| * Kw)) by A31, FINSEQ_1:def 3;

        now

          let k be Nat;

          assume

           A45: k in ( dom Dw);

          then (Dw . k) = ( ||.( partdiff (f,x,k)).|| * ||.s.||) by A32, A31;

          then (Dw . k) = ( ||.s.|| * (Kw . k)) by A26, A45, A31;

          hence (Dw . k) = (( ||.s.|| * Kw) . k) by RVSUM_1: 45;

        end;

        then Dw = ( ||.s.|| * Kw) by A44, FINSEQ_1: 13;

        then ( Sum Dw) = (( Sum Kw) * ||.s.||) by RVSUM_1: 87;

        hence thesis by A36, A43, XXREAL_0: 2;

      end;

      then

      reconsider L as Lipschitzian LinearOperator of ( product G), S by A27, RVSUM_1: 84, LOPBAN_1:def 8;

      take L;

      thus thesis by A5;

    end;

    theorem :: NDIFF_5:42

    

     Th42: for S be RealNormSpace, h,g be FinSequence of S st ( len h) = (( len g) + 1) & (for i be Nat st i in ( dom g) holds (g /. i) = ((h /. i) - (h /. (i + 1)))) holds ((h /. 1) - (h /. ( len h))) = ( Sum g)

    proof

      let S be RealNormSpace, h,g be FinSequence of S;

      assume that

       A1: ( len h) = (( len g) + 1) and

       A2: for i be Nat st i in ( dom g) holds (g /. i) = ((h /. i) - (h /. (i + 1)));

      consider F be sequence of the carrier of S such that

       A3: ( Sum g) = (F . ( len g)) & (F . 0 ) = ( 0. S) & for j be Nat, v be Element of S st j < ( len g) & v = (g . (j + 1)) holds (F . (j + 1)) = ((F . j) + v) by RLVECT_1:def 12;

      per cases ;

        suppose ( len g) = 0 ;

        hence thesis by A3, A1, RLVECT_1: 15;

      end;

        suppose

         A4: ( len g) > 0 ;

        defpred P[ Nat] means $1 <= ( len g) implies (F . $1) = ((h /. 1) - (h /. ($1 + 1)));

        

         A5: P[1]

        proof

          assume

           A6: 1 <= ( len g);

          then 1 in ( Seg ( len g));

          then

           A7: 1 in ( dom g) by FINSEQ_1:def 3;

          reconsider zz0 = 0 as Element of NAT ;

          (g /. 1) = (g . (zz0 + 1)) by A7, PARTFUN1:def 6;

          

          then (F . (zz0 + 1)) = ((F . 0 ) + (g /. 1)) by A3, A6

          .= (g /. 1) by A3, RLVECT_1: 4;

          hence (F . 1) = ((h /. 1) - (h /. (1 + 1))) by A7, A2;

        end;

        

         A8: for j be Nat st 1 <= j holds P[j] implies P[(j + 1)]

        proof

          let j be Nat;

          assume 1 <= j;

          assume

           A9: P[j];

          assume

           A10: (j + 1) <= ( len g);

          then

           A12: (j + 1) in ( dom g) by XREAL_1: 38, FINSEQ_3: 25;

          then

           A13: (g . (j + 1)) = (g /. (j + 1)) by PARTFUN1:def 6;

          (F . (j + 1)) = ((F . j) + (g /. (j + 1))) by A13, A10, A3, NAT_1: 13

          .= ((F . j) + ((h /. (j + 1)) - (h /. ((j + 1) + 1)))) by A2, A12

          .= ((((h /. 1) - (h /. (j + 1))) + (h /. (j + 1))) - (h /. ((j + 1) + 1))) by A9, A10, NAT_1: 13, RLVECT_1: 28

          .= (((h /. 1) - ((h /. (j + 1)) - (h /. (j + 1)))) - (h /. ((j + 1) + 1))) by RLVECT_1: 29

          .= (((h /. 1) - ( 0. S)) - (h /. ((j + 1) + 1))) by RLVECT_1: 15;

          hence thesis by RLVECT_1: 13;

        end;

        

         A14: 1 <= ( len g) by A4, NAT_1: 14;

        for i be Nat st 1 <= i holds P[i] from NAT_1:sch 8( A5, A8);

        hence thesis by A3, A1, A14;

      end;

    end;

    theorem :: NDIFF_5:43

    for G be RealNormSpace-Sequence, x,y be Element of ( product ( carr G)), Z be set holds (x +* (y | Z)) is Element of ( product ( carr G)) by CARD_3: 79;

    theorem :: NDIFF_5:44

    

     Th44: for G be RealNormSpace-Sequence, x,y be Point of ( product G), Z,x0 be Element of ( product ( carr G)), X be set st Z = ( 0. ( product G)) & x0 = x & y = (Z +* (x0 | X)) holds ||.y.|| <= ||.x.||

    proof

      let G be RealNormSpace-Sequence, x,y be Point of ( product G), Z,x0 be Element of ( product ( carr G)), X be set;

      assume

       A1: Z = ( 0. ( product G)) & x0 = x & y = (Z +* (x0 | X));

      reconsider y0 = y as Element of ( product ( carr G)) by Th10;

      

       A2: ||.y.|| = (( productnorm G) . y) by PRVECT_2:def 13

      .= |.( normsequence (G,y0)).| by PRVECT_2:def 12;

      

       A3: ||.x.|| = (( productnorm G) . x) by PRVECT_2:def 13

      .= |.( normsequence (G,x0)).| by A1, PRVECT_2:def 12;

      reconsider Ny = ( normsequence (G,y0)) as ( len G) -element FinSequence of REAL ;

      reconsider Nx = ( normsequence (G,x0)) as ( len G) -element FinSequence of REAL ;

      

       A4: ( len Nx) = ( len G) & ( len Ny) = ( len G) by CARD_1:def 7;

      for k be Element of NAT st k in ( Seg ( len Ny)) holds 0 <= (Ny . k) & (Ny . k) <= (Nx . k)

      proof

        let k be Element of NAT ;

        assume

         A5: k in ( Seg ( len Ny));

        then

        reconsider k1 = k as Element of ( dom G) by CARD_1:def 7, FINSEQ_1:def 3;

        x0 is Element of the carrier of ( product G) by Th10;

        then

        reconsider xx = x0 as ( len G) -element FinSequence;

        ( dom xx) = ( Seg ( len G)) by FINSEQ_1: 89;

        then

         A6: k in ( dom x0) by A5, CARD_1:def 7;

        reconsider yk = (y0 . k1), xk = (x0 . k1) as Element of the carrier of (G . k1);

        

         A7: (Nx . k) = (the normF of (G . k1) . (x0 . k1)) by PRVECT_2:def 11;

        

         A8: (Ny . k) = ||.yk.|| by PRVECT_2:def 11;

        hence 0 <= (Ny . k);

        

         A9: (Nx . k) = ||.xk.|| by PRVECT_2:def 11;

        per cases ;

          suppose k1 in X;

          then

           A10: k1 in ( dom (x0 | X)) by A6, RELAT_1: 57;

          then (y0 . k1) = ((x0 | X) . k1) by A1, FUNCT_4: 13;

          then (y0 . k1) = (x0 . k1) by A10, FUNCT_1: 47;

          hence (Ny . k) <= (Nx . k) by A7, PRVECT_2:def 11;

        end;

          suppose not k1 in X;

          then not k1 in ( dom (x0 | X));

          then (y0 . k1) = (Z . k1) by A1, FUNCT_4: 11;

          then (y0 . k1) = ( 0. (G . k1)) by A1, Th14;

          hence (Ny . k) <= (Nx . k) by A8, A9;

        end;

      end;

      hence ||.y.|| <= ||.x.|| by A2, A3, A4, PRVECT_2: 2;

    end;

    theorem :: NDIFF_5:45

    

     Th45: for G be RealNormSpace-Sequence, S be RealNormSpace, f be PartFunc of ( product G), S, x,y be Point of ( product G) holds ex h be FinSequence of ( product G), g be FinSequence of S, Z,y0 be Element of ( product ( carr G)) st y0 = y & Z = ( 0. ( product G)) & ( len h) = (( len G) + 1) & ( len g) = ( len G) & (for i be Nat st i in ( dom h) holds (h /. i) = (Z +* (y0 | ( Seg ((( len G) + 1) -' i))))) & (for i be Nat st i in ( dom g) holds (g /. i) = ((f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))))) & (for i be Nat, hi be Point of ( product G) st i in ( dom h) & (h /. i) = hi holds ||.hi.|| <= ||.y.||) & ((f /. (x + y)) - (f /. x)) = ( Sum g)

    proof

      let G be RealNormSpace-Sequence, S be RealNormSpace, f be PartFunc of ( product G), S, x,y be Point of ( product G);

      set m = ( len G);

      

       A1: the carrier of ( product G) = ( product ( carr G)) by Th10;

      reconsider Z0 = ( 0. ( product G)) as Element of ( product ( carr G)) by Th10;

      reconsider y0 = y as Element of ( product ( carr G)) by Th10;

      reconsider y1 = y as ( len G) -element FinSequence;

      reconsider Z1 = ( 0. ( product G)) as ( len G) -element FinSequence;

      ( len y1) = m by CARD_1:def 7;

      then

       A2: ( dom y1) = ( dom G) by FINSEQ_3: 29;

      ( len Z1) = m by CARD_1:def 7;

      then

       A3: ( dom Z1) = ( dom G) by FINSEQ_3: 29;

      defpred H[ Nat, set] means $2 = (Z0 +* (y0 | ( Seg ((( len G) + 1) -' $1))));

      

       A4: for k be Nat st k in ( Seg (m + 1)) holds ex x be Element of ( product G) st H[k, x]

      proof

        let k be Nat;

        assume k in ( Seg (m + 1));

        (Z0 +* (y0 | ( Seg ((( len G) + 1) -' k)))) is Element of ( product ( carr G)) by CARD_3: 79;

        hence thesis by A1;

      end;

      consider h be FinSequence of ( product G) such that

       A5: ( dom h) = ( Seg (m + 1)) & for j be Nat st j in ( Seg (m + 1)) holds H[j, (h . j)] from FINSEQ_1:sch 5( A4);

       A6:

      now

        let j be Nat;

        assume

         A7: j in ( dom h);

        then (h /. j) = (h . j) by PARTFUN1:def 6;

        hence (h /. j) = (Z0 +* (y0 | ( Seg ((( len G) + 1) -' j)))) by A7, A5;

      end;

      deffunc Z( Nat) = (f /. (x + (h /. $1)));

      consider z be FinSequence of S such that

       A8: ( len z) = (m + 1) & for j be Nat st j in ( dom z) holds (z . j) = Z(j) from FINSEQ_2:sch 1;

       A9:

      now

        let j be Nat;

        assume

         A10: j in ( dom z);

        then (z /. j) = (z . j) by PARTFUN1:def 6;

        hence (z /. j) = (f /. (x + (h /. j))) by A10, A8;

      end;

      deffunc G( Nat) = ((z /. $1) - (z /. ($1 + 1)));

      consider g be FinSequence of S such that

       A11: ( len g) = m & for j be Nat st j in ( dom g) holds (g . j) = G(j) from FINSEQ_2:sch 1;

       A12:

      now

        let j be Nat;

        assume

         A13: j in ( dom g);

        then (g /. j) = (g . j) by PARTFUN1:def 6;

        hence (g /. j) = ((z /. j) - (z /. (j + 1))) by A13, A11;

      end;

      

       A14: ((m + 1) -' 1) = ((m + 1) - 1) by NAT_1: 11, XREAL_1: 233;

      reconsider zz0 = 0 as Element of NAT ;

      1 <= (m + 1) by NAT_1: 11;

      then

       A15: 1 in ( dom h) by A5;

      

      then (h /. 1) = (Z0 +* (y0 | ( Seg ((( len G) + 1) -' 1)))) by A6

      .= (Z0 +* (y0 | ( dom G))) by A14, FINSEQ_1:def 3

      .= (Z0 +* y0) by A2;

      then

       A16: (h /. 1) = y by A2, A3, FUNCT_4: 19;

      

       A17: ((m + 1) -' ( len z)) = ((m + 1) - ( len z)) by A8, XREAL_1: 233;

      1 <= ( len z) & ( len z) <= (m + 1) by A8, NAT_1: 14;

      then

       A18: ( len z) in ( dom h) by A5;

      

      then

       A19: (h /. ( len z)) = (Z0 +* (y0 | ( Seg 0 ))) by A6, A17, A8

      .= ( 0. ( product G));

      

       A20: ( dom h) = ( dom z) by A5, A8, FINSEQ_1:def 3;

      then

       A21: (z /. 1) = (f /. (x + y)) by A9, A16, A15;

      (z /. ( len z)) = (f /. (x + (h /. ( len z)))) by A9, A20, A18;

      then

       A22: (z /. ( len z)) = (f /. x) by A19, RLVECT_1:def 4;

      take h, g, Z0, y0;

       A23:

      now

        let i be Nat;

        assume

         A24: i in ( dom g);

        then

         A25: i in ( Seg m) by A11, FINSEQ_1:def 3;

        then 1 <= i & i <= m by FINSEQ_1: 1;

        then

         A26: (i + 1) <= (m + 1) by XREAL_1: 6;

        ( Seg m) c= ( Seg (m + 1)) by NAT_1: 11, FINSEQ_1: 5;

        then

         A27: (z /. i) = (f /. (x + (h /. i))) by A9, A5, A25, A20;

        1 <= (i + 1) by NAT_1: 11;

        then (i + 1) in ( Seg (m + 1)) by A26;

        then (i + 1) in ( dom z) by A8, FINSEQ_1:def 3;

        then (z /. (i + 1)) = (f /. (x + (h /. (i + 1)))) by A9;

        hence (g /. i) = ((f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1))))) by A12, A24, A27;

      end;

      now

        let i be Nat, hi be Element of ( product G);

        assume

         A28: i in ( dom h) & (h /. i) = hi;

        then (h /. i) = (Z0 +* (y0 | ( Seg ((( len G) + 1) -' i)))) by A6;

        hence ||.hi.|| <= ||.y.|| by Th44, A28;

      end;

      hence thesis by A6, A21, A22, A23, A8, A12, Th42, A5, A11, FINSEQ_1:def 3;

    end;

    theorem :: NDIFF_5:46

    

     Th46: for G be RealNormSpace-Sequence, i be Element of ( dom G), x,y be Point of ( product G), xi be Point of (G . i) st y = (( reproj (i,x)) . xi) holds (( proj i) . y) = xi

    proof

      let G be RealNormSpace-Sequence, i be Element of ( dom G), x,y be Point of ( product G), xi be Point of (G . i);

      assume

       A1: y = (( reproj (i,x)) . xi);

      

       A2: y = (x +* (i,xi)) by A1, Def4;

      x in the carrier of ( product G);

      then x in ( product ( carr G)) by Th10;

      then

      consider g be Function such that

       A3: x = g & ( dom g) = ( dom ( carr G)) & for y be object st y in ( dom ( carr G)) holds (g . y) in (( carr G) . y) by CARD_3:def 5;

      

       A4: i in ( dom G);

      

       A5: i in ( dom x) by Lm1, A4, A3;

      y is Element of ( product ( carr G)) by Th10;

      then (( proj i) . y) = ((x +* (i,xi)) . i) by A2, Def3;

      hence (( proj i) . y) = xi by A5, FUNCT_7: 31;

    end;

    theorem :: NDIFF_5:47

    

     Th47: for G be RealNormSpace-Sequence, i be Element of ( dom G), y be Point of ( product G), q be Point of (G . i) st q = (( proj i) . y) holds y = (( reproj (i,y)) . q)

    proof

      let G be RealNormSpace-Sequence, i be Element of ( dom G), y be Point of ( product G), q be Point of (G . i);

      assume

       A1: q = (( proj i) . y);

      reconsider z1 = (( reproj (i,y)) . q) as ( len G) -element FinSequence;

      reconsider z2 = y as ( len G) -element FinSequence;

      

       A2: ( dom z1) = ( Seg ( len G)) by FINSEQ_1: 89

      .= ( dom z2) by FINSEQ_1: 89;

      for k be Nat st k in ( dom z1) holds (z1 . k) = (z2 . k)

      proof

        let k be Nat;

        assume k in ( dom z1);

        ( product G) = NORMSTR (# ( product ( carr G)), ( zeros G), [:( addop G):], [:( multop G):], ( productnorm G) #) by PRVECT_2: 6;

        then

         A3: q = (y . i) by A1, Def3;

        per cases ;

          suppose

           A4: k = i;

          then ((y +* (i,q)) . k) = q by A3, FUNCT_7: 35;

          hence (z1 . k) = (z2 . k) by A4, A3, Def4;

        end;

          suppose k <> i;

          then ((y +* (i,q)) . k) = (y . k) by FUNCT_7: 32;

          hence (z1 . k) = (z2 . k) by Def4;

        end;

      end;

      hence thesis by A2, FINSEQ_1: 13;

    end;

    theorem :: NDIFF_5:48

    

     Th48: for G be RealNormSpace-Sequence, i be Element of ( dom G), x,y be Point of ( product G), xi be Point of (G . i) st y = (( reproj (i,x)) . xi) holds ( reproj (i,x)) = ( reproj (i,y))

    proof

      let G be RealNormSpace-Sequence, i be Element of ( dom G), x,y be Point of ( product G), xi be Point of (G . i);

      assume

       A1: y = (( reproj (i,x)) . xi);

      for v be Element of (G . i) holds (( reproj (i,x)) . v) = (( reproj (i,y)) . v)

      proof

        let v be Element of (G . i);

        

         A2: (( reproj (i,x)) . v) = (x +* (i,v)) & (( reproj (i,y)) . v) = (y +* (i,v)) by Def4;

        reconsider xv = (( reproj (i,x)) . v), yv = (( reproj (i,y)) . v) as ( len G) -element FinSequence;

        

         A3: ( dom xv) = ( Seg ( len G)) & ( dom yv) = ( Seg ( len G)) by FINSEQ_1: 89;

        then

         A4: ( dom xv) = ( dom G) by FINSEQ_1:def 3;

        for k be Nat st k in ( dom xv) holds (xv . k) = (yv . k)

        proof

          let k be Nat;

          assume

           A5: k in ( dom xv);

          x in the carrier of ( product G) & y in the carrier of ( product G);

          then

           A6: x in ( product ( carr G)) & y in ( product ( carr G)) by Th10;

          then

          consider g be Function such that

           A7: x = g & ( dom g) = ( dom ( carr G)) & for i be object st i in ( dom ( carr G)) holds (g . i) in (( carr G) . i) by CARD_3:def 5;

          consider g1 be Function such that

           A8: y = g1 & ( dom g1) = ( dom ( carr G)) & for i be object st i in ( dom ( carr G)) holds (g1 . i) in (( carr G) . i) by A6, CARD_3:def 5;

          

           A9: k in ( dom y) & k in ( dom x) by A7, A8, Lm1, A5, A4;

          per cases ;

            suppose k = i;

            then ((y +* (i,v)) . k) = v & ((x +* (i,v)) . k) = v by A9, FUNCT_7: 31;

            hence (yv . k) = (xv . k) by A2;

          end;

            suppose

             A10: k <> i;

            

             A11: (yv . k) = (y . k) & (xv . k) = (x . k) by A2, A10, FUNCT_7: 32;

            y = (x +* (i,xi)) by A1, Def4;

            hence (yv . k) = (xv . k) by A11, A10, FUNCT_7: 32;

          end;

        end;

        hence (( reproj (i,x)) . v) = (( reproj (i,y)) . v) by A3, FINSEQ_1: 13;

      end;

      hence thesis;

    end;

    theorem :: NDIFF_5:49

    

     Th49: for G be RealNormSpace-Sequence, i,j be Element of ( dom G), x,y be Point of ( product G), xi be Point of (G . i) st y = (( reproj (i,x)) . xi) & i <> j holds (( proj j) . x) = (( proj j) . y)

    proof

      let G be RealNormSpace-Sequence, i,j be Element of ( dom G), x,y be Point of ( product G), xi be Point of (G . i);

      assume

       A1: y = (( reproj (i,x)) . xi) & i <> j;

      reconsider y1 = y as Element of ( product ( carr G)) by Th10;

      

       A2: y = (x +* (i,xi)) by A1, Def4;

      set ix = (i .--> xi);

      

       A3: the carrier of ( product G) = ( product ( carr G)) by Th10;

      (y1 . j) = (x . j) by A2, A1, FUNCT_7: 32;

      then (( proj j) . y) = (x . j) by Def3;

      hence thesis by A3, Def3;

    end;

    theorem :: NDIFF_5:50

    for G be RealNormSpace-Sequence, F be RealNormSpace, i be Element of ( dom G), x be Point of ( product G), xi be Point of (G . i), f be PartFunc of ( product G), F, g be PartFunc of (G . i), F st (( proj i) . x) = xi & g = (f * ( reproj (i,x))) holds ( diff (g,xi)) = ( partdiff (f,x,i))

    proof

      let G be RealNormSpace-Sequence, F be RealNormSpace, i be Element of ( dom G), x be Point of ( product G), xi be Point of (G . i), f be PartFunc of ( product G), F, g be PartFunc of (G . i), F;

      i = ( In (i,( dom G))) by SUBSET_1:def 8;

      hence thesis;

    end;

    theorem :: NDIFF_5:51

    

     Th51: for G be RealNormSpace-Sequence, F be RealNormSpace, f be PartFunc of ( product G), F, x be Point of ( product G), i be set, M be Real, L be Point of ( R_NormSpace_of_BoundedLinearOperators ((G . ( In (i,( dom G)))),F)), p,q be Point of (G . ( In (i,( dom G)))) st i in ( dom G) & (for h be Point of (G . ( In (i,( dom G)))) st h in ].p, q.[ holds ||.(( partdiff (f,(( reproj (( In (i,( dom G))),x)) . h),i)) - L).|| <= M) & (for h be Point of (G . ( In (i,( dom G)))) st h in [.p, q.] holds (( reproj (( In (i,( dom G))),x)) . h) in ( dom f)) & (for h be Point of (G . ( In (i,( dom G)))) st h in [.p, q.] holds f is_partial_differentiable_in ((( reproj (( In (i,( dom G))),x)) . h),i)) holds ||.(((f /. (( reproj (( In (i,( dom G))),x)) . q)) - (f /. (( reproj (( In (i,( dom G))),x)) . p))) - (L . (q - p))).|| <= (M * ||.(q - p).||)

    proof

      let G be RealNormSpace-Sequence, F be RealNormSpace, f be PartFunc of ( product G), F, x be Point of ( product G), i be set, M be Real, L be Point of ( R_NormSpace_of_BoundedLinearOperators ((G . ( In (i,( dom G)))),F)), p,q be Point of (G . ( In (i,( dom G))));

      assume

       A1: i in ( dom G) & (for h be Point of (G . ( In (i,( dom G)))) st h in ].p, q.[ holds ||.(( partdiff (f,(( reproj (( In (i,( dom G))),x)) . h),i)) - L).|| <= M) & (for h be Point of (G . ( In (i,( dom G)))) st h in [.p, q.] holds (( reproj (( In (i,( dom G))),x)) . h) in ( dom f)) & (for h be Point of (G . ( In (i,( dom G)))) st h in [.p, q.] holds f is_partial_differentiable_in ((( reproj (( In (i,( dom G))),x)) . h),i));

      per cases ;

        suppose

         B2: p = q;

        set S = (G . ( In (i,( dom G))));

        reconsider LL = L as Lipschitzian LinearOperator of S, F by LOPBAN_1:def 9;

        

         B3: (L . ( 0. S)) = (LL . ( 0 * ( 0. S))) by RLVECT_1: 10

        .= ( 0 * (LL . ( 0. S))) by LOPBAN_1:def 5

        .= ( 0. F) by RLVECT_1: 10;

        

         B4: ||.(((f /. (( reproj (( In (i,( dom G))),x)) . q)) - (f /. (( reproj (( In (i,( dom G))),x)) . p))) - (L . (q - p))).|| = ||.(( 0. F) - (L . (q - p))).|| by B2, RLVECT_1: 15

        .= ||.(( 0. F) - (L . ( 0. S))).|| by B2, RLVECT_1: 15

        .= ||.( 0. F).|| by B3, RLVECT_1: 13

        .= 0 ;

        (M * ||.(q - p).||) = (M * ||.( 0. S).||) by B2, RLVECT_1: 15

        .= 0 ;

        hence thesis by B4;

      end;

        suppose

         ASM: p <> q;

        set m = ( len G);

        set S = (G . ( In (i,( dom G))));

        set g = (f * ( reproj (( In (i,( dom G))),x)));

         A2:

        now

          let h be object;

          assume

           A3: h in [.p, q.];

          then

          reconsider h1 = h as Point of S;

          

           A4: ( dom ( reproj (( In (i,( dom G))),x))) = the carrier of S by FUNCT_2:def 1;

          (( reproj (( In (i,( dom G))),x)) . h1) in ( dom f) by A1, A3;

          hence h in ( dom g) by A4, FUNCT_1: 11;

        end;

        then

         A5: [.p, q.] c= ( dom g);

         A6:

        now

          let x0 be Point of S;

          assume

           A7: x0 in [.p, q.];

          set y = (( reproj (( In (i,( dom G))),x)) . x0);

          

           A8: (( proj ( In (i,( dom G)))) . y) = x0 by Th46;

          f is_partial_differentiable_in (y,i) by A1, A7;

          hence g is_differentiable_in x0 by A8, Th48;

        end;

        

         X1: ].p, q.[ = { (p + (t * (q - p))) where t be Real : 0 < t & t < 1 } by ASM, LMOPN;

        now

          let z be object;

          assume z in ].p, q.[;

          then

          consider z1 be Real such that

           A9: z = (p + (z1 * (q - p))) & 0 < z1 & z1 < 1 by X1;

          z = (((1 - z1) * p) + (z1 * q)) by A9, Lm2;

          then z in { (((1 - r1) * p) + (r1 * q)) where r1 be Real : 0 <= r1 & r1 <= 1 } by A9;

          hence z in [.p, q.] by RLTOPSP1:def 2;

        end;

        then

         A10: for x be Point of S st x in ].p, q.[ holds g is_differentiable_in x by A6;

        

         A11: for x be Point of S st x in [.p, q.] holds g is_continuous_in x by A6, NDIFF_1: 44;

         A12:

        now

          let h be Point of S;

          set y = (( reproj (( In (i,( dom G))),x)) . h);

          assume h in ].p, q.[;

          then

           A13: ||.(( partdiff (f,y,i)) - L).|| <= M by A1;

          (( proj ( In (i,( dom G)))) . y) = h by Th46;

          hence ||.(( diff (g,h)) - L).|| <= M by A13, Th48;

        end;

        

         A14: p in ( dom g) & q in ( dom g) by A2, RLTOPSP1: 68;

        (f /. (( reproj (( In (i,( dom G))),x)) . p)) = (f /. (( reproj (( In (i,( dom G))),x)) /. p)) & (f /. (( reproj (( In (i,( dom G))),x)) . q)) = (f /. (( reproj (( In (i,( dom G))),x)) /. q));

        then (f /. (( reproj (( In (i,( dom G))),x)) . p)) = (g /. p) & (f /. (( reproj (( In (i,( dom G))),x)) . q)) = (g /. q) by A14, PARTFUN2: 3;

        hence ||.(((f /. (( reproj (( In (i,( dom G))),x)) . q)) - (f /. (( reproj (( In (i,( dom G))),x)) . p))) - (L . (q - p))).|| <= (M * ||.(q - p).||) by A12, Th20, A5, A10, A11;

      end;

    end;

    theorem :: NDIFF_5:52

    

     Th52: for G be RealNormSpace-Sequence, x,y,z,w be Point of ( product G), i be Element of ( dom G), d be Real, p,q,r be Point of (G . i) st ||.(y - x).|| < d & ||.(z - x).|| < d & p = (( proj i) . y) & z = (( reproj (i,y)) . q) & r in [.p, q.] & w = (( reproj (i,y)) . r) holds ||.(w - x).|| < d

    proof

      let G be RealNormSpace-Sequence, x,y,z,w be Point of ( product G), i be Element of ( dom G), d be Real, p,q,r be Point of (G . i);

      assume that

       A1: ||.(y - x).|| < d & ||.(z - x).|| < d and

       A2: p = (( proj i) . y) & z = (( reproj (i,y)) . q) and

       A3: r in [.p, q.] and

       A4: w = (( reproj (i,y)) . r);

      set wx = (w - x);

      set yx = (y - x);

      set zx = (z - x);

      reconsider xi = (( proj i) . x) as Point of (G . i);

      r in { (((1 - t) * p) + (t * q)) where t be Real : 0 <= t & t <= 1 } by A3, RLTOPSP1:def 2;

      then

      consider t be Real such that

       A5: r = (((1 - t) * p) + (t * q)) & 0 <= t & t <= 1;

      

       A6: r = (p + (t * (q - p))) & 0 <= t & t <= 1 by A5, Lm2;

      reconsider wx0 = wx, yx0 = yx, zx0 = zx as Element of ( product ( carr G)) by Th10;

      reconsider Nwx = ( normsequence (G,wx0)) as ( len G) -element FinSequence of REAL ;

      reconsider Nyx = ( normsequence (G,yx0)) as ( len G) -element FinSequence of REAL ;

      reconsider Nzx = ( normsequence (G,zx0)) as ( len G) -element FinSequence of REAL ;

      set tyz = (((1 - t) * yx) + (t * zx));

      reconsider tyz0 = tyz as Element of ( product ( carr G)) by Th10;

      reconsider Ntyz = ( normsequence (G,tyz0)) as ( len G) -element FinSequence of REAL ;

      

       A7: 1 = ((1 - t) + t);

      r = (p + ((t * q) - (t * p))) by A6, RLVECT_1: 34

      .= ((p + ( - (t * p))) + (t * q)) by RLVECT_1:def 3

      .= (((1 * p) - (t * p)) + (t * q)) by RLVECT_1:def 8

      .= (((1 - t) * p) + (t * q)) by RLVECT_1: 35;

      

      then

       A8: (r - xi) = ((((1 - t) * p) + (t * q)) - (1 * xi)) by RLVECT_1:def 8

      .= ((((1 - t) * p) + (t * q)) - (((1 - t) * xi) + (t * xi))) by A7, RLVECT_1:def 6

      .= (((((1 - t) * p) + (t * q)) - (t * xi)) - ((1 - t) * xi)) by RLVECT_1: 27

      .= ((((1 - t) * p) + ((t * q) - (t * xi))) - ((1 - t) * xi)) by RLVECT_1: 28

      .= (((t * q) - (t * xi)) + (((1 - t) * p) - ((1 - t) * xi))) by RLVECT_1:def 3

      .= ((t * (q - xi)) + (((1 - t) * p) - ((1 - t) * xi))) by RLVECT_1: 34

      .= ((t * (q - xi)) + ((1 - t) * (p - xi))) by RLVECT_1: 34;

      reconsider Swx = wx as ( len G) -element FinSequence;

      reconsider Syz = (((1 - t) * yx) + (t * zx)) as ( len G) -element FinSequence;

      

       A9: ( dom Swx) = ( Seg ( len G)) & ( dom Syz) = ( Seg ( len G)) by FINSEQ_1: 89;

      

       A10: for k be Nat st k in ( dom Swx) holds (Swx . k) = (Syz . k)

      proof

        let k be Nat;

        assume k in ( dom Swx);

        then

        reconsider k0 = k as Element of ( dom G) by A9, FINSEQ_1:def 3;

        per cases ;

          suppose

           A11: k = i;

          then (Swx . k) = (( proj i) . wx0) by Def3;

          then

           A12: (Swx . k) = ((( proj i) . w) - (( proj i) . x)) by Th37;

          

           A13: (( proj i) . z) = q by A2, Th46;

          (Syz . k) = (( proj i) . tyz0) by A11, Def3;

          then (Syz . k) = ((( proj i) . ((1 - t) * yx)) + (( proj i) . (t * zx))) by Th35;

          then (Syz . k) = (((1 - t) * (( proj i) . yx)) + (( proj i) . (t * zx))) by Th40;

          then (Syz . k) = (((1 - t) * (( proj i) . yx)) + (t * (( proj i) . zx))) by Th40;

          then (Syz . k) = (((1 - t) * ((( proj i) . y) - (( proj i) . x))) + (t * (( proj i) . zx))) by Th37;

          then (Syz . k) = (((1 - t) * (p - xi)) + (t * (q - xi))) by A2, A13, Th37;

          hence (Swx . k) = (Syz . k) by A12, A8, A4, Th46;

        end;

          suppose k <> i;

          then

           A14: (( proj k0) . y) = (( proj k0) . w) & (( proj k0) . z) = (( proj k0) . y) by A2, A4, Th49;

          (Swx . k) = (( proj k0) . wx0) by Def3;

          then

           A15: (Swx . k) = ((( proj k0) . w) - (( proj k0) . x)) by Th37;

          (Syz . k) = (( proj k0) . tyz0) by Def3

          .= ((( proj k0) . ((1 - t) * yx)) + (( proj k0) . (t * zx))) by Th35

          .= (((1 - t) * (( proj k0) . yx)) + (( proj k0) . (t * zx))) by Th40

          .= (((1 - t) * (( proj k0) . yx)) + (t * (( proj k0) . zx))) by Th40;

          then (Syz . k) = (((1 - t) * ((( proj k0) . y) - (( proj k0) . x))) + (t * (( proj k0) . zx))) by Th37;

          then (Syz . k) = (((1 - t) * ((( proj k0) . y) - (( proj k0) . x))) + (t * ((( proj k0) . y) - (( proj k0) . x)))) by A14, Th37;

          then (Syz . k) = ((((1 - t) * (( proj k0) . y)) - ((1 - t) * (( proj k0) . x))) + (t * ((( proj k0) . y) - (( proj k0) . x)))) by RLVECT_1: 34;

          then (Syz . k) = ((((1 - t) * (( proj k0) . y)) - ((1 - t) * (( proj k0) . x))) + ((t * (( proj k0) . y)) - (t * (( proj k0) . x)))) by RLVECT_1: 34;

          then (Syz . k) = (((((1 - t) * (( proj k0) . y)) - ((1 - t) * (( proj k0) . x))) + (t * (( proj k0) . y))) - (t * (( proj k0) . x))) by RLVECT_1:def 3;

          then (Syz . k) = ((((1 - t) * (( proj k0) . y)) - (((1 - t) * (( proj k0) . x)) - (t * (( proj k0) . y)))) - (t * (( proj k0) . x))) by RLVECT_1: 29;

          then (Syz . k) = ((((1 - t) * (( proj k0) . y)) + ((t * (( proj k0) . y)) + ( - ((1 - t) * (( proj k0) . x))))) - (t * (( proj k0) . x))) by RLVECT_1: 33;

          then (Syz . k) = (((((1 - t) * (( proj k0) . y)) + (t * (( proj k0) . y))) + ( - ((1 - t) * (( proj k0) . x)))) - (t * (( proj k0) . x))) by RLVECT_1:def 3;

          then (Syz . k) = (((((1 - t) + t) * (( proj k0) . y)) + ( - ((1 - t) * (( proj k0) . x)))) - (t * (( proj k0) . x))) by RLVECT_1:def 6;

          then (Syz . k) = (((( proj k0) . y) + ( - ((1 - t) * (( proj k0) . x)))) - (t * (( proj k0) . x))) by RLVECT_1:def 8;

          then (Syz . k) = ((( proj k0) . y) + (( - ((1 - t) * (( proj k0) . x))) - (t * (( proj k0) . x)))) by RLVECT_1: 28;

          then (Syz . k) = ((( proj k0) . y) + ( - ((t * (( proj k0) . x)) + ((1 - t) * (( proj k0) . x))))) by RLVECT_1: 30;

          then (Syz . k) = ((( proj k0) . y) + ( - ((t + (1 - t)) * (( proj k0) . x)))) by RLVECT_1:def 6;

          hence (Swx . k) = (Syz . k) by A15, A14, RLVECT_1:def 8;

        end;

      end;

      

       A16: ( len Nwx) = ( len G) & ( len Ntyz) = ( len G) by CARD_1:def 7;

      for k be Element of NAT st k in ( Seg ( len Nwx)) holds 0 <= (Nwx . k) & (Nwx . k) <= (Ntyz . k)

      proof

        let k be Element of NAT ;

        assume

         A17: k in ( Seg ( len Nwx));

        then

        reconsider k1 = k as Element of ( dom G) by CARD_1:def 7, FINSEQ_1:def 3;

        reconsider wxk = (wx0 . k1) as Element of (G . k1);

        

         A18: (Nwx . k) = ||.wxk.|| by PRVECT_2:def 11;

        (wx0 . k1) = (Syz . k) by A10, A17, A16, A9;

        hence thesis by A18, PRVECT_2:def 11;

      end;

      then

       A19: |.Nwx.| <= |.Ntyz.| by A16, PRVECT_2: 2;

      

       A20: ||.(w - x).|| = (( productnorm G) . wx) by PRVECT_2:def 13;

       ||.(((1 - t) * yx) + (t * zx)).|| = (( productnorm G) . tyz) by PRVECT_2:def 13

      .= |.( normsequence (G,tyz0)).| by PRVECT_2:def 12;

      then

       A21: ||.(w - x).|| <= ||.(((1 - t) * yx) + (t * zx)).|| by A19, A20, PRVECT_2:def 12;

      

       A22: ||.(((1 - t) * yx) + (t * zx)).|| <= (( |.(1 - t).| * ||.(y - x).||) + ( |.t.| * ||.(z - x).||)) by NORMSP_1: 5;

      

       A23: |.(1 - t).| = (1 - t) & |.t.| = t by A5, ABSVALUE:def 1, XREAL_1: 48;

      (( |.(1 - t).| * ||.(y - x).||) + ( |.t.| * ||.(z - x).||)) < d

      proof

        per cases ;

          suppose t = 1 or t = 0 ;

          hence thesis by A1, A23;

        end;

          suppose t <> 1 & t <> 0 ;

          then 0 < t & t < 1 by A5, XXREAL_0: 1;

          then 0 < t & (1 - t) > 0 by XREAL_1: 50;

          then ( |.(1 - t).| * ||.(y - x).||) < ((1 - t) * d) & ( |.t.| * ||.(z - x).||) < (t * d) by A1, A23, XREAL_1: 68;

          then (( |.(1 - t).| * ||.(y - x).||) + ( |.t.| * ||.(z - x).||)) < (((1 - t) * d) + (t * d)) by XREAL_1: 8;

          hence thesis;

        end;

      end;

      then ||.(((1 - t) * yx) + (t * zx)).|| < d by A22, XXREAL_0: 2;

      hence ||.(w - x).|| < d by A21, XXREAL_0: 2;

    end;

    theorem :: NDIFF_5:53

    

     Th53: for G be RealNormSpace-Sequence, S be RealNormSpace, f be PartFunc of ( product G), S, X be Subset of ( product G), x,y,z be Point of ( product G), i be set, p,q be Point of (G . ( In (i,( dom G)))), d,r be Real st i in ( dom G) & X is open & x in X & ||.(y - x).|| < d & ||.(z - x).|| < d & X c= ( dom f) & (for x be Point of ( product G) st x in X holds f is_partial_differentiable_in (x,i)) & (for z be Point of ( product G) st ||.(z - x).|| < d holds z in X) & (for z be Point of ( product G) st ||.(z - x).|| < d holds ||.(( partdiff (f,z,i)) - ( partdiff (f,x,i))).|| <= r) & z = (( reproj (( In (i,( dom G))),y)) . p) & q = (( proj ( In (i,( dom G)))) . y) holds ||.(((f /. z) - (f /. y)) - (( partdiff (f,x,i)) . (p - q))).|| <= ( ||.(p - q).|| * r)

    proof

      let G be RealNormSpace-Sequence, S be RealNormSpace, f be PartFunc of ( product G), S, X be Subset of ( product G), x,y,z be Point of ( product G), i0 be set, p,q be Point of (G . ( In (i0,( dom G)))), d,r be Real;

      assume

       A1: i0 in ( dom G) & X is open & x in X & ||.(y - x).|| < d & ||.(z - x).|| < d & X c= ( dom f) & (for x be Point of ( product G) st x in X holds f is_partial_differentiable_in (x,i0)) & (for z be Point of ( product G) st ||.(z - x).|| < d holds z in X) & (for z be Point of ( product G) st ||.(z - x).|| < d holds ||.(( partdiff (f,z,i0)) - ( partdiff (f,x,i0))).|| <= r) & z = (( reproj (( In (i0,( dom G))),y)) . p) & q = (( proj ( In (i0,( dom G)))) . y);

      set i = ( In (i0,( dom G)));

      

       A2: y = (( reproj (i,y)) . q) by A1, Th47;

       A3:

      now

        let h be Point of (G . i);

        assume h in [.q, p.];

        then ||.((( reproj (i,y)) . h) - x).|| < d by A1, Th52;

        hence (( reproj (i,y)) . h) in ( dom f) by A1;

      end;

       A4:

      now

        let h be Point of (G . i);

        assume h in [.q, p.];

        then ||.((( reproj (i,y)) . h) - x).|| < d by A1, Th52;

        hence f is_partial_differentiable_in ((( reproj (i,y)) . h),i0) by A1;

      end;

      for h be Point of (G . i) st h in ].q, p.[ holds ||.(( partdiff (f,(( reproj (i,y)) . h),i0)) - ( partdiff (f,x,i0))).|| <= r

      proof

        let h be Point of (G . i);

        assume

         A5: h in ].q, p.[;

         ].q, p.[ c= [.q, p.] by XBOOLE_1: 36;

        then ||.((( reproj (i,y)) . h) - x).|| < d by A1, A5, Th52;

        hence ||.(( partdiff (f,(( reproj (i,y)) . h),i0)) - ( partdiff (f,x,i0))).|| <= r by A1;

      end;

      hence thesis by A2, A1, Th51, A3, A4;

    end;

    theorem :: NDIFF_5:54

    

     Th54: for G be RealNormSpace-Sequence, h be FinSequence of ( product G), y,x be Point of ( product G), y0,Z be Element of ( product ( carr G)), j be Element of NAT st y = y0 & Z = ( 0. ( product G)) & ( len h) = (( len G) + 1) & 1 <= j & j <= ( len G) & (for i be Nat st i in ( dom h) holds (h /. i) = (Z +* (y0 | ( Seg ((( len G) + 1) -' i))))) holds (x + (h /. j)) = (( reproj (( In (((( len G) + 1) -' j),( dom G))),(x + (h /. (j + 1))))) . (( proj ( In (((( len G) + 1) -' j),( dom G)))) . (x + y)))

    proof

      let G be RealNormSpace-Sequence, h be FinSequence of ( product G), y,x be Point of ( product G), y0,Z be Element of ( product ( carr G)), j be Element of NAT ;

      assume that

       A1: y = y0 and

       A2: Z = ( 0. ( product G)) and

       A3: ( len h) = (( len G) + 1) and

       A4: 1 <= j & j <= ( len G) and

       A5: for i be Nat st i in ( dom h) holds (h /. i) = (Z +* (y0 | ( Seg ((( len G) + 1) -' i))));

      ( len G) <= ( len h) by A3, NAT_1: 11;

      then j <= ( len h) by A4, XXREAL_0: 2;

      then j in ( Seg ( len h)) by A4;

      then j in ( dom h) by FINSEQ_1:def 3;

      then

       A6: (h /. j) = (Z +* (y0 | ( Seg ((( len G) + 1) -' j)))) by A5;

      1 <= (j + 1) & (j + 1) <= ( len h) by A3, A4, NAT_1: 12, XREAL_1: 6;

      then (j + 1) in ( Seg ( len h));

      then (j + 1) in ( dom h) by FINSEQ_1:def 3;

      then

       A7: (h /. (j + 1)) = (Z +* (y0 | ( Seg ((( len G) + 1) -' (j + 1))))) by A5;

      j in ( Seg ( len G)) by A4;

      then ((( len G) -' j) + 1) in ( Seg ( len G)) by NAT_2: 6;

      then ((( len G) + 1) -' j) in ( Seg ( len G)) by A4, NAT_D: 38;

      then ((( len G) + 1) -' j) in ( dom G) by FINSEQ_1:def 3;

      then

       A8: ( In (((( len G) + 1) -' j),( dom G))) = ((( len G) + 1) -' j) by SUBSET_1:def 8;

      set xh = (x + (h /. (j + 1)));

      reconsider x1 = x, y1 = y as Element of ( product ( carr G)) by Th10;

      reconsider xy = (x + y) as Element of ( product ( carr G)) by Th10;

      xh is Element of ( product ( carr G)) by Th10;

      then

      consider g be Function such that

       A9: xh = g & ( dom g) = ( dom ( carr G)) & for y be object st y in ( dom ( carr G)) holds (g . y) in (( carr G) . y) by CARD_3:def 5;

      

       A10: ( dom xh) = ( dom G) by A9, Lm1;

      (( proj ( In (((( len G) + 1) -' j),( dom G)))) . (x + y)) = (xy . ((( len G) + 1) -' j)) by A8, Def3;

      

      then

       A11: (( reproj (( In (((( len G) + 1) -' j),( dom G))),(x + (h /. (j + 1))))) . (( proj ( In (((( len G) + 1) -' j),( dom G)))) . (x + y))) = (xh +* (( In (((( len G) + 1) -' j),( dom G))),(xy . ((( len G) + 1) -' j)))) by Def4

      .= (xh +* (( In (((( len G) + 1) -' j),( dom G))) .--> (xy . ((( len G) + 1) -' j)))) by A10, FUNCT_7:def 3

      .= (xh +* ( {((( len G) + 1) -' j)} --> (xy . ((( len G) + 1) -' j)))) by A8, FUNCOP_1:def 9;

      reconsider F1 = (x + (h /. j)) as ( len G) -element FinSequence;

      reconsider F2 = (( reproj (( In (((( len G) + 1) -' j),( dom G))),(x + (h /. (j + 1))))) . (( proj ( In (((( len G) + 1) -' j),( dom G)))) . (x + y))) as ( len G) -element FinSequence;

      reconsider h1 = (h /. j) as Element of ( product ( carr G)) by Th10;

      reconsider xh1 = (x + (h /. j)) as Element of ( product ( carr G)) by Th10;

      reconsider h2 = (h /. (j + 1)) as Element of ( product ( carr G)) by Th10;

      

       A12: ( len F1) = ( len G) & ( len F2) = ( len G) by CARD_1:def 7;

      for k be Nat st 1 <= k & k <= ( len F1) holds (F1 . k) = (F2 . k)

      proof

        let k be Nat;

        assume

         A13: 1 <= k & k <= ( len F1);

        then

         A14: k in ( Seg ( len F1));

        then

        reconsider k1 = k as Element of ( dom G) by CARD_1:def 7, FINSEQ_1:def 3;

        (( proj k1) . xh1) = ((( proj k1) . x) + (( proj k1) . (h /. j))) by Th35;

        then

         A15: (F1 . k) = ((( proj k1) . x) + (( proj k1) . (h /. j))) by Def3;

        y0 is Element of the carrier of ( product G) by Th10;

        then

         A16: ( dom y0) = ( Seg ( len G)) by FINSEQ_1: 89;

        

         A17: (( proj k1) . (h /. j)) = (h1 . k) by Def3;

        

         A18: ( dom (y0 | ( Seg ((( len G) + 1) -' j)))) = (( dom y0) /\ ( Seg ((( len G) + 1) -' j))) by RELAT_1: 61;

        

         A19: the carrier of ( product G) = ( product ( carr G)) by Th10;

        per cases ;

          suppose

           A20: not k in ( Seg ((( len G) + 1) -' j));

          then not k in ( dom (y0 | ( Seg ((( len G) + 1) -' j)))) by A18, XBOOLE_0:def 4;

          then (( proj k1) . (h /. j)) = (Z . k) by A17, A6, FUNCT_4: 11;

          then

           A21: (( proj k1) . (h /. j)) = (( proj k1) . ( 0. ( product G))) by A2, Def3;

           not 1 <= k or not k <= ((( len G) + 1) -' j) by A20;

          then not k in ( dom ( {((( len G) + 1) -' j)} --> (xy . ((( len G) + 1) -' j)))) by A13, TARSKI:def 1;

          then ((xh +* ( {((( len G) + 1) -' j)} --> (xy . ((( len G) + 1) -' j)))) . k1) = (xh . k1) by FUNCT_4: 11;

          then

           A22: (F2 . k) = (( proj k1) . (x + (h /. (j + 1)))) by A19, A11, Def3;

          

           A23: (( proj k1) . (h /. (j + 1))) = (h2 . k) by Def3;

          ((( len G) + 1) -' (j + 1)) <= ((( len G) + 1) -' j) by NAT_1: 11, NAT_D: 41;

          then ( Seg ((( len G) + 1) -' (j + 1))) c= ( Seg ((( len G) + 1) -' j)) by FINSEQ_1: 5;

          then not k in ( Seg ((( len G) + 1) -' (j + 1))) by A20;

          then not k in (( dom y0) /\ ( Seg ((( len G) + 1) -' (j + 1)))) by XBOOLE_0:def 4;

          then not k in ( dom (y0 | ( Seg ((( len G) + 1) -' (j + 1))))) by RELAT_1: 61;

          then ((Z +* (y0 | ( Seg ((( len G) + 1) -' (j + 1))))) . k) = (Z . k) by FUNCT_4: 11;

          then (( proj k1) . (h /. (j + 1))) = (( proj k1) . ( 0. ( product G))) by A2, A23, A7, Def3;

          hence (F1 . k) = (F2 . k) by A21, A15, A22, Th35;

        end;

          suppose

           A24: k in ( Seg ((( len G) + 1) -' j));

          then

           A25: k in ( dom (y0 | ( Seg ((( len G) + 1) -' j)))) by A18, A14, A16, A12, XBOOLE_0:def 4;

          then (( proj k1) . (h /. j)) = ((y0 | ( Seg ((( len G) + 1) -' j))) . k) by A17, A6, FUNCT_4: 13;

          then (( proj k1) . (h /. j)) = (y0 . k) by A25, FUNCT_1: 47;

          then

           A26: (( proj k1) . (h /. j)) = (( proj k1) . y) by A1, Def3;

          then

           A27: (F1 . k) = (( proj k1) . (x + y)) by A15, Th35;

          per cases ;

            suppose

             A28: k = ((( len G) + 1) -' j);

            

             A29: k in {k} by TARSKI:def 1;

            then k in ( dom ( {k} --> (xy . k)));

            then ((xh +* ( {k} --> (xy . k))) . k1) = (( {k} --> (xy . k)) . k) by FUNCT_4: 13;

            then (F2 . k) = (xy . k) by A11, A29, A28, FUNCOP_1: 7;

            hence (F1 . k) = (F2 . k) by A27, Def3;

          end;

            suppose

             A30: k <> ((( len G) + 1) -' j);

            then not k in ( dom ( {((( len G) + 1) -' j)} --> (xy . ((( len G) + 1) -' j)))) by TARSKI:def 1;

            then (F2 . k) = (xh . k) by A11, FUNCT_4: 11;

            then

             A31: (F2 . k) = (( proj k1) . (x + (h /. (j + 1)))) by A19, Def3;

            k <= ((( len G) + 1) -' j) by A24, FINSEQ_1: 1;

            then k < ((( len G) + 1) -' j) by A30, XXREAL_0: 1;

            then k <= (((( len G) + 1) -' j) -' 1) by NAT_D: 49;

            then k <= ((( len G) + 1) -' (j + 1)) by NAT_2: 30;

            then k in ( Seg ((( len G) + 1) -' (j + 1))) by A13;

            then

             A32: k in ( dom (y0 | ( Seg ((( len G) + 1) -' (j + 1))))) by A14, A16, A12, RELAT_1: 57;

            (( proj k1) . (h /. (j + 1))) = (h2 . k) by Def3;

            then (( proj k1) . (h /. (j + 1))) = ((y0 | ( Seg ((( len G) + 1) -' (j + 1)))) . k1) by A7, A32, FUNCT_4: 13;

            then (( proj k1) . (h /. (j + 1))) = (y0 . k) by A32, FUNCT_1: 47;

            then (( proj k1) . (h /. (j + 1))) = (( proj k1) . y) by A1, Def3;

            hence (F1 . k) = (F2 . k) by A26, A15, A31, Th35;

          end;

        end;

      end;

      hence thesis by A12;

    end;

    theorem :: NDIFF_5:55

    

     Th55: for G be RealNormSpace-Sequence, h be FinSequence of ( product G), y,x be Point of ( product G), y0,Z be Element of ( product ( carr G)), j be Element of NAT st y = y0 & Z = ( 0. ( product G)) & ( len h) = (( len G) + 1) & 1 <= j & j <= ( len G) & (for i be Nat st i in ( dom h) holds (h /. i) = (Z +* (y0 | ( Seg ((( len G) + 1) -' i))))) holds ((( proj ( In (((( len G) + 1) -' j),( dom G)))) . (x + y)) - (( proj ( In (((( len G) + 1) -' j),( dom G)))) . (x + (h /. (j + 1))))) = (( proj ( In (((( len G) + 1) -' j),( dom G)))) . y)

    proof

      let G be RealNormSpace-Sequence, h be FinSequence of ( product G), y,x be Point of ( product G), y0,Z be Element of ( product ( carr G)), j be Element of NAT ;

      assume that

       A1: y = y0 and

       A2: Z = ( 0. ( product G)) and

       A3: ( len h) = (( len G) + 1) & 1 <= j & j <= ( len G) and

       A4: for i be Nat st i in ( dom h) holds (h /. i) = (Z +* (y0 | ( Seg ((( len G) + 1) -' i))));

      (x + (h /. j)) = (( reproj (( In (((( len G) + 1) -' j),( dom G))),(x + (h /. (j + 1))))) . (( proj ( In (((( len G) + 1) -' j),( dom G)))) . (x + y))) by A1, A2, A3, A4, Th54;

      then (( proj ( In (((( len G) + 1) -' j),( dom G)))) . (x + (h /. j))) = (( proj ( In (((( len G) + 1) -' j),( dom G)))) . (x + y)) by Th46;

      then

       A5: ((( proj ( In (((( len G) + 1) -' j),( dom G)))) . (x + y)) - (( proj ( In (((( len G) + 1) -' j),( dom G)))) . (x + (h /. (j + 1))))) = (( proj ( In (((( len G) + 1) -' j),( dom G)))) . ((x + (h /. j)) - (x + (h /. (j + 1))))) by Th37;

      ((x + (h /. j)) - (x + (h /. (j + 1)))) = ((((h /. j) + x) - x) - (h /. (j + 1))) by RLVECT_1: 27

      .= (((h /. j) + (x - x)) - (h /. (j + 1))) by RLVECT_1: 28

      .= (((h /. j) + ( 0. ( product G))) - (h /. (j + 1))) by RLVECT_1: 15

      .= ((h /. j) - (h /. (j + 1))) by RLVECT_1: 4;

      then

       A6: ((( proj ( In (((( len G) + 1) -' j),( dom G)))) . (x + y)) - (( proj ( In (((( len G) + 1) -' j),( dom G)))) . (x + (h /. (j + 1))))) = ((( proj ( In (((( len G) + 1) -' j),( dom G)))) . (h /. j)) - (( proj ( In (((( len G) + 1) -' j),( dom G)))) . (h /. (j + 1)))) by A5, Th37;

      y0 is Element of the carrier of ( product G) by Th10;

      then

       A7: ( dom y0) = ( Seg ( len G)) by FINSEQ_1: 89;

      j in ( Seg ( len G)) by A3;

      then ((( len G) -' j) + 1) in ( Seg ( len G)) by NAT_2: 6;

      then

       A8: ((( len G) + 1) -' j) in ( Seg ( len G)) by A3, NAT_D: 38;

      

       A9: j < (( len G) + 1) by A3, NAT_1: 13;

      then ((( len G) + 1) -' j) in ( Seg ((( len G) + 1) -' j)) by FINSEQ_1: 3, NAT_D: 36;

      then

       A10: ((( len G) + 1) -' j) in ( dom (y0 | ( Seg ((( len G) + 1) -' j)))) by A7, A8, RELAT_1: 57;

      ((( len G) + 1) -' j) = (((( len G) + 1) -' (j + 1)) + 1) by A9, NAT_2: 7;

      then

       A11: ((( len G) + 1) -' (j + 1)) < ((( len G) + 1) -' j) by NAT_1: 13;

      ( dom (y0 | ( Seg ((( len G) + 1) -' (j + 1))))) c= ( Seg ((( len G) + 1) -' (j + 1))) by RELAT_1: 58;

      then

       A12: not ((( len G) + 1) -' j) in ( dom (y0 | ( Seg ((( len G) + 1) -' (j + 1))))) by A11, FINSEQ_1: 1;

      reconsider h1 = (h /. j) as Element of ( product ( carr G)) by Th10;

      reconsider h2 = (h /. (j + 1)) as Element of ( product ( carr G)) by Th10;

      j in ( Seg ( len h)) by A3, A9;

      then j in ( dom h) by FINSEQ_1:def 3;

      then

       A13: (h /. j) = (Z +* (y0 | ( Seg ((( len G) + 1) -' j)))) by A4;

      ((( len G) + 1) -' j) in ( dom G) by A8, FINSEQ_1:def 3;

      then

       A14: ( In (((( len G) + 1) -' j),( dom G))) = ((( len G) + 1) -' j) by SUBSET_1:def 8;

      

      then

       A15: (( proj ( In (((( len G) + 1) -' j),( dom G)))) . (h /. j)) = (h1 . ((( len G) + 1) -' j)) by Def3

      .= ((y0 | ( Seg ((( len G) + 1) -' j))) . ((( len G) + 1) -' j)) by A10, A13, FUNCT_4: 13

      .= (y0 . ((( len G) + 1) -' j)) by A10, FUNCT_1: 47

      .= (( proj ( In (((( len G) + 1) -' j),( dom G)))) . y) by A1, A14, Def3;

      1 <= (j + 1) & (j + 1) <= ( len h) by A3, NAT_1: 12, XREAL_1: 6;

      then (j + 1) in ( Seg ( len h));

      then (j + 1) in ( dom h) by FINSEQ_1:def 3;

      then

       A16: (h /. (j + 1)) = (Z +* (y0 | ( Seg ((( len G) + 1) -' (j + 1))))) by A4;

      (( proj ( In (((( len G) + 1) -' j),( dom G)))) . (h /. (j + 1))) = (h2 . ((( len G) + 1) -' j)) by A14, Def3

      .= (Z . ((( len G) + 1) -' j)) by A16, A12, FUNCT_4: 11

      .= (( proj ( In (((( len G) + 1) -' j),( dom G)))) . ( 0. ( product G))) by A14, A2, Def3;

      

      hence ((( proj ( In (((( len G) + 1) -' j),( dom G)))) . (x + y)) - (( proj ( In (((( len G) + 1) -' j),( dom G)))) . (x + (h /. (j + 1))))) = (( proj ( In (((( len G) + 1) -' j),( dom G)))) . (y - ( 0. ( product G)))) by A6, A15, Th37

      .= (( proj ( In (((( len G) + 1) -' j),( dom G)))) . y) by RLVECT_1: 13;

    end;

    theorem :: NDIFF_5:56

    

     Th56: for G be RealNormSpace-Sequence, S be RealNormSpace, f be PartFunc of ( product G), S, X be Subset of ( product G), x be Point of ( product G) st X is open & x in X & (for i be set st i in ( dom G) holds f is_partial_differentiable_on (X,i) & (f `partial| (X,i)) is_continuous_on X) holds f is_differentiable_in x & for h be Point of ( product G) holds ex w be FinSequence of S st ( dom w) = ( dom G) & (for i be set st i in ( dom G) holds (w . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . h))) & (( diff (f,x)) . h) = ( Sum w)

    proof

      let G be RealNormSpace-Sequence, S be RealNormSpace, f be PartFunc of ( product G), S, X be Subset of ( product G), x be Point of ( product G);

      assume

       A1: X is open & x in X & (for i be set st i in ( dom G) holds f is_partial_differentiable_on (X,i) & (f `partial| (X,i)) is_continuous_on X);

      set m = ( len G);

      

       A2: ( dom G) = ( Seg m) by FINSEQ_1:def 3;

      reconsider Z0 = ( 0. ( product G)) as Element of ( product ( carr G)) by Th10;

      reconsider x0 = x as Element of ( product ( carr G)) by Th10;

      reconsider x1 = x as ( len G) -element FinSequence;

      reconsider Z1 = ( 0. ( product G)) as ( len G) -element FinSequence;

      consider L be Lipschitzian LinearOperator of ( product G), S such that

       A3: for h be Point of ( product G) holds ex w be FinSequence of S st ( dom w) = ( Seg m) & (for i be Element of NAT st i in ( Seg m) holds (w . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . h))) & (L . h) = ( Sum w) by Lm5;

      

       A4: for h be Point of ( product G) holds ex w be FinSequence of S st ( dom w) = ( dom G) & (for i be set st i in ( dom G) holds (w . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . h))) & (L . h) = ( Sum w)

      proof

        let h be Point of ( product G);

        consider w be FinSequence of S such that

         A5: ( dom w) = ( Seg m) & (for i be Element of NAT st i in ( Seg m) holds (w . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . h))) & (L . h) = ( Sum w) by A3;

        take w;

        thus ( dom w) = ( dom G) by A5, FINSEQ_1:def 3;

        thus thesis by A5, A2;

      end;

      consider d0 be Real such that

       A6: d0 > 0 and

       A7: { y where y be Element of ( product G) : ||.(y - x).|| < d0 } c= X by A1, NDIFF_1: 3;

      set N = { y where y be Element of ( product G) : ||.(y - x).|| < d0 };

      N c= the carrier of ( product G) by A7, XBOOLE_1: 1;

      then

       A8: N is Neighbourhood of x by A6, NFCONT_1:def 1;

      

       A9: 1 <= m by NAT_1: 14;

      then m in ( dom G) by A2;

      then f is_partial_differentiable_on (X,m) by A1;

      then X c= ( dom f);

      then

       A10: N c= ( dom f) by A7;

      deffunc RF( Element of ( product G)) = (((f /. (x + $1)) - (f /. x)) - (L . $1));

      consider R be Function of the carrier of ( product G), the carrier of S such that

       A11: for h be Element of the carrier of ( product G) holds (R . h) = RF(h) from FUNCT_2:sch 4;

      now

        let r0 be Real;

        assume

         A12: r0 > 0 ;

        set r1 = (r0 / 2);

        set r = (r1 / m);

        defpred DSQ[ Nat, Real] means ex k be Element of NAT st $1 = k & 0 < $2 & for q be Element of ( product G) st q in X & ||.(q - x).|| < $2 holds ||.(( partdiff (f,q,k)) - ( partdiff (f,x,k))).|| < r;

        

         A13: for k0 be Nat st k0 in ( Seg m) holds ex d be Element of REAL st DSQ[k0, d]

        proof

          let k0 be Nat;

          assume

           A14: k0 in ( Seg m);

          reconsider k = k0 as Element of NAT by ORDINAL1:def 12;

          (f `partial| (X,k)) is_continuous_on X by A2, A14, A1;

          then

          consider d be Real such that

           A15: 0 < d & for q be Point of ( product G) st q in X & ||.(q - x).|| < d holds ||.(((f `partial| (X,k)) /. q) - ((f `partial| (X,k)) /. x)).|| < r by A12, A1, NFCONT_1: 19;

          reconsider d as Element of REAL by XREAL_0:def 1;

          take d;

          for q be Point of ( product G) st q in X & ||.(q - x).|| < d holds ||.(( partdiff (f,q,k)) - ( partdiff (f,x,k))).|| < r

          proof

            let q be Point of ( product G);

            assume

             A16: q in X & ||.(q - x).|| < d;

            then

             A17: ||.(((f `partial| (X,k)) /. q) - ((f `partial| (X,k)) /. x)).|| < r by A15;

            

             A18: f is_partial_differentiable_on (X,k) by A1, A14, A2;

            then ((f `partial| (X,k)) /. q) = ( partdiff (f,q,k)) by A16, Def9;

            hence ||.(( partdiff (f,q,k)) - ( partdiff (f,x,k))).|| < r by A17, A18, A1, Def9;

          end;

          hence ex k be Element of NAT st k0 = k & 0 < d & for q be Element of ( product G) st q in X & ||.(q - x).|| < d holds ||.(( partdiff (f,q,k)) - ( partdiff (f,x,k))).|| < r by A15;

        end;

        consider Dseq be FinSequence of REAL such that

         A19: ( dom Dseq) = ( Seg m) & for i be Nat st i in ( Seg m) holds DSQ[i, (Dseq . i)] from FINSEQ_1:sch 5( A13);

        m in ( Seg m) by A9;

        then

        reconsider rDseq = ( rng Dseq) as non empty ext-real-membered set by A19, FUNCT_1: 3;

        reconsider rDseq as left_end right_end non empty ext-real-membered set;

        

         A20: ( min rDseq) in ( rng Dseq) by XXREAL_2:def 7;

        reconsider d1 = ( min rDseq) as Real;

        set d = ( min (d0,d1));

        

         A21: d <= d0 & d <= d1 by XXREAL_0: 17;

        consider i1 be object such that

         A22: i1 in ( dom Dseq) & d1 = (Dseq . i1) by A20, FUNCT_1:def 3;

        reconsider i1 as Nat by A22;

        

         A23: ex k be Element of NAT st i1 = k & 0 < (Dseq . i1) & for q be Element of ( product G) st q in X & ||.(q - x).|| < (Dseq . i1) holds ||.(( partdiff (f,q,k)) - ( partdiff (f,x,k))).|| < r by A19, A22;

         A24:

        now

          let q be Element of ( product G);

          assume ||.(q - x).|| < d;

          then ||.(q - x).|| < d0 by A21, XXREAL_0: 2;

          then q in { y where y be Element of ( product G) : ||.(y - x).|| < d0 };

          hence q in X by A7;

        end;

         A25:

        now

          let q be Element of ( product G), i be Element of NAT ;

          assume

           A26: ||.(q - x).|| < d & i in ( Seg m);

          reconsider i0 = i as Nat;

          consider k be Element of NAT such that

           A27: i0 = k & 0 < (Dseq . i0) & for q be Element of ( product G) st q in X & ||.(q - x).|| < (Dseq . i0) holds ||.(( partdiff (f,q,k)) - ( partdiff (f,x,k))).|| < r by A19, A26;

          (Dseq . i0) in ( rng Dseq) by A19, A26, FUNCT_1: 3;

          then d1 <= (Dseq . i0) by XXREAL_2:def 7;

          then d <= (Dseq . i0) by A21, XXREAL_0: 2;

          then ||.(q - x).|| < (Dseq . i0) by A26, XXREAL_0: 2;

          hence ||.(( partdiff (f,q,i)) - ( partdiff (f,x,i))).|| < r by A24, A26, A27;

        end;

        take d;

        thus 0 < d by A6, A22, A23, XXREAL_0: 21;

        thus for y be Point of ( product G) st y <> ( 0. ( product G)) & ||.y.|| < d holds (( ||.y.|| " ) * ||.(R /. y).||) < r0

        proof

          let y be Point of ( product G);

          assume

           A28: y <> ( 0. ( product G)) & ||.y.|| < d;

          set z = (R /. y);

          consider h be FinSequence of ( product G), g be FinSequence of S, Z,y0 be Element of ( product ( carr G)) such that

           A30: y0 = y & Z = ( 0. ( product G)) & ( len h) = (( len G) + 1) & ( len g) = ( len G) & (for i be Nat st i in ( dom h) holds (h /. i) = (Z +* (y0 | ( Seg ((( len G) + 1) -' i))))) & (for i be Nat st i in ( dom g) holds (g /. i) = ((f /. (x + (h /. i))) - (f /. (x + (h /. (i + 1)))))) & (for i be Nat, hi be Point of ( product G) st i in ( dom h) & (h /. i) = hi holds ||.hi.|| <= ||.y.||) & ((f /. (x + y)) - (f /. x)) = ( Sum g) by Th45;

          consider w be FinSequence of S such that

           A31: ( dom w) = ( Seg m) & (for i be Element of NAT st i in ( Seg m) holds (w . i) = (( partdiff (f,x,i)) . (( proj ( In (i,( dom G)))) . y))) & (L . y) = ( Sum w) by A3;

          

           A32: ( dom ( idseq m)) = ( Seg m) & ( rng ( idseq m)) = ( Seg m);

          then

           A33: ( dom ( Rev ( idseq m))) = ( Seg m) & ( rng ( Rev ( idseq m))) = ( Seg m) by FINSEQ_5: 57;

          then

          reconsider Ri = ( Rev ( idseq m)) as Function of ( Seg m), ( Seg m) by FUNCT_2: 1;

          Ri is one-to-one onto by A32, FINSEQ_5: 57;

          then

          reconsider Ri = ( Rev ( idseq m)) as Permutation of ( dom w) by A31;

          

           A34: ( len ( idseq m)) = m & ( len w) = m by A31, A32, FINSEQ_1:def 3;

          ( dom (w * Ri)) = ( dom Ri) by A33, RELAT_1: 27;

          then

           A35: ( dom (w * Ri)) = ( dom ( Rev w)) by A33, A31, FINSEQ_5: 57;

          reconsider wRi = (w * Ri) as FinSequence of S by FINSEQ_2: 47;

          now

            let k be Nat;

            assume

             A36: k in ( dom ( Rev w));

            then

             A37: k in ( dom ( Rev ( idseq m))) by A33, A31, FINSEQ_5: 57;

            then

             A38: 1 <= k & k <= m by A33, FINSEQ_1: 1;

            then

            reconsider mk = (m - k) as Nat by NAT_1: 21;

            reconsider zr0 = 0 as Nat;

             0 <= mk;

            then

             A39: (zr0 + 1) <= ((m - k) + 1) by XREAL_1: 6;

            (k - 1) >= (1 - 1) by A38, XREAL_1: 9;

            then (m - (k - 1)) <= m by XREAL_1: 43;

            then

             A40: (mk + 1) in ( Seg m) by A39;

            (( Rev w) . k) = (w . ((( len ( idseq m)) - k) + 1)) by A34, A36, FINSEQ_5:def 3

            .= (w . (( idseq m) . ((( len ( idseq m)) - k) + 1))) by A40, A34, FINSEQ_2: 49

            .= (w . (( Rev ( idseq m)) . k)) by A37, FINSEQ_5:def 3;

            hence (( Rev w) . k) = (wRi . k) by A36, A35, FUNCT_1: 12;

          end;

          then

           A41: ( Sum ( Rev w)) = ( Sum w) by A35, FINSEQ_1: 13, RLVECT_2: 7;

          deffunc GW( Nat) = ((g /. $1) - (( Rev w) /. $1));

          consider gw be FinSequence of S such that

           A42: ( len gw) = m & for j be Nat st j in ( dom gw) holds (gw . j) = GW(j) from FINSEQ_2:sch 1;

           A43:

          now

            let j be Nat;

            assume j in ( dom g);

            then j in ( Seg m) by A30, FINSEQ_1:def 3;

            then j in ( dom gw) by A42, FINSEQ_1:def 3;

            hence (gw . j) = ((g /. j) - (( Rev w) /. j)) by A42;

          end;

          ( len ( Rev w)) = ( len g) by A30, A34, FINSEQ_5:def 3;

          then ( Sum gw) = (( Sum g) - ( Sum ( Rev w))) by A30, A42, A43, RLVECT_2: 5;

          then

           A44: (R /. y) = ( Sum gw) by A11, A30, A31, A41;

          

           A45: for j be Element of NAT st j in ( dom gw) holds ||.(gw /. j).|| <= ( ||.y.|| * r)

          proof

            let j be Element of NAT ;

            assume

             A46: j in ( dom gw);

            then

             A47: j in ( Seg m) by A42, FINSEQ_1:def 3;

            then

             A48: j in ( dom g) by A30, FINSEQ_1:def 3;

            then

             A49: (g /. j) = ((f /. (x + (h /. j))) - (f /. (x + (h /. (j + 1))))) by A30;

            

             A50: 1 <= j & j <= m by A47, FINSEQ_1: 1;

            then (m + 1) <= (m + j) & (j + 1) <= (m + 1) by XREAL_1: 6;

            then ((m + 1) - j) <= m & 1 <= ((m + 1) - j) by XREAL_1: 19, XREAL_1: 20;

            then ((m + 1) -' j) <= m & 1 <= ((m + 1) -' j) by A50, NAT_D: 37;

            then

             A52: ((m + 1) -' j) in ( Seg m);

            then f is_partial_differentiable_on (X,((m + 1) -' j)) by A1, A2;

            then

             A53: X c= ( dom f) & for x be Element of ( product G) st x in X holds f is_partial_differentiable_in (x,((m + 1) -' j)) by Th24, A1;

            (w /. ((m + 1) -' j)) = (w . ((m + 1) -' j)) by A31, A52, PARTFUN1:def 6;

            then

             A54: (w /. ((m + 1) -' j)) = (( partdiff (f,x,((m + 1) -' j))) . (( proj ( In (((m + 1) -' j),( dom G)))) . y)) by A52, A31;

             A55:

            now

              let j be Element of NAT ;

              reconsider hj = (h /. j) as Element of ( product G);

              assume 1 <= j & j <= (m + 1);

              then

               A56: ||.hj.|| <= ||.y.|| by A30, FINSEQ_3: 25;

              ((x + (h /. j)) - x) = ((h /. j) + (x - x)) by RLVECT_1: 28

              .= ((h /. j) + ( 0. ( product G))) by RLVECT_1: 15;

              then ((x + (h /. j)) - x) = (h /. j) by RLVECT_1: 4;

              hence ||.((x + (h /. j)) - x).|| < d by A56, A28, XXREAL_0: 2;

            end;

            ( Seg m) c= ( Seg (m + 1)) by FINSEQ_1: 5, NAT_1: 11;

            then 1 <= j & j <= (m + 1) by A47, FINSEQ_1: 1;

            then

             A57: ||.((x + (h /. j)) - x).|| < d by A55;

            1 <= (j + 1) by NAT_1: 11;

            then

             A58: ||.((x + (h /. (j + 1))) - x).|| < d by A50, A55, XREAL_1: 6;

            

             A59: (x + (h /. j)) = (( reproj (( In (((m + 1) -' j),( dom G))),(x + (h /. (j + 1))))) . (( proj ( In (((m + 1) -' j),( dom G)))) . (x + y))) by Th54, A30, A50;

            

             A60: ((( proj ( In (((m + 1) -' j),( dom G)))) . (x + y)) - (( proj ( In (((m + 1) -' j),( dom G)))) . (x + (h /. (j + 1))))) = (( proj ( In (((m + 1) -' j),( dom G)))) . y) by Th55, A30, A50;

            for z be Point of ( product G) st ||.(z - x).|| < d holds ||.(( partdiff (f,z,((m + 1) -' j))) - ( partdiff (f,x,((m + 1) -' j)))).|| <= r by A25, A52;

            then

             A61: ||.(((f /. (x + (h /. j))) - (f /. (x + (h /. (j + 1))))) - (( partdiff (f,x,((m + 1) -' j))) . (( proj ( In (((m + 1) -' j),( dom G)))) . y))).|| <= ( ||.(( proj ( In (((m + 1) -' j),( dom G)))) . y).|| * r) by A1, A53, A52, A2, A24, A57, A58, A59, A60, Th53;

            

             A62: ((m + 1) -' j) = ((m + 1) - j) by A50, NAT_1: 12, XREAL_1: 233;

            j in ( Seg ( len ( Rev w))) by A42, A46, A34, FINSEQ_1:def 3, FINSEQ_5:def 3;

            then

             A63: j in ( dom ( Rev w)) by FINSEQ_1:def 3;

            

            then

             A64: (( Rev w) /. j) = (( Rev w) . j) by PARTFUN1:def 6

            .= (w . ((m - j) + 1)) by A34, A63, FINSEQ_5:def 3

            .= (w /. ((m + 1) -' j)) by A62, A52, A31, PARTFUN1:def 6;

            

             A65: (gw /. j) = (gw . j) by A46, PARTFUN1:def 6

            .= (((f /. (x + (h /. j))) - (f /. (x + (h /. (j + 1))))) - (( partdiff (f,x,((m + 1) -' j))) . (( proj ( In (((m + 1) -' j),( dom G)))) . y))) by A54, A49, A64, A48, A43;

            ( ||.(( proj ( In (((m + 1) -' j),( dom G)))) . y).|| * r) <= ( ||.y.|| * r) by A12, Th31, XREAL_1: 64;

            hence ||.(gw /. j).|| <= ( ||.y.|| * r) by A65, A61, XXREAL_0: 2;

          end;

          defpred YSQ[ set, set] means $2 = ||.(gw /. $1).||;

          

           A66: for k be Nat st k in ( Seg m) holds ex x be Element of REAL st YSQ[k, x];

          consider yseq be FinSequence of REAL such that

           A67: ( dom yseq) = ( Seg m) & for i be Nat st i in ( Seg m) holds YSQ[i, (yseq . i)] from FINSEQ_1:sch 5( A66);

          

           A68: ( len gw) = ( len yseq) by A42, A67, FINSEQ_1:def 3;

           A69:

          now

            let i be Element of NAT ;

            assume i in ( dom gw);

            then i in ( Seg m) by A42, FINSEQ_1:def 3;

            hence (yseq . i) = ||.(gw /. i).|| by A67;

          end;

          reconsider yseq as Element of ( REAL m) by A68, A42, FINSEQ_2: 92;

          

           A70: ||.( Sum gw).|| <= ( Sum yseq) by A69, A68, Th7;

          reconsider yr = ( ||.y.|| * r) as Element of REAL by XREAL_0:def 1;

          for j be Nat st j in ( Seg m) holds (yseq . j) <= ((m |-> yr) . j)

          proof

            let j be Nat;

            assume

             A71: j in ( Seg m);

            then j in ( dom gw) by A42, FINSEQ_1:def 3;

            then

             A72: ||.(gw /. j).|| <= ( ||.y.|| * r) by A45;

            (yseq . j) = ||.(gw /. j).|| by A67, A71;

            hence (yseq . j) <= ((m |-> yr) . j) by A71, A72, FINSEQ_2: 57;

          end;

          then ( Sum yseq) <= ( Sum (m |-> yr)) by RVSUM_1: 82;

          then ( Sum yseq) <= (m * ( ||.y.|| * r)) by RVSUM_1: 80;

          then ||.z.|| <= (m * ( ||.y.|| * r)) by A44, A70, XXREAL_0: 2;

          then ( ||.z.|| * ( ||.y.|| " )) <= (((m * ||.y.||) * r) * ( ||.y.|| " )) by XREAL_1: 64;

          then ( ||.z.|| * ( ||.y.|| " )) <= (m * ((r * ||.y.||) * ( ||.y.|| " )));

          then (( ||.y.|| " ) * ||.z.||) <= (m * r) by A28, NORMSP_0:def 5, XCMPLX_1: 203;

          then

           A73: (( ||.y.|| " ) * ||.z.||) <= r1 by XCMPLX_1: 87;

          r1 < r0 by A12, XREAL_1: 216;

          hence (( ||.y.|| " ) * ||.z.||) < r0 by A73, XXREAL_0: 2;

        end;

      end;

      then

      reconsider R as RestFunc of ( product G), S by NDIFF_1: 23;

      reconsider L as Point of ( R_NormSpace_of_BoundedLinearOperators (( product G),S)) by LOPBAN_1:def 9;

      

       A74: for y be Point of ( product G) st y in N holds ((f /. y) - (f /. x)) = ((L . (y - x)) + (R /. (y - x)))

      proof

        let y be Point of ( product G);

        assume y in N;

        (y - x) in the carrier of ( product G);

        then (y - x) in ( dom R) by PARTFUN1:def 2;

        then (R /. (y - x)) = (R . (y - x)) by PARTFUN1:def 6;

        then (R /. (y - x)) = (((f /. (x + (y - x))) - (f /. x)) - (L . (y - x))) by A11;

        

        hence ((L . (y - x)) + (R /. (y - x))) = (((f /. (x + (y - x))) - (f /. x)) - ((L . (y - x)) - (L . (y - x)))) by RLVECT_1: 29

        .= (((f /. (x + (y - x))) - (f /. x)) - ( 0. S)) by RLVECT_1: 5

        .= ((f /. (x + (y - x))) - (f /. x)) by RLVECT_1: 13

        .= ((f /. (y - (x - x))) - (f /. x)) by RLVECT_1: 29

        .= ((f /. (y - ( 0. ( product G)))) - (f /. x)) by RLVECT_1: 5

        .= ((f /. y) - (f /. x)) by RLVECT_1: 13;

      end;

      then f is_differentiable_in x by A10, A8, NDIFF_1:def 6;

      then ( diff (f,x)) = L by A8, A10, A74, NDIFF_1:def 7;

      hence thesis by A4, A74, A10, A8, NDIFF_1:def 6;

    end;

    theorem :: NDIFF_5:57

    for G be RealNormSpace-Sequence, F be RealNormSpace, f be PartFunc of ( product G), F, X be Subset of ( product G) st X is open holds (for i be set st i in ( dom G) holds f is_partial_differentiable_on (X,i) & (f `partial| (X,i)) is_continuous_on X) iff f is_differentiable_on X & (f `| X) is_continuous_on X

    proof

      let G be RealNormSpace-Sequence, F be RealNormSpace, f be PartFunc of ( product G), F, X be Subset of ( product G);

      assume

       A1: X is open;

      set m = ( len G);

      

       A2: ( dom G) = ( Seg m) by FINSEQ_1:def 3;

      hereby

        assume

         A3: for i be set st i in ( dom G) holds f is_partial_differentiable_on (X,i) & (f `partial| (X,i)) is_continuous_on X;

         A4:

        now

          let i be Element of NAT ;

          assume 1 <= i & i <= m;

          then i in ( Seg m);

          then (f `partial| (X,i)) is_continuous_on X by A3, A2;

          hence X c= ( dom (f `partial| (X,i))) & for y0 be Point of ( product G), r be Real st y0 in X & 0 < r holds ex s be Real st 0 < s & for y1 be Point of ( product G) st y1 in X & ||.(y1 - y0).|| < s holds ||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r by NFCONT_1: 19;

        end;

        

         A5: 1 <= m by NAT_1: 14;

        then m in ( dom G) by A2;

        then

         A6: f is_partial_differentiable_on (X,m) by A3;

        for x be Point of ( product G) st x in X holds f is_differentiable_in x by A1, A3, Th56;

        hence

         A7: f is_differentiable_on X by A1, A6, NDIFF_1: 31;

        then

         A8: ( dom (f `| X)) = X by NDIFF_1:def 9;

        for y0 be Point of ( product G), r be Real st y0 in X & 0 < r holds ex s be Real st 0 < s & for y1 be Point of ( product G) st y1 in X & ||.(y1 - y0).|| < s holds ||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r

        proof

          let y0 be Point of ( product G), r be Real;

          assume

           A9: y0 in X & 0 < r;

          defpred P[ Nat, Real] means for i be Element of NAT st i = $1 holds ( 0 < $2 & for y1 be Point of ( product G) st y1 in X & ||.(y1 - y0).|| < $2 holds ||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < (r / (2 * m)));

           A10:

          now

            let i be Nat;

            reconsider j = i as Element of NAT by ORDINAL1:def 12;

            assume i in ( Seg m);

            then 1 <= j & j <= m by FINSEQ_1: 1;

            then

            consider s be Real such that

             A11: 0 < s & for y1 be Point of ( product G) st y1 in X & ||.(y1 - y0).|| < s holds ||.(((f `partial| (X,j)) /. y1) - ((f `partial| (X,j)) /. y0)).|| < (r / (2 * m)) by A9, A4;

            reconsider s as Element of REAL by XREAL_0:def 1;

            take s;

            thus P[i, s] by A11;

          end;

          consider S be FinSequence of REAL such that

           A12: ( dom S) = ( Seg m) & for i be Nat st i in ( Seg m) holds P[i, (S . i)] from FINSEQ_1:sch 5( A10);

          take s = ( min S);

          

           A13: ( len S) = m by A12, FINSEQ_1:def 3;

          then ( min_p S) in ( dom S) by RFINSEQ2:def 2;

          hence s > 0 by A12;

          let y1 be Point of ( product G);

          assume

           A14: y1 in X & ||.(y1 - y0).|| < s;

          reconsider DD = (( diff (f,y1)) - ( diff (f,y0))) as Lipschitzian LinearOperator of ( product G), F by LOPBAN_1:def 9;

          

           A15: ( upper_bound ( PreNorms DD)) = ||.(( diff (f,y1)) - ( diff (f,y0))).|| by LOPBAN_1: 30;

          now

            let mt be Real;

            assume mt in ( PreNorms DD);

            then

            consider t be VECTOR of ( product G) such that

             A16: mt = ||.(DD . t).|| & ||.t.|| <= 1;

            consider w0 be FinSequence of F such that

             A17: ( dom w0) = ( dom G) & (for i be set st i in ( dom G) holds (w0 . i) = (( partdiff (f,y0,i)) . (( proj ( In (i,( dom G)))) . t))) & (( diff (f,y0)) . t) = ( Sum w0) by A1, A3, Th56, A9;

            reconsider Sw0 = ( Sum w0) as Point of F;

            consider w1 be FinSequence of F such that

             A18: ( dom w1) = ( dom G) & (for i be set st i in ( dom G) holds (w1 . i) = (( partdiff (f,y1,i)) . (( proj ( In (i,( dom G)))) . t))) & (( diff (f,y1)) . t) = ( Sum w1) by A1, A3, Th56, A14;

            reconsider Sw1 = ( Sum w1) as Point of F;

            deffunc F( set) = ((w1 /. $1) - (w0 /. $1));

            consider w2 be FinSequence of F such that

             A19: ( len w2) = m & for i be Nat st i in ( dom w2) holds (w2 . i) = F(i) from FINSEQ_2:sch 1;

            

             A20: ( len w1) = m & ( len w0) = m by A2, A17, A18, FINSEQ_1:def 3;

            now

              let i be Nat;

              assume i in ( dom w1);

              then i in ( dom w2) by A19, A2, A18, FINSEQ_1:def 3;

              hence (w2 . i) = F(i) by A19;

            end;

            then ( Sum w2) = (( Sum w1) - ( Sum w0)) by A19, A20, RLVECT_2: 5;

            then

             A21: mt = ||.( Sum w2).|| by A16, A18, A17, LOPBAN_1: 40;

            deffunc F( Nat) = ( In ( ||.(w2 /. $1).||, REAL ));

            consider ys be FinSequence of REAL such that

             A22: ( len ys) = m & for j be Nat st j in ( dom ys) holds (ys . j) = F(j) from FINSEQ_2:sch 1;

             A23:

            now

              let i be Element of NAT ;

              assume i in ( dom w2);

              then i in ( Seg m) by A19, FINSEQ_1:def 3;

              then i in ( dom ys) by A22, FINSEQ_1:def 3;

              

              hence (ys . i) = F(i) by A22

              .= ||.(w2 /. i).||;

            end;

            then

             A24: ||.( Sum w2).|| <= ( Sum ys) by A19, A22, Th7;

            reconsider rm = (r / (2 * m)) as Element of REAL by XREAL_0:def 1;

            deffunc F( Nat) = rm;

            consider rs be FinSequence of REAL such that

             A25: ( len rs) = m & for j be Nat st j in ( dom rs) holds (rs . j) = F(j) from FINSEQ_2:sch 1;

            

             A26: ( dom rs) = ( Seg m) by A25, FINSEQ_1:def 3;

            now

              let a be object;

              assume a in ( rng rs);

              then

              consider b be object such that

               A27: b in ( dom rs) & a = (rs . b) by FUNCT_1:def 3;

              reconsider b as Nat by A27;

              (rs . b) = rm by A27, A25;

              hence a in {rm} by A27, TARSKI:def 1;

            end;

            then

             A28: ( rng rs) c= {rm};

            now

              let a be object;

              assume a in {rm};

              then

               A29: a = rm by TARSKI:def 1;

              

               A30: 1 in ( dom rs) by A5, A26;

              then a = (rs . 1) by A29, A25;

              hence a in ( rng rs) by A30, FUNCT_1: 3;

            end;

            then {rm} c= ( rng rs);

            then rs = (m |-> (r / (2 * m))) by A26, A28, XBOOLE_0:def 10, FUNCOP_1: 9;

            

            then ( Sum rs) = (m * (r / (2 * m))) by RVSUM_1: 80

            .= (m * ((r / 2) / m)) by XCMPLX_1: 78;

            then

             A31: ( Sum rs) = (r / 2) by XCMPLX_1: 87;

            now

              let i be Element of NAT ;

              assume i in ( dom ys);

              then

               A32: i in ( Seg m) by A22, FINSEQ_1:def 3;

              then

               A33: i in ( dom w2) & i in ( dom w1) & i in ( dom w0) by A17, A18, A19, FINSEQ_1:def 3;

              then

               A34: (ys . i) = ||.(w2 /. i).|| & (w2 /. i) = (w2 . i) by A23, PARTFUN1:def 6;

              

               A35: i in ( dom rs) by A25, A32, FINSEQ_1:def 3;

              reconsider p1 = ( partdiff (f,y1,i)), p0 = ( partdiff (f,y0,i)) as Lipschitzian LinearOperator of (G . ( In (i,( dom G)))), F by LOPBAN_1:def 9;

              reconsider P1 = (p1 . (( proj ( In (i,( dom G)))) . t)) as VECTOR of F;

              reconsider P0 = (p0 . (( proj ( In (i,( dom G)))) . t)) as VECTOR of F;

              (w0 /. i) = (w0 . i) & (w1 /. i) = (w1 . i) by A33, PARTFUN1:def 6;

              then (w0 /. i) = P0 & (w1 /. i) = P1 by A2, A17, A18, A32;

              then

               A36: (w2 . i) = (P1 - P0) by A33, A19;

              1 <= i & i <= ( len S) by A13, A32, FINSEQ_1: 1;

              then

               A37: s <= (S . i) & f is_partial_differentiable_on (X,i) by A2, A32, A3, RFINSEQ2: 2;

              then ||.(y1 - y0).|| < (S . i) by A14, XXREAL_0: 2;

              then ||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < (r / (2 * m)) by A12, A32, A14;

              then ||.(( partdiff (f,y1,i)) - ((f `partial| (X,i)) /. y0)).|| < (r / (2 * m)) by Def9, A14, A37;

              then

               A38: ||.(( partdiff (f,y1,i)) - ( partdiff (f,y0,i))).|| < (r / (2 * m)) by Def9, A9, A37;

              reconsider PP = (( partdiff (f,y1,i)) - ( partdiff (f,y0,i))) as Lipschitzian LinearOperator of (G . ( In (i,( dom G)))), F by LOPBAN_1:def 9;

              

               A39: ( upper_bound ( PreNorms PP)) = ||.(( partdiff (f,y1,i)) - ( partdiff (f,y0,i))).|| by LOPBAN_1: 30;

              reconsider pt = (( proj ( In (i,( dom G)))) . t) as VECTOR of (G . ( In (i,( dom G))));

              

               A40: (PP . pt) = (P1 - P0) by LOPBAN_1: 40;

               ||.pt.|| <= ||.t.|| by Th31;

              then ||.pt.|| <= 1 by A16, XXREAL_0: 2;

              then ||.(PP . pt).|| in ( PreNorms PP) & ( PreNorms PP) is non empty bounded_above by LOPBAN_1: 27;

              then ||.(PP . pt).|| <= ( upper_bound ( PreNorms PP)) by SEQ_4:def 1;

              then ||.(P1 - P0).|| <= (r / (2 * m)) by A40, A38, A39, XXREAL_0: 2;

              hence (ys . i) <= (rs . i) by A34, A25, A35, A36;

            end;

            then ( Sum ys) <= (r / 2) by A31, A25, A22, INTEGRA5: 3;

            hence mt <= (r / 2) by A21, A24, XXREAL_0: 2;

          end;

          then ||.(( diff (f,y1)) - ( diff (f,y0))).|| <= (r / 2) & (r / 2) < r by A15, A9, SEQ_4: 45, XREAL_1: 216;

          then ||.(( diff (f,y1)) - ( diff (f,y0))).|| < r by XXREAL_0: 2;

          then ||.(( diff (f,y1)) - ((f `| X) /. y0)).|| < r by A7, A9, NDIFF_1:def 9;

          hence ||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r by A7, A14, NDIFF_1:def 9;

        end;

        hence (f `| X) is_continuous_on X by A8, NFCONT_1: 19;

      end;

      assume

       A41: f is_differentiable_on X & (f `| X) is_continuous_on X;

      then

       A42: X c= ( dom f) & for x be Point of ( product G) st x in X holds f is_differentiable_in x by A1, NDIFF_1: 31;

      thus for i be set st i in ( dom G) holds f is_partial_differentiable_on (X,i) & (f `partial| (X,i)) is_continuous_on X

      proof

        let i be set;

        assume i in ( dom G);

        then

        reconsider i0 = i as Element of NAT ;

        now

          let x be Point of ( product G);

          assume x in X;

          then f is_differentiable_in x by A41, A1, NDIFF_1: 31;

          hence f is_partial_differentiable_in (x,i) & ( partdiff (f,x,i)) = (( diff (f,x)) * ( reproj (( In (i,( dom G))),( 0. ( product G))))) by Th41;

        end;

        then for x be Point of ( product G) st x in X holds f is_partial_differentiable_in (x,i);

        hence

         A44: f is_partial_differentiable_on (X,i) by A1, Th32, A42;

        then

         A45: ( dom (f `partial| (X,i))) = X by Def9;

        for y0 be Point of ( product G), r be Real st y0 in X & 0 < r holds ex s be Real st 0 < s & for y1 be Point of ( product G) st y1 in X & ||.(y1 - y0).|| < s holds ||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r

        proof

          let y0 be Point of ( product G), r be Real;

          assume

           A46: y0 in X & 0 < r;

          then

          consider s be Real such that

           A47: 0 < s & for y1 be Point of ( product G) st y1 in X & ||.(y1 - y0).|| < s holds ||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r by A41, NFCONT_1: 19;

          take s;

          thus 0 < s by A47;

          let y1 be Point of ( product G);

          assume

           A48: y1 in X & ||.(y1 - y0).|| < s;

          then ||.(((f `| X) /. y1) - ((f `| X) /. y0)).|| < r by A47;

          then ||.(( diff (f,y1)) - ((f `| X) /. y0)).|| < r by A48, A41, NDIFF_1:def 9;

          then

           A49: ||.(( diff (f,y1)) - ( diff (f,y0))).|| < r by A46, A41, NDIFF_1:def 9;

          f is_differentiable_in y1 & f is_differentiable_in y0 by A41, A1, A48, A46, NDIFF_1: 31;

          then

           A50: ( partdiff (f,y1,i)) = (( diff (f,y1)) * ( reproj (( In (i,( dom G))),( 0. ( product G))))) & ( partdiff (f,y0,i)) = (( diff (f,y0)) * ( reproj (( In (i,( dom G))),( 0. ( product G))))) by Th41;

          reconsider PP = (( partdiff (f,y1,i)) - ( partdiff (f,y0,i))) as Lipschitzian LinearOperator of (G . ( In (i,( dom G)))), F by LOPBAN_1:def 9;

          reconsider DD = (( diff (f,y1)) - ( diff (f,y0))) as Lipschitzian LinearOperator of ( product G), F by LOPBAN_1:def 9;

          

           A51: ( upper_bound ( PreNorms PP)) = ||.(( partdiff (f,y1,i)) - ( partdiff (f,y0,i))).|| & ( upper_bound ( PreNorms DD)) = ||.(( diff (f,y1)) - ( diff (f,y0))).|| by LOPBAN_1: 30;

          

           A52: ( PreNorms PP) is bounded_above & ( PreNorms DD) is bounded_above by LOPBAN_1: 28;

          now

            let a be object;

            assume a in ( PreNorms PP);

            then

            consider t be VECTOR of (G . ( In (i,( dom G)))) such that

             A53: a = ||.(PP . t).|| & ||.t.|| <= 1;

            

             A54: ( dom ( reproj (( In (i,( dom G))),( 0. ( product G))))) = the carrier of (G . ( In (i,( dom G)))) by FUNCT_2:def 1;

            reconsider tm = (( reproj (( In (i,( dom G))),( 0. ( product G)))) . t) as Point of ( product G);

            

             A55: ||.tm.|| <= 1 by A53, Th21;

            (( partdiff (f,y1,i)) . t) = (( diff (f,y1)) . tm) & (( partdiff (f,y0,i)) . t) = (( diff (f,y0)) . tm) by A54, A50, FUNCT_1: 13;

            then ||.(PP . t).|| = ||.((( diff (f,y1)) . tm) - (( diff (f,y0)) . tm)).|| by LOPBAN_1: 40;

            then ||.(PP . t).|| = ||.(DD . tm).|| by LOPBAN_1: 40;

            hence a in ( PreNorms DD) by A53, A55;

          end;

          then ( PreNorms PP) c= ( PreNorms DD);

          then ||.(( partdiff (f,y1,i)) - ( partdiff (f,y0,i))).|| <= ||.(( diff (f,y1)) - ( diff (f,y0))).|| by A52, A51, SEQ_4: 48;

          then ||.(( partdiff (f,y1,i)) - ( partdiff (f,y0,i))).|| < r by A49, XXREAL_0: 2;

          then ||.(( partdiff (f,y1,i)) - ((f `partial| (X,i)) /. y0)).|| < r by Def9, A46, A44;

          hence ||.(((f `partial| (X,i)) /. y1) - ((f `partial| (X,i)) /. y0)).|| < r by Def9, A48, A44;

        end;

        hence (f `partial| (X,i)) is_continuous_on X by A45, NFCONT_1: 19;

      end;

    end;