Article NDIFF_1, MML version 4.99.1005

:: NDIFF_1:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of b1
for b3, b4 being Neighbourhood of b2 holds
ex b5 being Neighbourhood of b2 st
   b5 c= b3 & b5 c= b4;

:: NDIFF_1:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of b1
   st b2 is open(b1)
for b3 being Element of the carrier of b1
      st b3 in b2
   holds ex b4 being Neighbourhood of b3 st
      b4 c= b2;

:: NDIFF_1:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of b1
   st b2 is open(b1)
for b3 being Element of the carrier of b1
      st b3 in b2
   holds ex b4 being Element of REAL st
      0 < b4 &
       {b5 where b5 is Element of the carrier of b1: ||.b5 - b3.|| < b4} c= b2;

:: NDIFF_1:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of b1
      st for b3 being Element of the carrier of b1
              st b3 in b2
           holds ex b4 being Neighbourhood of b3 st
              b4 c= b2
   holds b2 is open(b1);

:: NDIFF_1:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of b1 holds
      for b3 being Element of the carrier of b1
            st b3 in b2
         holds ex b4 being Neighbourhood of b3 st
            b4 c= b2
   iff
      b2 is open(b1);

:: NDIFF_1:attrnot 1 => NDIFF_1:attr 1
definition
  let a1 be ZeroStr;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  attr a2 is being_not_0 means
    rng a2 c= (the carrier of a1) \ {0. a1};
end;

:: NDIFF_1:dfs 1
definiens
  let a1 be ZeroStr;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
     a2 is being_not_0
it is sufficient to prove
  thus rng a2 c= (the carrier of a1) \ {0. a1};

:: NDIFF_1:def 1
theorem
for b1 being ZeroStr
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is being_not_0(b1)
   iff
      rng b2 c= (the carrier of b1) \ {0. b1};

:: NDIFF_1:prednot 1 => NDIFF_1:attr 1
notation
  let a1 be ZeroStr;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  synonym a2 is_not_0 for being_not_0;
end;

:: NDIFF_1:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is being_not_0(b1)
   iff
      for b3 being set
            st b3 in NAT
         holds b2 . b3 <> 0. b1;

:: NDIFF_1:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is being_not_0(b1)
   iff
      for b3 being Element of NAT holds
         b2 . b3 <> 0. b1;

:: NDIFF_1:funcnot 1 => NDIFF_1:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  let a3 be Function-like quasi_total Relation of NAT,REAL;
  func A3 (#) A2 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    for b1 being Element of NAT holds
       it . b1 = (a3 . b1) * (a2 . b1);
end;

:: NDIFF_1:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b4 = b3 (#) b2
   iff
      for b5 being Element of NAT holds
         b4 . b5 = (b3 . b5) * (b2 . b5);

:: NDIFF_1:funcnot 2 => NDIFF_1:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Element of the carrier of a1;
  let a3 be Function-like quasi_total Relation of NAT,REAL;
  func A3 * A2 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    for b1 being Element of NAT holds
       it . b1 = (a3 . b1) * a2;
end;

:: NDIFF_1:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b4 = b3 * b2
   iff
      for b5 being Element of NAT holds
         b4 . b5 = (b3 . b5) * b2;

:: NDIFF_1:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Function-like quasi_total Relation of NAT,REAL holds
(b3 + b4) (#) b2 = (b3 (#) b2) + (b4 (#) b2);

:: NDIFF_1:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 (#) (b3 + b4) = (b2 (#) b3) + (b2 (#) b4);

:: NDIFF_1:th 10
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,REAL holds
   b1 * (b4 (#) b3) = b4 (#) (b1 * b3);

:: NDIFF_1:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3, b4 being Function-like quasi_total Relation of NAT,REAL holds
(b3 - b4) (#) b2 = (b3 (#) b2) - (b4 (#) b2);

:: NDIFF_1:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 (#) (b3 - b4) = (b2 (#) b3) - (b2 (#) b4);

:: NDIFF_1:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent & b3 is convergent(b1)
   holds b2 (#) b3 is convergent(b1);

:: NDIFF_1:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent & b3 is convergent(b1)
   holds lim (b2 (#) b3) = (lim b2) * lim b3;

:: NDIFF_1:th 15
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
(b3 + b4) ^\ b1 = (b3 ^\ b1) + (b4 ^\ b1);

:: NDIFF_1:th 16
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
(b3 - b4) ^\ b1 = (b3 ^\ b1) - (b4 ^\ b1);

:: NDIFF_1:th 17
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st b3 is being_not_0(b2)
   holds b3 ^\ b1 is being_not_0(b2);

:: NDIFF_1:th 18
theorem
for b1 being Element of NAT
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
   b3 ^\ b1 is subsequence of b3;

:: NDIFF_1:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is constant & b3 is subsequence of b2
   holds b3 is constant;

:: NDIFF_1:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is constant & b3 is subsequence of b2
   holds b2 = b3;

:: NDIFF_1:attrnot 2 => NDIFF_1:attr 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  attr a2 is convergent_to_0 means
    a2 is being_not_0(a1) & a2 is convergent(a1) & lim a2 = 0. a1;
end;

:: NDIFF_1:dfs 4
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
     a2 is convergent_to_0
it is sufficient to prove
  thus a2 is being_not_0(a1) & a2 is convergent(a1) & lim a2 = 0. a1;

:: NDIFF_1:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is convergent_to_0(b1)
   iff
      b2 is being_not_0(b1) & b2 is convergent(b1) & lim b2 = 0. b1;

:: NDIFF_1:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is constant
   holds b2 is convergent(b1) &
    (for b3 being Element of NAT holds
       lim b2 = b2 . b3);

:: NDIFF_1:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st 0 < b4 &
         (for b5 being Element of NAT holds
            b2 . b5 = (1 / (b5 + b4)) * b3)
   holds b2 is convergent(b1);

:: NDIFF_1:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st 0 < b4 &
         (for b5 being Element of NAT holds
            b2 . b5 = (1 / (b5 + b4)) * b3)
   holds lim b2 = 0. b1;

:: NDIFF_1:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
for b3 being Element of the carrier of b1
      st b3 <> 0. b1
   holds b2 * b3 is convergent_to_0(b1);

:: NDIFF_1:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of b1 holds
      for b3 being Element of the carrier of b1 holds
            b3 in b2
         iff
            b3 in the carrier of b1
   iff
      b2 = the carrier of b1;

:: NDIFF_1:exreg 1
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster Relation-like Function-like non empty total quasi_total convergent_to_0 Relation of NAT,the carrier of a1;
end;

:: NDIFF_1:exreg 2
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster Relation-like Function-like constant non empty total quasi_total Relation of NAT,the carrier of a1;
end;

:: NDIFF_1:attrnot 3 => NDIFF_1:attr 3
definition
  let a1, a2 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
  attr a3 is REST-like means
    a3 is total(the carrier of a1, the carrier of a2) &
     (for b1 being Function-like quasi_total convergent_to_0 Relation of NAT,the carrier of a1 holds
        ||.b1.|| " (#) (a3 * b1) is convergent(a2) &
         lim (||.b1.|| " (#) (a3 * b1)) = 0. a2);
end;

:: NDIFF_1:dfs 5
definiens
  let a1, a2 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is REST-like
it is sufficient to prove
  thus a3 is total(the carrier of a1, the carrier of a2) &
     (for b1 being Function-like quasi_total convergent_to_0 Relation of NAT,the carrier of a1 holds
        ||.b1.|| " (#) (a3 * b1) is convergent(a2) &
         lim (||.b1.|| " (#) (a3 * b1)) = 0. a2);

:: NDIFF_1:def 5
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2 holds
      b3 is REST-like(b1, b2)
   iff
      b3 is total(the carrier of b1, the carrier of b2) &
       (for b4 being Function-like quasi_total convergent_to_0 Relation of NAT,the carrier of b1 holds
          ||.b4.|| " (#) (b3 * b4) is convergent(b2) &
           lim (||.b4.|| " (#) (b3 * b4)) = 0. b2);

:: NDIFF_1:exreg 3
registration
  let a1, a2 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster Relation-like Function-like REST-like Relation of the carrier of a1,the carrier of a2;
end;

:: NDIFF_1:modenot 1
definition
  let a1, a2 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  mode REST of a1,a2 is Function-like REST-like Relation of the carrier of a1,the carrier of a2;
end;

:: NDIFF_1:th 26
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
      st b3 is total(the carrier of b1, the carrier of b2)
   holds    b3 is REST-like(b1, b2)
   iff
      for b4 being Element of REAL
            st 0 < b4
         holds ex b5 being Element of REAL st
            0 < b5 &
             (for b6 being Element of the carrier of b1
                   st b6 <> 0. b1 & ||.b6.|| < b5
                holds ||.b6.|| " * ||.b3 /. b6.|| < b4);

:: NDIFF_1:th 27
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like REST-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total convergent_to_0 Relation of NAT,the carrier of b1 holds
   b3 * b4 is convergent(b2) & lim (b3 * b4) = 0. b2;

:: NDIFF_1:th 28
theorem
for b1 being Element of NAT
for b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
   rng (b3 ^\ b1) c= rng b3;

:: NDIFF_1:th 29
theorem
for b1 being Element of NAT
for b2, b3 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
for b5 being Function-like quasi_total Relation of NAT,the carrier of b2
      st rng b5 c= dom b4
   holds (b4 * b5) ^\ b1 = b4 * (b5 ^\ b1);

:: NDIFF_1:th 30
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is total(the carrier of b1, the carrier of b2) & b4 is total(the carrier of b1, the carrier of b2)
   holds (b3 + b4) * b5 = (b3 * b5) + (b4 * b5) &
    (b3 - b4) * b5 = (b3 * b5) - (b4 * b5);

:: NDIFF_1:th 31
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
for b5 being Element of REAL
      st b3 is total(the carrier of b1, the carrier of b2)
   holds (b5 (#) b3) * b4 = b5 * (b3 * b4);

:: NDIFF_1:th 32
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Element of the carrier of b2 holds
      b3 is_continuous_in b4
   iff
      b4 in dom b3 &
       (for b5 being Function-like quasi_total Relation of NAT,the carrier of b2
             st rng b5 c= dom b3 &
                b5 is convergent(b2) &
                lim b5 = b4 &
                (for b6 being Element of NAT holds
                   b5 . b6 <> b4)
          holds b3 * b5 is convergent(b1) & b3 /. b4 = lim (b3 * b5));

:: NDIFF_1:th 33
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like REST-like Relation of the carrier of b2,the carrier of b1 holds
b3 + b4 is Function-like REST-like Relation of the carrier of b2,the carrier of b1 &
 b3 - b4 is Function-like REST-like Relation of the carrier of b2,the carrier of b1;

:: NDIFF_1:th 34
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Element of REAL
for b4 being Function-like REST-like Relation of the carrier of b2,the carrier of b1 holds
   b3 (#) b4 is Function-like REST-like Relation of the carrier of b2,the carrier of b1;

:: NDIFF_1:prednot 2 => NDIFF_1:pred 1
definition
  let a1, a2 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
  let a4 be Element of the carrier of a1;
  pred A3 is_differentiable_in A4 means
    ex b1 being Neighbourhood of a4 st
       b1 c= dom a3 &
        (ex b2 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(a1,a2) st
           ex b3 being Function-like REST-like Relation of the carrier of a1,the carrier of a2 st
              for b4 being Element of the carrier of a1
                    st b4 in b1
                 holds (a3 /. b4) - (a3 /. a4) = (b2 . (b4 - a4)) + (b3 /. (b4 - a4)));
end;

:: NDIFF_1:dfs 6
definiens
  let a1, a2 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
  let a4 be Element of the carrier of a1;
To prove
     a3 is_differentiable_in a4
it is sufficient to prove
  thus ex b1 being Neighbourhood of a4 st
       b1 c= dom a3 &
        (ex b2 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(a1,a2) st
           ex b3 being Function-like REST-like Relation of the carrier of a1,the carrier of a2 st
              for b4 being Element of the carrier of a1
                    st b4 in b1
                 holds (a3 /. b4) - (a3 /. a4) = (b2 . (b4 - a4)) + (b3 /. (b4 - a4)));

:: NDIFF_1:def 6
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
      b3 is_differentiable_in b4
   iff
      ex b5 being Neighbourhood of b4 st
         b5 c= dom b3 &
          (ex b6 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2) st
             ex b7 being Function-like REST-like Relation of the carrier of b1,the carrier of b2 st
                for b8 being Element of the carrier of b1
                      st b8 in b5
                   holds (b3 /. b8) - (b3 /. b4) = (b6 . (b8 - b4)) + (b7 /. (b8 - b4)));

:: NDIFF_1:funcnot 3 => NDIFF_1:func 3
definition
  let a1, a2 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
  let a4 be Element of the carrier of a1;
  assume a3 is_differentiable_in a4;
  func diff(A3,A4) -> Element of the carrier of R_NormSpace_of_BoundedLinearOperators(a1,a2) means
    ex b1 being Neighbourhood of a4 st
       b1 c= dom a3 &
        (ex b2 being Function-like REST-like Relation of the carrier of a1,the carrier of a2 st
           for b3 being Element of the carrier of a1
                 st b3 in b1
              holds (a3 /. b3) - (a3 /. a4) = (it . (b3 - a4)) + (b2 /. (b3 - a4)));
end;

:: NDIFF_1:def 7
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1
   st b3 is_differentiable_in b4
for b5 being Element of the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2) holds
      b5 = diff(b3,b4)
   iff
      ex b6 being Neighbourhood of b4 st
         b6 c= dom b3 &
          (ex b7 being Function-like REST-like Relation of the carrier of b1,the carrier of b2 st
             for b8 being Element of the carrier of b1
                   st b8 in b6
                holds (b3 /. b8) - (b3 /. b4) = (b5 . (b8 - b4)) + (b7 /. (b8 - b4)));

:: NDIFF_1:prednot 3 => NDIFF_1:pred 2
definition
  let a1 be set;
  let a2, a3 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a4 be Function-like Relation of the carrier of a2,the carrier of a3;
  pred A4 is_differentiable_on A1 means
    a1 c= dom a4 &
     (for b1 being Element of the carrier of a2
           st b1 in a1
        holds a4 | a1 is_differentiable_in b1);
end;

:: NDIFF_1:dfs 8
definiens
  let a1 be set;
  let a2, a3 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a4 be Function-like Relation of the carrier of a2,the carrier of a3;
To prove
     a4 is_differentiable_on a1
it is sufficient to prove
  thus a1 c= dom a4 &
     (for b1 being Element of the carrier of a2
           st b1 in a1
        holds a4 | a1 is_differentiable_in b1);

:: NDIFF_1:def 8
theorem
for b1 being set
for b2, b3 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3 holds
      b4 is_differentiable_on b1
   iff
      b1 c= dom b4 &
       (for b5 being Element of the carrier of b2
             st b5 in b1
          holds b4 | b1 is_differentiable_in b5);

:: NDIFF_1:th 35
theorem
for b1 being set
for b2, b3 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
      st b4 is_differentiable_on b1
   holds b1 is Element of bool the carrier of b2;

:: NDIFF_1:th 36
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
      st b4 is open(b1)
   holds    b3 is_differentiable_on b4
   iff
      b4 c= dom b3 &
       (for b5 being Element of the carrier of b1
             st b5 in b4
          holds b3 is_differentiable_in b5);

:: NDIFF_1:th 37
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
      st b3 is_differentiable_on b4
   holds b4 is open(b1);

:: NDIFF_1:funcnot 4 => NDIFF_1:func 4
definition
  let a1, a2 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be Function-like Relation of the carrier of a1,the carrier of a2;
  let a4 be set;
  assume a3 is_differentiable_on a4;
  func A3 `| A4 -> Function-like Relation of the carrier of a1,the carrier of R_NormSpace_of_BoundedLinearOperators(a1,a2) means
    dom it = a4 &
     (for b1 being Element of the carrier of a1
           st b1 in a4
        holds it /. b1 = diff(a3,b1));
end;

:: NDIFF_1:def 9
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being set
   st b3 is_differentiable_on b4
for b5 being Function-like Relation of the carrier of b1,the carrier of R_NormSpace_of_BoundedLinearOperators(b1,b2) holds
      b5 = b3 `| b4
   iff
      dom b5 = b4 &
       (for b6 being Element of the carrier of b1
             st b6 in b4
          holds b5 /. b6 = diff(b3,b6));

:: NDIFF_1:th 38
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
      st b4 is open(b1) &
         b4 c= dom b3 &
         (ex b5 being Element of the carrier of b2 st
            rng b3 = {b5})
   holds b3 is_differentiable_on b4 &
    (for b5 being Element of the carrier of b1
          st b5 in b4
       holds (b3 `| b4) /. b5 = 0. R_NormSpace_of_BoundedLinearOperators(b1,b2));

:: NDIFF_1:funcreg 1
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like quasi_total convergent_to_0 Relation of NAT,the carrier of a1;
  let a3 be Element of NAT;
  cluster a2 ^\ a3 -> Function-like quasi_total convergent_to_0;
end;

:: NDIFF_1:funcreg 2
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like constant quasi_total Relation of NAT,the carrier of a1;
  let a3 be Element of NAT;
  cluster a2 ^\ a3 -> Function-like constant quasi_total;
end;

:: NDIFF_1:th 39
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of the carrier of b1
for b5 being Neighbourhood of b4
   st b3 is_differentiable_in b4 & b5 c= dom b3
for b6 being Function-like quasi_total convergent_to_0 Relation of NAT,the carrier of b1
for b7 being Function-like constant quasi_total Relation of NAT,the carrier of b1
      st rng b7 = {b4} & rng (b6 + b7) c= b5
   holds (b3 * (b6 + b7)) - (b3 * b7) is convergent(b2) &
    lim ((b3 * (b6 + b7)) - (b3 * b7)) = 0. b2;

:: NDIFF_1:th 40
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of the carrier of b2,the carrier of b1
for b5 being Element of the carrier of b2
      st b3 is_differentiable_in b5 & b4 is_differentiable_in b5
   holds b3 + b4 is_differentiable_in b5 &
    diff(b3 + b4,b5) = (diff(b3,b5)) + diff(b4,b5);

:: NDIFF_1:th 41
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of the carrier of b2,the carrier of b1
for b5 being Element of the carrier of b2
      st b3 is_differentiable_in b5 & b4 is_differentiable_in b5
   holds b3 - b4 is_differentiable_in b5 &
    diff(b3 - b4,b5) = (diff(b3,b5)) - diff(b4,b5);

:: NDIFF_1:th 42
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Element of REAL
for b4 being Function-like Relation of the carrier of b2,the carrier of b1
for b5 being Element of the carrier of b2
      st b4 is_differentiable_in b5
   holds b3 (#) b4 is_differentiable_in b5 &
    diff(b3 (#) b4,b5) = b3 * diff(b4,b5);

:: NDIFF_1:th 43
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,the carrier of b1
for b3 being Element of bool the carrier of b1
      st b3 is open(b1) & b3 c= dom b2 & b2 | b3 = id b3
   holds b2 is_differentiable_on b3 &
    (for b4 being Element of the carrier of b1
          st b4 in b3
       holds (b2 `| b3) /. b4 = id the carrier of b1);

:: NDIFF_1:th 44
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Element of bool the carrier of b1
   st b3 is open(b1)
for b4, b5 being Function-like Relation of the carrier of b1,the carrier of b2
      st b3 c= dom (b4 + b5) & b4 is_differentiable_on b3 & b5 is_differentiable_on b3
   holds b4 + b5 is_differentiable_on b3 &
    (for b6 being Element of the carrier of b1
          st b6 in b3
       holds ((b4 + b5) `| b3) /. b6 = (diff(b4,b6)) + diff(b5,b6));

:: NDIFF_1:th 45
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Element of bool the carrier of b1
   st b3 is open(b1)
for b4, b5 being Function-like Relation of the carrier of b1,the carrier of b2
      st b3 c= dom (b4 - b5) & b4 is_differentiable_on b3 & b5 is_differentiable_on b3
   holds b4 - b5 is_differentiable_on b3 &
    (for b6 being Element of the carrier of b1
          st b6 in b3
       holds ((b4 - b5) `| b3) /. b6 = (diff(b4,b6)) - diff(b5,b6));

:: NDIFF_1:th 46
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Element of bool the carrier of b1
   st b3 is open(b1)
for b4 being Element of REAL
for b5 being Function-like Relation of the carrier of b1,the carrier of b2
      st b3 c= dom (b4 (#) b5) & b5 is_differentiable_on b3
   holds b4 (#) b5 is_differentiable_on b3 &
    (for b6 being Element of the carrier of b1
          st b6 in b3
       holds ((b4 (#) b5) `| b3) /. b6 = b4 * diff(b5,b6));

:: NDIFF_1:th 47
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
      st b4 is open(b1) & b4 c= dom b3 & b3 is_constant_on b4
   holds b3 is_differentiable_on b4 &
    (for b5 being Element of the carrier of b1
          st b5 in b4
       holds (b3 `| b4) /. b5 = 0. R_NormSpace_of_BoundedLinearOperators(b1,b2));

:: NDIFF_1:th 48
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,the carrier of b1
for b3 being Element of REAL
for b4 being Element of the carrier of b1
for b5 being Element of bool the carrier of b1
      st b5 is open(b1) &
         b5 c= dom b2 &
         (for b6 being Element of the carrier of b1
               st b6 in b5
            holds b2 /. b6 = (b3 * b6) + b4)
   holds b2 is_differentiable_on b5 &
    (for b6 being Element of the carrier of b1
          st b6 in b5
       holds (b2 `| b5) /. b6 = b3 * FuncUnit b1);

:: NDIFF_1:th 49
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Element of the carrier of b2
      st b3 is_differentiable_in b4
   holds b3 is_continuous_in b4;

:: NDIFF_1:th 50
theorem
for b1 being set
for b2, b3 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
      st b4 is_differentiable_on b1
   holds b4 is_continuous_on b1;

:: NDIFF_1:th 51
theorem
for b1 being set
for b2, b3 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b3,the carrier of b2
for b5 being Element of bool the carrier of b3
      st b5 is open(b3) & b4 is_differentiable_on b1 & b5 c= b1
   holds b4 is_differentiable_on b5;

:: NDIFF_1:th 52
theorem
for b1, b2 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Element of the carrier of b2
      st b3 is_differentiable_in b4
   holds ex b5 being Neighbourhood of b4 st
      b5 c= dom b3 &
       (ex b6 being Function-like REST-like Relation of the carrier of b2,the carrier of b1 st
          b6 /. 0. b2 = 0. b1 &
           b6 is_continuous_in 0. b2 &
           (for b7 being Element of the carrier of b2
                 st b7 in b5
              holds (b3 /. b7) - (b3 /. b4) = ((diff(b3,b4)) . (b7 - b4)) + (b6 /. (b7 - b4))));