Article FDIFF_3, MML version 4.99.1005

:: FDIFF_3:th 1
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
      st ex b3 being Element of REAL st
           0 < b3 & [.b2 - b3,b2.] c= dom b1
   holds ex b3 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL st
      ex b4 being Function-like constant quasi_total Relation of NAT,REAL st
         rng b4 = {b2} &
          rng (b3 + b4) c= dom b1 &
          (for b5 being Element of NAT holds
             b3 . b5 < 0);

:: FDIFF_3:th 2
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
      st ex b3 being Element of REAL st
           0 < b3 & [.b2,b2 + b3.] c= dom b1
   holds ex b3 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL st
      ex b4 being Function-like constant quasi_total Relation of NAT,REAL st
         rng b4 = {b2} &
          rng (b3 + b4) c= dom b1 &
          (for b5 being Element of NAT holds
             0 < b3 . b5);

:: FDIFF_3:th 3
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
   st (for b3 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
      for b4 being Function-like constant quasi_total Relation of NAT,REAL
            st rng b4 = {b2} &
               rng (b3 + b4) c= dom b1 &
               (for b5 being Element of NAT holds
                  b3 . b5 < 0)
         holds b3 " (#) ((b1 * (b3 + b4)) - (b1 * b4)) is convergent) &
      {b2} c= dom b1
for b3, b4 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
for b5 being Function-like constant quasi_total Relation of NAT,REAL
      st rng b5 = {b2} &
         rng (b3 + b5) c= dom b1 &
         (for b6 being Element of NAT holds
            b3 . b6 < 0) &
         rng (b4 + b5) c= dom b1 &
         (for b6 being Element of NAT holds
            b4 . b6 < 0)
   holds lim (b3 " (#) ((b1 * (b3 + b5)) - (b1 * b5))) = lim (b4 " (#) ((b1 * (b4 + b5)) - (b1 * b5)));

:: FDIFF_3:th 4
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
   st (for b3 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
      for b4 being Function-like constant quasi_total Relation of NAT,REAL
            st rng b4 = {b2} &
               rng (b3 + b4) c= dom b1 &
               (for b5 being Element of NAT holds
                  0 < b3 . b5)
         holds b3 " (#) ((b1 * (b3 + b4)) - (b1 * b4)) is convergent) &
      {b2} c= dom b1
for b3, b4 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
for b5 being Function-like constant quasi_total Relation of NAT,REAL
      st rng b5 = {b2} &
         rng (b3 + b5) c= dom b1 &
         rng (b4 + b5) c= dom b1 &
         (for b6 being Element of NAT holds
            0 < b3 . b6) &
         (for b6 being Element of NAT holds
            0 < b4 . b6)
   holds lim (b3 " (#) ((b1 * (b3 + b5)) - (b1 * b5))) = lim (b4 " (#) ((b1 * (b4 + b5)) - (b1 * b5)));

:: FDIFF_3:prednot 1 => FDIFF_3:pred 1
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  pred A1 is_Lcontinuous_in A2 means
    a2 in dom a1 &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st rng b1 c= (halfline a2) /\ dom a1 & b1 is convergent & lim b1 = a2
        holds a1 * b1 is convergent & a1 . a2 = lim (a1 * b1));
end;

:: FDIFF_3:dfs 1
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
To prove
     a1 is_Lcontinuous_in a2
it is sufficient to prove
  thus a2 in dom a1 &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st rng b1 c= (halfline a2) /\ dom a1 & b1 is convergent & lim b1 = a2
        holds a1 * b1 is convergent & a1 . a2 = lim (a1 * b1));

:: FDIFF_3:def 1
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
      b1 is_Lcontinuous_in b2
   iff
      b2 in dom b1 &
       (for b3 being Function-like quasi_total Relation of NAT,REAL
             st rng b3 c= (halfline b2) /\ dom b1 & b3 is convergent & lim b3 = b2
          holds b1 * b3 is convergent & b1 . b2 = lim (b1 * b3));

:: FDIFF_3:prednot 2 => FDIFF_3:pred 2
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  pred A1 is_Rcontinuous_in A2 means
    a2 in dom a1 &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st rng b1 c= (right_open_halfline a2) /\ dom a1 & b1 is convergent & lim b1 = a2
        holds a1 * b1 is convergent & a1 . a2 = lim (a1 * b1));
end;

:: FDIFF_3:dfs 2
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
To prove
     a1 is_Rcontinuous_in a2
it is sufficient to prove
  thus a2 in dom a1 &
     (for b1 being Function-like quasi_total Relation of NAT,REAL
           st rng b1 c= (right_open_halfline a2) /\ dom a1 & b1 is convergent & lim b1 = a2
        holds a1 * b1 is convergent & a1 . a2 = lim (a1 * b1));

:: FDIFF_3:def 2
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
      b1 is_Rcontinuous_in b2
   iff
      b2 in dom b1 &
       (for b3 being Function-like quasi_total Relation of NAT,REAL
             st rng b3 c= (right_open_halfline b2) /\ dom b1 & b3 is convergent & lim b3 = b2
          holds b1 * b3 is convergent & b1 . b2 = lim (b1 * b3));

:: FDIFF_3:prednot 3 => FDIFF_3:pred 3
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  pred A1 is_right_differentiable_in A2 means
    (ex b1 being Element of REAL st
        0 < b1 & [.a2,a2 + b1.] c= dom a1) &
     (for b1 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
     for b2 being Function-like constant quasi_total Relation of NAT,REAL
           st rng b2 = {a2} &
              rng (b1 + b2) c= dom a1 &
              (for b3 being Element of NAT holds
                 0 < b1 . b3)
        holds b1 " (#) ((a1 * (b1 + b2)) - (a1 * b2)) is convergent);
end;

:: FDIFF_3:dfs 3
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
To prove
     a1 is_right_differentiable_in a2
it is sufficient to prove
  thus (ex b1 being Element of REAL st
        0 < b1 & [.a2,a2 + b1.] c= dom a1) &
     (for b1 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
     for b2 being Function-like constant quasi_total Relation of NAT,REAL
           st rng b2 = {a2} &
              rng (b1 + b2) c= dom a1 &
              (for b3 being Element of NAT holds
                 0 < b1 . b3)
        holds b1 " (#) ((a1 * (b1 + b2)) - (a1 * b2)) is convergent);

:: FDIFF_3:def 3
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
      b1 is_right_differentiable_in b2
   iff
      (ex b3 being Element of REAL st
          0 < b3 & [.b2,b2 + b3.] c= dom b1) &
       (for b3 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
       for b4 being Function-like constant quasi_total Relation of NAT,REAL
             st rng b4 = {b2} &
                rng (b3 + b4) c= dom b1 &
                (for b5 being Element of NAT holds
                   0 < b3 . b5)
          holds b3 " (#) ((b1 * (b3 + b4)) - (b1 * b4)) is convergent);

:: FDIFF_3:prednot 4 => FDIFF_3:pred 4
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
  pred A1 is_left_differentiable_in A2 means
    (ex b1 being Element of REAL st
        0 < b1 & [.a2 - b1,a2.] c= dom a1) &
     (for b1 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
     for b2 being Function-like constant quasi_total Relation of NAT,REAL
           st rng b2 = {a2} &
              rng (b1 + b2) c= dom a1 &
              (for b3 being Element of NAT holds
                 b1 . b3 < 0)
        holds b1 " (#) ((a1 * (b1 + b2)) - (a1 * b2)) is convergent);
end;

:: FDIFF_3:dfs 4
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of REAL;
To prove
     a1 is_left_differentiable_in a2
it is sufficient to prove
  thus (ex b1 being Element of REAL st
        0 < b1 & [.a2 - b1,a2.] c= dom a1) &
     (for b1 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
     for b2 being Function-like constant quasi_total Relation of NAT,REAL
           st rng b2 = {a2} &
              rng (b1 + b2) c= dom a1 &
              (for b3 being Element of NAT holds
                 b1 . b3 < 0)
        holds b1 " (#) ((a1 * (b1 + b2)) - (a1 * b2)) is convergent);

:: FDIFF_3:def 4
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
      b1 is_left_differentiable_in b2
   iff
      (ex b3 being Element of REAL st
          0 < b3 & [.b2 - b3,b2.] c= dom b1) &
       (for b3 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
       for b4 being Function-like constant quasi_total Relation of NAT,REAL
             st rng b4 = {b2} &
                rng (b3 + b4) c= dom b1 &
                (for b5 being Element of NAT holds
                   b3 . b5 < 0)
          holds b3 " (#) ((b1 * (b3 + b4)) - (b1 * b4)) is convergent);

:: FDIFF_3:th 5
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
      st b1 is_left_differentiable_in b2
   holds b1 is_Lcontinuous_in b2;

:: FDIFF_3:th 6
theorem
for b1 being Function-like Relation of REAL,REAL
for b2, b3 being Element of REAL
      st b1 is_Lcontinuous_in b2 &
         b1 . b2 <> b3 &
         (ex b4 being Element of REAL st
            0 < b4 & [.b2 - b4,b2.] c= dom b1)
   holds ex b4 being Element of REAL st
      0 < b4 &
       [.b2 - b4,b2.] c= dom b1 &
       (for b5 being Element of REAL
             st b5 in [.b2 - b4,b2.]
          holds b1 . b5 <> b3);

:: FDIFF_3:th 7
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
      st b1 is_right_differentiable_in b2
   holds b1 is_Rcontinuous_in b2;

:: FDIFF_3:th 8
theorem
for b1 being Function-like Relation of REAL,REAL
for b2, b3 being Element of REAL
      st b1 is_Rcontinuous_in b2 &
         b1 . b2 <> b3 &
         (ex b4 being Element of REAL st
            0 < b4 & [.b2,b2 + b4.] c= dom b1)
   holds ex b4 being Element of REAL st
      0 < b4 &
       [.b2,b2 + b4.] c= dom b1 &
       (for b5 being Element of REAL
             st b5 in [.b2,b2 + b4.]
          holds b1 . b5 <> b3);

:: FDIFF_3:funcnot 1 => FDIFF_3:func 1
definition
  let a1 be Element of REAL;
  let a2 be Function-like Relation of REAL,REAL;
  assume a2 is_left_differentiable_in a1;
  func Ldiff(A2,A1) -> Element of REAL means
    for b1 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
    for b2 being Function-like constant quasi_total Relation of NAT,REAL
          st rng b2 = {a1} &
             rng (b1 + b2) c= dom a2 &
             (for b3 being Element of NAT holds
                b1 . b3 < 0)
       holds it = lim (b1 " (#) ((a2 * (b1 + b2)) - (a2 * b2)));
end;

:: FDIFF_3:def 5
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
   st b2 is_left_differentiable_in b1
for b3 being Element of REAL holds
      b3 = Ldiff(b2,b1)
   iff
      for b4 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
      for b5 being Function-like constant quasi_total Relation of NAT,REAL
            st rng b5 = {b1} &
               rng (b4 + b5) c= dom b2 &
               (for b6 being Element of NAT holds
                  b4 . b6 < 0)
         holds b3 = lim (b4 " (#) ((b2 * (b4 + b5)) - (b2 * b5)));

:: FDIFF_3:funcnot 2 => FDIFF_3:func 2
definition
  let a1 be Element of REAL;
  let a2 be Function-like Relation of REAL,REAL;
  assume a2 is_right_differentiable_in a1;
  func Rdiff(A2,A1) -> Element of REAL means
    for b1 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
    for b2 being Function-like constant quasi_total Relation of NAT,REAL
          st rng b2 = {a1} &
             rng (b1 + b2) c= dom a2 &
             (for b3 being Element of NAT holds
                0 < b1 . b3)
       holds it = lim (b1 " (#) ((a2 * (b1 + b2)) - (a2 * b2)));
end;

:: FDIFF_3:def 6
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
   st b2 is_right_differentiable_in b1
for b3 being Element of REAL holds
      b3 = Rdiff(b2,b1)
   iff
      for b4 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
      for b5 being Function-like constant quasi_total Relation of NAT,REAL
            st rng b5 = {b1} &
               rng (b4 + b5) c= dom b2 &
               (for b6 being Element of NAT holds
                  0 < b4 . b6)
         holds b3 = lim (b4 " (#) ((b2 * (b4 + b5)) - (b2 * b5)));

:: FDIFF_3:th 9
theorem
for b1 being Function-like Relation of REAL,REAL
for b2, b3 being Element of REAL holds
   b1 is_left_differentiable_in b2 & Ldiff(b1,b2) = b3
iff
   (ex b4 being Element of REAL st
       0 < b4 & [.b2 - b4,b2.] c= dom b1) &
    (for b4 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
    for b5 being Function-like constant quasi_total Relation of NAT,REAL
          st rng b5 = {b2} &
             rng (b4 + b5) c= dom b1 &
             (for b6 being Element of NAT holds
                b4 . b6 < 0)
       holds b4 " (#) ((b1 * (b4 + b5)) - (b1 * b5)) is convergent &
        lim (b4 " (#) ((b1 * (b4 + b5)) - (b1 * b5))) = b3);

:: FDIFF_3:th 10
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
      st b1 is_left_differentiable_in b3 & b2 is_left_differentiable_in b3
   holds b1 + b2 is_left_differentiable_in b3 &
    Ldiff(b1 + b2,b3) = (Ldiff(b1,b3)) + Ldiff(b2,b3);

:: FDIFF_3:th 11
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
      st b1 is_left_differentiable_in b3 & b2 is_left_differentiable_in b3
   holds b1 - b2 is_left_differentiable_in b3 &
    Ldiff(b1 - b2,b3) = (Ldiff(b1,b3)) - Ldiff(b2,b3);

:: FDIFF_3:th 12
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
      st b1 is_left_differentiable_in b3 & b2 is_left_differentiable_in b3
   holds b1 (#) b2 is_left_differentiable_in b3 &
    Ldiff(b1 (#) b2,b3) = ((Ldiff(b1,b3)) * (b2 . b3)) + ((Ldiff(b2,b3)) * (b1 . b3));

:: FDIFF_3:th 13
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
      st b1 is_left_differentiable_in b3 & b2 is_left_differentiable_in b3 & b2 . b3 <> 0
   holds b1 / b2 is_left_differentiable_in b3 &
    Ldiff(b1 / b2,b3) = (((Ldiff(b1,b3)) * (b2 . b3)) - ((Ldiff(b2,b3)) * (b1 . b3))) / ((b2 . b3) ^2);

:: FDIFF_3:th 14
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
      st b1 is_left_differentiable_in b2 & b1 . b2 <> 0
   holds b1 ^ is_left_differentiable_in b2 &
    Ldiff(b1 ^,b2) = - ((Ldiff(b1,b2)) / ((b1 . b2) ^2));

:: FDIFF_3:th 15
theorem
for b1 being Function-like Relation of REAL,REAL
for b2, b3 being Element of REAL holds
   b1 is_right_differentiable_in b2 & Rdiff(b1,b2) = b3
iff
   (ex b4 being Element of REAL st
       0 < b4 & [.b2,b2 + b4.] c= dom b1) &
    (for b4 being Function-like quasi_total convergent_to_0 Relation of NAT,REAL
    for b5 being Function-like constant quasi_total Relation of NAT,REAL
          st rng b5 = {b2} &
             rng (b4 + b5) c= dom b1 &
             (for b6 being Element of NAT holds
                0 < b4 . b6)
       holds b4 " (#) ((b1 * (b4 + b5)) - (b1 * b5)) is convergent &
        lim (b4 " (#) ((b1 * (b4 + b5)) - (b1 * b5))) = b3);

:: FDIFF_3:th 16
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
      st b1 is_right_differentiable_in b3 & b2 is_right_differentiable_in b3
   holds b1 + b2 is_right_differentiable_in b3 &
    Rdiff(b1 + b2,b3) = (Rdiff(b1,b3)) + Rdiff(b2,b3);

:: FDIFF_3:th 17
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
      st b1 is_right_differentiable_in b3 & b2 is_right_differentiable_in b3
   holds b1 - b2 is_right_differentiable_in b3 &
    Rdiff(b1 - b2,b3) = (Rdiff(b1,b3)) - Rdiff(b2,b3);

:: FDIFF_3:th 18
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
      st b1 is_right_differentiable_in b3 & b2 is_right_differentiable_in b3
   holds b1 (#) b2 is_right_differentiable_in b3 &
    Rdiff(b1 (#) b2,b3) = ((Rdiff(b1,b3)) * (b2 . b3)) + ((Rdiff(b2,b3)) * (b1 . b3));

:: FDIFF_3:th 19
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
      st b1 is_right_differentiable_in b3 & b2 is_right_differentiable_in b3 & b2 . b3 <> 0
   holds b1 / b2 is_right_differentiable_in b3 &
    Rdiff(b1 / b2,b3) = (((Rdiff(b1,b3)) * (b2 . b3)) - ((Rdiff(b2,b3)) * (b1 . b3))) / ((b2 . b3) ^2);

:: FDIFF_3:th 20
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
      st b1 is_right_differentiable_in b2 & b1 . b2 <> 0
   holds b1 ^ is_right_differentiable_in b2 &
    Rdiff(b1 ^,b2) = - ((Rdiff(b1,b2)) / ((b1 . b2) ^2));

:: FDIFF_3:th 21
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
      st b1 is_right_differentiable_in b2 & b1 is_left_differentiable_in b2 & Rdiff(b1,b2) = Ldiff(b1,b2)
   holds b1 is_differentiable_in b2 & diff(b1,b2) = Rdiff(b1,b2) & diff(b1,b2) = Ldiff(b1,b2);

:: FDIFF_3:th 22
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
      st b1 is_differentiable_in b2
   holds b1 is_right_differentiable_in b2 & b1 is_left_differentiable_in b2 & diff(b1,b2) = Rdiff(b1,b2) & diff(b1,b2) = Ldiff(b1,b2);