Article DIFF_2, MML version 4.99.1005

:: DIFF_2:th 1
theorem
for b1, b2 being Element of REAL
for b3 being Function-like quasi_total Relation of REAL,REAL holds
   [!b3,b1,b1 + b2!] = (((forward_difference(b3,b2)) . 1) . b1) / b2;

:: DIFF_2:th 2
theorem
for b1, b2 being Element of REAL
for b3 being Function-like quasi_total Relation of REAL,REAL
      st b1 <> 0
   holds [!b3,b2,b2 + b1,b2 + (2 * b1)!] = (((forward_difference(b3,b1)) . 2) . b2) / (2 * (b1 ^2));

:: DIFF_2:th 3
theorem
for b1, b2 being Element of REAL
for b3 being Function-like quasi_total Relation of REAL,REAL holds
   [!b3,b1 - b2,b1!] = (((backward_difference(b3,b2)) . 1) . b1) / b2;

:: DIFF_2:th 4
theorem
for b1, b2 being Element of REAL
for b3 being Function-like quasi_total Relation of REAL,REAL
      st b1 <> 0
   holds [!b3,b2 - (2 * b1),b2 - b1,b2!] = (((backward_difference(b3,b1)) . 2) . b2) / (2 * (b1 ^2));

:: DIFF_2:th 5
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Function-like quasi_total Relation of REAL,REAL holds
   [!b1 (#) b5,b2,b3,b4!] = b1 * [!b5,b2,b3,b4!];

:: DIFF_2:th 6
theorem
for b1, b2, b3 being Element of REAL
for b4, b5 being Function-like quasi_total Relation of REAL,REAL holds
[!b4 + b5,b1,b2,b3!] = [!b4,b1,b2,b3!] + [!b5,b1,b2,b3!];

:: DIFF_2:th 7
theorem
for b1, b2, b3, b4, b5 being Element of REAL
for b6, b7 being Function-like quasi_total Relation of REAL,REAL holds
[!(b1 (#) b6) + (b2 (#) b7),b3,b4,b5!] = (b1 * [!b6,b3,b4,b5!]) + (b2 * [!b7,b3,b4,b5!]);

:: DIFF_2:th 8
theorem
for b1, b2, b3, b4, b5 being Element of REAL
for b6 being Function-like quasi_total Relation of REAL,REAL holds
   [!b1 (#) b6,b2,b3,b4,b5!] = b1 * [!b6,b2,b3,b4,b5!];

:: DIFF_2:th 9
theorem
for b1, b2, b3, b4 being Element of REAL
for b5, b6 being Function-like quasi_total Relation of REAL,REAL holds
[!b5 + b6,b1,b2,b3,b4!] = [!b5,b1,b2,b3,b4!] + [!b6,b1,b2,b3,b4!];

:: DIFF_2:th 10
theorem
for b1, b2, b3, b4, b5, b6 being Element of REAL
for b7, b8 being Function-like quasi_total Relation of REAL,REAL holds
[!(b1 (#) b7) + (b2 (#) b8),b3,b4,b5,b6!] = (b1 * [!b7,b3,b4,b5,b6!]) + (b2 * [!b8,b3,b4,b5,b6!]);

:: DIFF_2:funcnot 1 => DIFF_2:func 1
definition
  let a1 be Relation-like Function-like real-valued set;
  let a2, a3, a4, a5, a6 be real set;
  func [!A1,A2,A3,A4,A5,A6!] -> Element of REAL equals
    ([!a1,a2,a3,a4,a5!] - [!a1,a3,a4,a5,a6!]) / (a2 - a6);
end;

:: DIFF_2:def 1
theorem
for b1 being Relation-like Function-like real-valued set
for b2, b3, b4, b5, b6 being real set holds
[!b1,b2,b3,b4,b5,b6!] = ([!b1,b2,b3,b4,b5!] - [!b1,b3,b4,b5,b6!]) / (b2 - b6);

:: DIFF_2:th 11
theorem
for b1, b2, b3, b4, b5, b6 being Element of REAL
for b7 being Function-like quasi_total Relation of REAL,REAL holds
   [!b1 (#) b7,b2,b3,b4,b5,b6!] = b1 * [!b7,b2,b3,b4,b5,b6!];

:: DIFF_2:th 12
theorem
for b1, b2, b3, b4, b5 being Element of REAL
for b6, b7 being Function-like quasi_total Relation of REAL,REAL holds
[!b6 + b7,b1,b2,b3,b4,b5!] = [!b6,b1,b2,b3,b4,b5!] + [!b7,b1,b2,b3,b4,b5!];

:: DIFF_2:th 13
theorem
for b1, b2, b3, b4, b5, b6, b7 being Element of REAL
for b8, b9 being Function-like quasi_total Relation of REAL,REAL holds
[!(b1 (#) b8) + (b2 (#) b9),b3,b4,b5,b6,b7!] = (b1 * [!b8,b3,b4,b5,b6,b7!]) + (b2 * [!b9,b3,b4,b5,b6,b7!]);

:: DIFF_2:funcnot 2 => DIFF_2:func 2
definition
  let a1 be Relation-like Function-like real-valued set;
  let a2, a3, a4, a5, a6, a7 be real set;
  func [!A1,A2,A3,A4,A5,A6,A7!] -> Element of REAL equals
    ([!a1,a2,a3,a4,a5,a6!] - [!a1,a3,a4,a5,a6,a7!]) / (a2 - a7);
end;

:: DIFF_2:def 2
theorem
for b1 being Relation-like Function-like real-valued set
for b2, b3, b4, b5, b6, b7 being real set holds
[!b1,b2,b3,b4,b5,b6,b7!] = ([!b1,b2,b3,b4,b5,b6!] - [!b1,b3,b4,b5,b6,b7!]) / (b2 - b7);

:: DIFF_2:th 14
theorem
for b1, b2, b3, b4, b5, b6, b7 being Element of REAL
for b8 being Function-like quasi_total Relation of REAL,REAL holds
   [!b1 (#) b8,b2,b3,b4,b5,b6,b7!] = b1 * [!b8,b2,b3,b4,b5,b6,b7!];

:: DIFF_2:th 15
theorem
for b1, b2, b3, b4, b5, b6 being Element of REAL
for b7, b8 being Function-like quasi_total Relation of REAL,REAL holds
[!b7 + b8,b1,b2,b3,b4,b5,b6!] = [!b7,b1,b2,b3,b4,b5,b6!] + [!b8,b1,b2,b3,b4,b5,b6!];

:: DIFF_2:th 16
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being Element of REAL
for b9, b10 being Function-like quasi_total Relation of REAL,REAL holds
[!(b1 (#) b9) + (b2 (#) b10),b3,b4,b5,b6,b7,b8!] = (b1 * [!b9,b3,b4,b5,b6,b7,b8!]) + (b2 * [!b10,b3,b4,b5,b6,b7,b8!]);

:: DIFF_2:th 17
theorem
for b1, b2, b3 being Element of REAL
for b4 being Function-like quasi_total Relation of REAL,REAL
      st b1,b2,b3 are_mutually_different
   holds [!b4,b1,b2,b3!] = (((b4 . b1) / ((b1 - b2) * (b1 - b3))) + ((b4 . b2) / ((b2 - b1) * (b2 - b3)))) + ((b4 . b3) / ((b3 - b1) * (b3 - b2)));

:: DIFF_2:th 18
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Function-like quasi_total Relation of REAL,REAL
      st b1,b2,b3,b4 are_mutually_different
   holds [!b5,b1,b2,b3,b4!] = [!b5,b2,b3,b4,b1!] &
    [!b5,b1,b2,b3,b4!] = [!b5,b4,b3,b2,b1!];

:: DIFF_2:th 19
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Function-like quasi_total Relation of REAL,REAL
      st b1,b2,b3,b4 are_mutually_different
   holds [!b5,b1,b2,b3,b4!] = [!b5,b2,b1,b3,b4!] &
    [!b5,b1,b2,b3,b4!] = [!b5,b2,b3,b1,b4!];

:: DIFF_2:th 20
theorem
for b1, b2, b3 being Element of REAL
for b4 being Function-like quasi_total Relation of REAL,REAL
      st b4 is constant
   holds [!b4,b1,b2,b3!] = 0;

:: DIFF_2:th 21
theorem
for b1, b2, b3, b4 being Element of REAL
      st b1 <> b2
   holds [!AffineMap(b3,b4),b1,b2!] = b3;

:: DIFF_2:th 22
theorem
for b1, b2, b3, b4, b5 being Element of REAL
      st b1,b2,b3 are_mutually_different
   holds [!AffineMap(b4,b5),b1,b2,b3!] = 0;

:: DIFF_2:th 23
theorem
for b1, b2, b3, b4, b5, b6 being Element of REAL
      st b1,b2,b3,b4 are_mutually_different
   holds [!AffineMap(b5,b6),b1,b2,b3,b4!] = 0;

:: DIFF_2:th 24
theorem
for b1, b2, b3, b4 being Element of REAL holds
(fD(AffineMap(b1,b2),b3)) . b4 = b1 * b3;

:: DIFF_2:th 25
theorem
for b1, b2, b3, b4 being Element of REAL holds
(bD(AffineMap(b1,b2),b3)) . b4 = b1 * b3;

:: DIFF_2:th 26
theorem
for b1, b2, b3, b4 being Element of REAL holds
(cD(AffineMap(b1,b2),b3)) . b4 = b1 * b3;

:: DIFF_2:th 27
theorem
for b1, b2, b3, b4, b5 being Element of REAL
for b6 being Function-like quasi_total Relation of REAL,REAL
      st (for b7 being Element of REAL holds
            b6 . b7 = ((b1 * (b7 ^2)) + (b2 * b7)) + b3) &
         b4 <> b5
   holds [!b6,b4,b5!] = (b1 * (b4 + b5)) + b2;

:: DIFF_2:th 28
theorem
for b1, b2, b3, b4, b5, b6 being Element of REAL
for b7 being Function-like quasi_total Relation of REAL,REAL
      st (for b8 being Element of REAL holds
            b7 . b8 = ((b1 * (b8 ^2)) + (b2 * b8)) + b3) &
         b4,b5,b6 are_mutually_different
   holds [!b7,b4,b5,b6!] = b1;

:: DIFF_2:th 29
theorem
for b1, b2, b3, b4, b5, b6, b7 being Element of REAL
for b8 being Function-like quasi_total Relation of REAL,REAL
      st (for b9 being Element of REAL holds
            b8 . b9 = ((b1 * (b9 ^2)) + (b2 * b9)) + b3) &
         b4,b5,b6,b7 are_mutually_different
   holds [!b8,b4,b5,b6,b7!] = 0;

:: DIFF_2:th 30
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being Element of REAL
for b9 being Function-like quasi_total Relation of REAL,REAL
      st (for b10 being Element of REAL holds
            b9 . b10 = ((b1 * (b10 ^2)) + (b2 * b10)) + b3) &
         b4,b5,b6,b7,b8 are_mutually_different
   holds [!b9,b4,b5,b6,b7,b8!] = 0;

:: DIFF_2:th 31
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Function-like quasi_total Relation of REAL,REAL
   st for b6 being Element of REAL holds
        b5 . b6 = ((b1 * (b6 ^2)) + (b2 * b6)) + b3
for b6 being Element of REAL holds
   (fD(b5,b4)) . b6 = ((((2 * b1) * b4) * b6) + (b1 * (b4 ^2))) + (b2 * b4);

:: DIFF_2:th 32
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Function-like quasi_total Relation of REAL,REAL
   st for b6 being Element of REAL holds
        b5 . b6 = ((b1 * (b6 ^2)) + (b2 * b6)) + b3
for b6 being Element of REAL holds
   (bD(b5,b4)) . b6 = ((((2 * b1) * b4) * b6) - (b1 * (b4 ^2))) + (b2 * b4);

:: DIFF_2:th 33
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Function-like quasi_total Relation of REAL,REAL
   st for b6 being Element of REAL holds
        b5 . b6 = ((b1 * (b6 ^2)) + (b2 * b6)) + b3
for b6 being Element of REAL holds
   (cD(b5,b4)) . b6 = (((2 * b1) * b4) * b6) + (b2 * b4);

:: DIFF_2:th 34
theorem
for b1, b2, b3 being Element of REAL
for b4 being Function-like quasi_total Relation of REAL,REAL
      st (for b5 being Element of REAL holds
            b4 . b5 = b1 / b5) &
         b2 <> b3 &
         b2 <> 0 &
         b3 <> 0
   holds [!b4,b2,b3!] = - (b1 / (b2 * b3));

:: DIFF_2:th 35
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Function-like quasi_total Relation of REAL,REAL
      st (for b6 being Element of REAL holds
            b5 . b6 = b1 / b6) &
         b2 <> 0 &
         b3 <> 0 &
         b4 <> 0 &
         b2,b3,b4 are_mutually_different
   holds [!b5,b2,b3,b4!] = b1 / ((b2 * b3) * b4);

:: DIFF_2:th 36
theorem
for b1, b2, b3, b4, b5 being Element of REAL
for b6 being Function-like quasi_total Relation of REAL,REAL
      st (for b7 being Element of REAL holds
            b6 . b7 = b1 / b7) &
         b2 <> 0 &
         b3 <> 0 &
         b4 <> 0 &
         b5 <> 0 &
         b2,b3,b4,b5 are_mutually_different
   holds [!b6,b2,b3,b4,b5!] = - (b1 / (((b2 * b3) * b4) * b5));

:: DIFF_2:th 37
theorem
for b1, b2, b3, b4, b5, b6 being Element of REAL
for b7 being Function-like quasi_total Relation of REAL,REAL
      st (for b8 being Element of REAL holds
            b7 . b8 = b1 / b8) &
         b2 <> 0 &
         b3 <> 0 &
         b4 <> 0 &
         b5 <> 0 &
         b6 <> 0 &
         b2,b3,b4,b5,b6 are_mutually_different
   holds [!b7,b2,b3,b4,b5,b6!] = b1 / ((((b2 * b3) * b4) * b5) * b6);

:: DIFF_2:th 38
theorem
for b1, b2 being Element of REAL
for b3 being Function-like quasi_total Relation of REAL,REAL
   st for b4 being Element of REAL holds
        b3 . b4 = b1 / b4 & b4 <> 0 & b4 + b2 <> 0
for b4 being Element of REAL holds
   (fD(b3,b2)) . b4 = (- (b1 * b2)) / ((b4 + b2) * b4);

:: DIFF_2:th 39
theorem
for b1, b2 being Element of REAL
for b3 being Function-like quasi_total Relation of REAL,REAL
   st for b4 being Element of REAL holds
        b3 . b4 = b1 / b4 & b4 <> 0 & b4 - b2 <> 0
for b4 being Element of REAL holds
   (bD(b3,b2)) . b4 = (- (b1 * b2)) / ((b4 - b2) * b4);

:: DIFF_2:th 40
theorem
for b1, b2 being Element of REAL
for b3 being Function-like quasi_total Relation of REAL,REAL
   st for b4 being Element of REAL holds
        b3 . b4 = b1 / b4 & b4 + (b2 / 2) <> 0 & b4 - (b2 / 2) <> 0
for b4 being Element of REAL holds
   (cD(b3,b2)) . b4 = (- (b1 * b2)) / ((b4 - (b2 / 2)) * (b4 + (b2 / 2)));

:: DIFF_2:th 41
theorem
for b1, b2 being Element of REAL holds
[!sin,b1,b2!] = ((2 * cos ((b1 + b2) / 2)) * sin ((b1 - b2) / 2)) / (b1 - b2);

:: DIFF_2:th 42
theorem
for b1, b2 being Element of REAL holds
(fD(sin,b1)) . b2 = 2 * ((cos (((2 * b2) + b1) / 2)) * sin (b1 / 2));

:: DIFF_2:th 43
theorem
for b1, b2 being Element of REAL holds
(bD(sin,b1)) . b2 = 2 * ((cos (((2 * b2) - b1) / 2)) * sin (b1 / 2));

:: DIFF_2:th 44
theorem
for b1, b2 being Element of REAL holds
(cD(sin,b1)) . b2 = 2 * ((cos b2) * sin (b1 / 2));

:: DIFF_2:th 45
theorem
for b1, b2 being Element of REAL holds
[!cos,b1,b2!] = - (((2 * sin ((b1 + b2) / 2)) * sin ((b1 - b2) / 2)) / (b1 - b2));

:: DIFF_2:th 46
theorem
for b1, b2 being Element of REAL holds
(fD(cos,b1)) . b2 = - (2 * ((sin (((2 * b2) + b1) / 2)) * sin (b1 / 2)));

:: DIFF_2:th 47
theorem
for b1, b2 being Element of REAL holds
(bD(cos,b1)) . b2 = - (2 * ((sin (((2 * b2) - b1) / 2)) * sin (b1 / 2)));

:: DIFF_2:th 48
theorem
for b1, b2 being Element of REAL holds
(cD(cos,b1)) . b2 = - (2 * ((sin b2) * sin (b1 / 2)));

:: DIFF_2:th 49
theorem
for b1, b2 being Element of REAL holds
[!sin (#) sin,b1,b2!] = ((1 / 2) * ((cos (2 * b2)) - cos (2 * b1))) / (b1 - b2);

:: DIFF_2:th 50
theorem
for b1, b2 being Element of REAL holds
(fD(sin (#) sin,b1)) . b2 = (1 / 2) * ((cos (2 * b2)) - cos (2 * (b2 + b1)));

:: DIFF_2:th 51
theorem
for b1, b2 being Element of REAL holds
(bD(sin (#) sin,b1)) . b2 = (1 / 2) * ((cos (2 * (b2 - b1))) - cos (2 * b2));

:: DIFF_2:th 52
theorem
for b1, b2 being Element of REAL holds
(cD(sin (#) sin,b1)) . b2 = (1 / 2) * ((cos ((2 * b2) - b1)) - cos ((2 * b2) + b1));

:: DIFF_2:th 53
theorem
for b1, b2 being Element of REAL holds
[!sin (#) cos,b1,b2!] = ((1 / 2) * ((sin (2 * b1)) - sin (2 * b2))) / (b1 - b2);

:: DIFF_2:th 54
theorem
for b1, b2 being Element of REAL holds
(fD(sin (#) cos,b1)) . b2 = (1 / 2) * ((sin (2 * (b2 + b1))) - sin (2 * b2));

:: DIFF_2:th 55
theorem
for b1, b2 being Element of REAL holds
(bD(sin (#) cos,b1)) . b2 = (1 / 2) * ((sin (2 * b2)) - sin (2 * (b2 - b1)));

:: DIFF_2:th 56
theorem
for b1, b2 being Element of REAL holds
(cD(sin (#) cos,b1)) . b2 = (1 / 2) * ((sin ((2 * b2) + b1)) - sin ((2 * b2) - b1));

:: DIFF_2:th 57
theorem
for b1, b2 being Element of REAL holds
[!cos (#) cos,b1,b2!] = ((1 / 2) * ((cos (2 * b1)) - cos (2 * b2))) / (b1 - b2);

:: DIFF_2:th 58
theorem
for b1, b2 being Element of REAL holds
(fD(cos (#) cos,b1)) . b2 = (1 / 2) * ((cos (2 * (b2 + b1))) - cos (2 * b2));

:: DIFF_2:th 59
theorem
for b1, b2 being Element of REAL holds
(bD(cos (#) cos,b1)) . b2 = (1 / 2) * ((cos (2 * b2)) - cos (2 * (b2 - b1)));

:: DIFF_2:th 60
theorem
for b1, b2 being Element of REAL holds
(cD(cos (#) cos,b1)) . b2 = (1 / 2) * ((cos ((2 * b2) + b1)) - cos ((2 * b2) - b1));

:: DIFF_2:th 61
theorem
for b1, b2 being Element of REAL holds
[!(sin (#) sin) (#) cos,b1,b2!] = - (((1 / 2) * (((sin ((3 * (b2 + b1)) / 2)) * sin ((3 * (b2 - b1)) / 2)) + ((sin ((b1 + b2) / 2)) * sin ((b1 - b2) / 2)))) / (b1 - b2));

:: DIFF_2:th 62
theorem
for b1, b2 being Element of REAL holds
(fD((sin (#) sin) (#) cos,b1)) . b2 = (1 / 2) * (((sin (((6 * b2) + (3 * b1)) / 2)) * sin ((3 * b1) / 2)) - ((sin (((2 * b2) + b1) / 2)) * sin (b1 / 2)));

:: DIFF_2:th 63
theorem
for b1, b2 being Element of REAL holds
(bD((sin (#) sin) (#) cos,b1)) . b2 = ((1 / 2) * ((sin (((6 * b2) - (3 * b1)) / 2)) * sin ((3 * b1) / 2))) - ((1 / 2) * ((sin (((2 * b2) - b1) / 2)) * sin (b1 / 2)));

:: DIFF_2:th 64
theorem
for b1, b2 being Element of REAL holds
(cD((sin (#) sin) (#) cos,b1)) . b2 = (- ((1 / 2) * ((sin b2) * sin (b1 / 2)))) + ((1 / 2) * ((sin (3 * b2)) * sin ((3 * b1) / 2)));

:: DIFF_2:th 65
theorem
for b1, b2 being Element of REAL holds
[!(sin (#) cos) (#) cos,b1,b2!] = ((1 / 2) * (((cos ((b1 + b2) / 2)) * sin ((b1 - b2) / 2)) + ((cos ((3 * (b1 + b2)) / 2)) * sin ((3 * (b1 - b2)) / 2)))) / (b1 - b2);

:: DIFF_2:th 66
theorem
for b1, b2 being Element of REAL holds
(fD((sin (#) cos) (#) cos,b1)) . b2 = (1 / 2) * (((cos (((2 * b2) + b1) / 2)) * sin (b1 / 2)) + ((cos (((6 * b2) + (3 * b1)) / 2)) * sin ((3 * b1) / 2)));

:: DIFF_2:th 67
theorem
for b1, b2 being Element of REAL holds
(bD((sin (#) cos) (#) cos,b1)) . b2 = (1 / 2) * (((cos (((2 * b2) - b1) / 2)) * sin (b1 / 2)) + ((cos (((6 * b2) - (3 * b1)) / 2)) * sin ((3 * b1) / 2)));

:: DIFF_2:th 68
theorem
for b1, b2 being Element of REAL holds
(cD((sin (#) cos) (#) cos,b1)) . b2 = (1 / 2) * (((cos b2) * sin (b1 / 2)) + ((cos (3 * b2)) * sin ((3 * b1) / 2)));

:: DIFF_2:th 69
theorem
for b1, b2 being Element of REAL
      st b1 in proj1 tan & b2 in proj1 tan
   holds [!tan,b1,b2!] = (sin (b1 - b2)) / (((cos b1) * cos b2) * (b1 - b2));

:: DIFF_2:th 70
theorem
for b1, b2 being Element of REAL
      st b1 in proj1 cot & b2 in proj1 cot
   holds [!cot,b1,b2!] = - ((sin (b1 - b2)) / (((sin b1) * sin b2) * (b1 - b2)));

:: DIFF_2:th 71
theorem
for b1, b2 being Element of REAL
      st b1 in proj1 cosec & b2 in proj1 cosec
   holds [!cosec,b1,b2!] = ((2 * cos ((b2 + b1) / 2)) * sin ((b2 - b1) / 2)) / (((sin b2) * sin b1) * (b1 - b2));

:: DIFF_2:th 72
theorem
for b1, b2 being Element of REAL
      st b1 in proj1 sec & b2 in proj1 sec
   holds [!sec,b1,b2!] = - (((2 * sin ((b2 + b1) / 2)) * sin ((b2 - b1) / 2)) / (((cos b2) * cos b1) * (b1 - b2)));