Article EUCLID_5, MML version 4.99.1005

:: EUCLID_5:th 1
theorem
for b1 being Element of the carrier of TOP-REAL 3 holds
   ex b2, b3, b4 being Element of REAL st
      b1 = <*b2,b3,b4*>;

:: EUCLID_5:funcnot 1 => EUCLID_5:func 1
definition
  let a1 be Element of the carrier of TOP-REAL 3;
  func A1 `1 -> Element of REAL equals
    a1 . 1;
end;

:: EUCLID_5:def 1
theorem
for b1 being Element of the carrier of TOP-REAL 3 holds
   b1 `1 = b1 . 1;

:: EUCLID_5:funcnot 2 => EUCLID_5:func 2
definition
  let a1 be Element of the carrier of TOP-REAL 3;
  func A1 `2 -> Element of REAL equals
    a1 . 2;
end;

:: EUCLID_5:def 2
theorem
for b1 being Element of the carrier of TOP-REAL 3 holds
   b1 `2 = b1 . 2;

:: EUCLID_5:funcnot 3 => EUCLID_5:func 3
definition
  let a1 be Element of the carrier of TOP-REAL 3;
  func A1 `3 -> Element of REAL equals
    a1 . 3;
end;

:: EUCLID_5:def 3
theorem
for b1 being Element of the carrier of TOP-REAL 3 holds
   b1 `3 = b1 . 3;

:: EUCLID_5:funcnot 4 => FINSEQ_1:func 11
notation
  let a1, a2, a3 be real set;
  synonym |[a1,a2,a3]| for <*a1,a2,a3*>;
end;

:: EUCLID_5:funcnot 5 => EUCLID_5:func 4
definition
  let a1, a2, a3 be real set;
  redefine func |[a1, a2, a3]| -> Element of the carrier of TOP-REAL 3;
end;

:: EUCLID_5:th 2
theorem
for b1, b2, b3 being Element of REAL holds
|[b1,b2,b3]| `1 = b1 & |[b1,b2,b3]| `2 = b2 & |[b1,b2,b3]| `3 = b3;

:: EUCLID_5:th 3
theorem
for b1 being Element of the carrier of TOP-REAL 3 holds
   b1 = |[b1 `1,b1 `2,b1 `3]|;

:: EUCLID_5:th 4
theorem
0.REAL 3 = |[0,0,0]|;

:: EUCLID_5:th 5
theorem
for b1, b2 being Element of the carrier of TOP-REAL 3 holds
b1 + b2 = |[b1 `1 + (b2 `1),b1 `2 + (b2 `2),b1 `3 + (b2 `3)]|;

:: EUCLID_5:th 6
theorem
for b1, b2, b3, b4, b5, b6 being Element of REAL holds
|[b1,b2,b3]| + |[b4,b5,b6]| = |[b1 + b4,b2 + b5,b3 + b6]|;

:: EUCLID_5:th 7
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 3 holds
   b1 * b2 = |[b1 * (b2 `1),b1 * (b2 `2),b1 * (b2 `3)]|;

:: EUCLID_5:th 8
theorem
for b1, b2, b3, b4 being Element of REAL holds
b1 * |[b2,b3,b4]| = |[b1 * b2,b1 * b3,b1 * b4]|;

:: EUCLID_5:th 9
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 3 holds
   (b1 * b2) `1 = b1 * (b2 `1) & (b1 * b2) `2 = b1 * (b2 `2) & (b1 * b2) `3 = b1 * (b2 `3);

:: EUCLID_5:th 10
theorem
for b1 being Element of the carrier of TOP-REAL 3 holds
   - b1 = |[- (b1 `1),- (b1 `2),- (b1 `3)]|;

:: EUCLID_5:th 11
theorem
for b1, b2, b3 being Element of REAL holds
- |[b1,b2,b3]| = |[- b1,- b2,- b3]|;

:: EUCLID_5:th 12
theorem
for b1, b2 being Element of the carrier of TOP-REAL 3 holds
b1 - b2 = |[b1 `1 - (b2 `1),b1 `2 - (b2 `2),b1 `3 - (b2 `3)]|;

:: EUCLID_5:th 13
theorem
for b1, b2, b3, b4, b5, b6 being Element of REAL holds
|[b1,b2,b3]| - |[b4,b5,b6]| = |[b1 - b4,b2 - b5,b3 - b6]|;

:: EUCLID_5:funcnot 6 => EUCLID_5:func 5
definition
  let a1, a2 be Element of the carrier of TOP-REAL 3;
  func A1 <X> A2 -> Element of the carrier of TOP-REAL 3 equals
    |[(a1 `2 * (a2 `3)) - (a1 `3 * (a2 `2)),(a1 `3 * (a2 `1)) - (a1 `1 * (a2 `3)),(a1 `1 * (a2 `2)) - (a1 `2 * (a2 `1))]|;
end;

:: EUCLID_5:def 5
theorem
for b1, b2 being Element of the carrier of TOP-REAL 3 holds
b1 <X> b2 = |[(b1 `2 * (b2 `3)) - (b1 `3 * (b2 `2)),(b1 `3 * (b2 `1)) - (b1 `1 * (b2 `3)),(b1 `1 * (b2 `2)) - (b1 `2 * (b2 `1))]|;

:: EUCLID_5:th 14
theorem
for b1, b2, b3 being Element of REAL
for b4 being Element of the carrier of TOP-REAL 3
      st b4 = |[b1,b2,b3]|
   holds b4 `1 = b1 & b4 `2 = b2 & b4 `3 = b3;

:: EUCLID_5:th 15
theorem
for b1, b2, b3, b4, b5, b6 being Element of REAL holds
|[b1,b2,b3]| <X> |[b4,b5,b6]| = |[(b2 * b6) - (b3 * b5),(b3 * b4) - (b1 * b6),(b1 * b5) - (b2 * b4)]|;

:: EUCLID_5:th 16
theorem
for b1 being Element of REAL
for b2, b3 being Element of the carrier of TOP-REAL 3 holds
(b1 * b2) <X> b3 = b1 * (b2 <X> b3) &
 (b1 * b2) <X> b3 = b2 <X> (b1 * b3);

:: EUCLID_5:th 17
theorem
for b1, b2 being Element of the carrier of TOP-REAL 3 holds
b1 <X> b2 = - (b2 <X> b1);

:: EUCLID_5:th 18
theorem
for b1, b2 being Element of the carrier of TOP-REAL 3 holds
(- b1) <X> b2 = b1 <X> - b2;

:: EUCLID_5:th 19
theorem
for b1, b2, b3 being Element of REAL holds
|[0,0,0]| <X> |[b1,b2,b3]| = 0.REAL 3;

:: EUCLID_5:th 20
theorem
for b1, b2 being Element of REAL holds
|[b1,0,0]| <X> |[b2,0,0]| = 0.REAL 3;

:: EUCLID_5:th 21
theorem
for b1, b2 being Element of REAL holds
|[0,b1,0]| <X> |[0,b2,0]| = 0.REAL 3;

:: EUCLID_5:th 22
theorem
for b1, b2 being Element of REAL holds
|[0,0,b1]| <X> |[0,0,b2]| = 0.REAL 3;

:: EUCLID_5:th 23
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 3 holds
b1 <X> (b2 + b3) = (b1 <X> b2) + (b1 <X> b3);

:: EUCLID_5:th 24
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 3 holds
(b1 + b2) <X> b3 = (b1 <X> b3) + (b2 <X> b3);

:: EUCLID_5:th 25
theorem
for b1 being Element of the carrier of TOP-REAL 3 holds
   b1 <X> b1 = 0.REAL 3;

:: EUCLID_5:th 26
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 3 holds
(b1 + b2) <X> (b3 + b4) = (((b1 <X> b3) + (b1 <X> b4)) + (b2 <X> b3)) + (b2 <X> b4);

:: EUCLID_5:th 27
theorem
for b1 being Element of the carrier of TOP-REAL 3 holds
   b1 = <*b1 `1,b1 `2,b1 `3*>;

:: EUCLID_5:th 28
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = 3 & len b2 = 3
   holds mlt(b1,b2) = <*(b1 . 1) * (b2 . 1),(b1 . 2) * (b2 . 2),(b1 . 3) * (b2 . 3)*>;

:: EUCLID_5:th 29
theorem
for b1, b2 being Element of the carrier of TOP-REAL 3 holds
|(b1,b2)| = ((b1 `1 * (b2 `1)) + (b1 `2 * (b2 `2))) + (b1 `3 * (b2 `3));

:: EUCLID_5:th 30
theorem
for b1, b2, b3, b4, b5, b6 being Element of REAL holds
|(|[b3,b4,b1]|,|[b5,b6,b2]|)| = ((b3 * b5) + (b4 * b6)) + (b1 * b2);

:: EUCLID_5:funcnot 7 => EUCLID_5:func 6
definition
  let a1, a2, a3 be Element of the carrier of TOP-REAL 3;
  func |{A1,A2,A3}| -> real set equals
    |(a1,a2 <X> a3)|;
end;

:: EUCLID_5:def 6
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 3 holds
|{b1,b2,b3}| = |(b1,b2 <X> b3)|;

:: EUCLID_5:th 31
theorem
for b1, b2 being Element of the carrier of TOP-REAL 3 holds
|{b1,b1,b2}| = 0 & |{b2,b1,b2}| = 0;

:: EUCLID_5:th 32
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 3 holds
b1 <X> (b2 <X> b3) = (|(b1,b3)| * b2) - (|(b1,b2)| * b3);

:: EUCLID_5:th 33
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 3 holds
|{b1,b2,b3}| = |{b2,b3,b1}|;

:: EUCLID_5:th 34
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 3 holds
|{b1,b2,b3}| = |{b3,b1,b2}|;

:: EUCLID_5:th 35
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 3 holds
|{b1,b2,b3}| = |(b1 <X> b2,b3)|;