Article EUCLID_2, MML version 4.99.1005

:: EUCLID_2:th 1
theorem
for b1 being Element of NAT
for b2 being Element of b1 -tuples_on REAL
for b3 being Element of NAT
      st b3 in Seg b1
   holds (mlt(b2,0* b1)) . b3 = 0;

:: EUCLID_2:th 2
theorem
for b1 being Element of NAT
for b2 being Element of b1 -tuples_on REAL holds
   mlt(b2,0* b1) = 0* b1;

:: EUCLID_2:th 3
theorem
for b1 being FinSequence of REAL holds
   (- 1) * b1 = - b1;

:: EUCLID_2:th 4
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2
   holds b1 - b2 = b1 + - b2;

:: EUCLID_2:th 5
theorem
for b1 being FinSequence of REAL holds
   len - b1 = len b1;

:: EUCLID_2:th 6
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2
   holds len (b1 + b2) = len b1;

:: EUCLID_2:th 7
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2
   holds len (b1 - b2) = len b1;

:: EUCLID_2:th 8
theorem
for b1 being real set
for b2 being FinSequence of REAL holds
   len (b1 * b2) = len b2;

:: EUCLID_2:th 9
theorem
for b1, b2, b3 being FinSequence of REAL
      st len b1 = len b2 & len b2 = len b3
   holds mlt(b1 + b2,b3) = (mlt(b1,b3)) + mlt(b2,b3);

:: EUCLID_2:funcnot 1 => EUCLID_2:func 1
definition
  let a1, a2 be Relation-like Function-like FinSequence-like real-valued set;
  func |(A1,A2)| -> real set equals
    Sum mlt(a1,a2);
  commutativity;
::  for a1, a2 being Relation-like Function-like FinSequence-like real-valued set holds
::  |(a1,a2)| = |(a2,a1)|;
end;

:: EUCLID_2:def 1
theorem
for b1, b2 being Relation-like Function-like FinSequence-like real-valued set holds
|(b1,b2)| = Sum mlt(b1,b2);

:: EUCLID_2:funcnot 2 => EUCLID_2:func 2
definition
  let a1, a2 be Relation-like Function-like FinSequence-like real-valued set;
  redefine func |(a1, a2)| -> Element of REAL;
  commutativity;
::  for a1, a2 being Relation-like Function-like FinSequence-like real-valued set holds
::  |(a1,a2)| = |(a2,a1)|;
end;

:: EUCLID_2:th 10
theorem
for b1 being Element of NAT
for b2, b3 being FinSequence of REAL
for b4, b5 being Element of REAL b1
      st b4 = b2 & b5 = b3
   holds |(b2,b3)| = (1 / 4) * (|.b4 + b5.| ^2 - (|.b4 - b5.| ^2));

:: EUCLID_2:th 11
theorem
for b1 being FinSequence of REAL holds
   0 <= |(b1,b1)|;

:: EUCLID_2:th 12
theorem
for b1 being FinSequence of REAL holds
   |.b1.| ^2 = |(b1,b1)|;

:: EUCLID_2:th 13
theorem
for b1 being FinSequence of REAL holds
   |.b1.| = sqrt |(b1,b1)|;

:: EUCLID_2:th 14
theorem
for b1 being FinSequence of REAL holds
   0 <= |.b1.|;

:: EUCLID_2:th 15
theorem
for b1 being FinSequence of REAL holds
      |(b1,b1)| = 0
   iff
      b1 = 0* len b1;

:: EUCLID_2:th 16
theorem
for b1 being FinSequence of REAL holds
      |(b1,b1)| = 0
   iff
      |.b1.| = 0;

:: EUCLID_2:th 17
theorem
for b1 being FinSequence of REAL holds
   |(b1,0* len b1)| = 0;

:: EUCLID_2:th 18
theorem
for b1 being FinSequence of REAL holds
   |(0* len b1,b1)| = 0;

:: EUCLID_2:th 19
theorem
for b1, b2, b3 being FinSequence of REAL
      st len b1 = len b2 & len b2 = len b3
   holds |(b1 + b2,b3)| = |(b1,b3)| + |(b2,b3)|;

:: EUCLID_2:th 20
theorem
for b1, b2 being FinSequence of REAL
for b3 being real set
      st len b1 = len b2
   holds |(b3 * b1,b2)| = b3 * |(b1,b2)|;

:: EUCLID_2:th 21
theorem
for b1, b2 being FinSequence of REAL
for b3 being real set
      st len b1 = len b2
   holds |(b1,b3 * b2)| = b3 * |(b1,b2)|;

:: EUCLID_2:th 22
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2
   holds |(- b1,b2)| = - |(b1,b2)|;

:: EUCLID_2:th 23
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2
   holds |(b1,- b2)| = - |(b1,b2)|;

:: EUCLID_2:th 24
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2
   holds |(- b1,- b2)| = |(b1,b2)|;

:: EUCLID_2:th 25
theorem
for b1, b2, b3 being FinSequence of REAL
      st len b1 = len b2 & len b2 = len b3
   holds |(b1 - b2,b3)| = |(b1,b3)| - |(b2,b3)|;

:: EUCLID_2:th 26
theorem
for b1, b2 being real set
for b3, b4, b5 being FinSequence of REAL
      st len b3 = len b4 & len b4 = len b5
   holds |((b1 * b3) + (b2 * b4),b5)| = (b1 * |(b3,b5)|) + (b2 * |(b4,b5)|);

:: EUCLID_2:th 27
theorem
for b1, b2, b3 being FinSequence of REAL
      st len b1 = len b2 & len b2 = len b3
   holds |(b1,b2 + b3)| = |(b1,b2)| + |(b1,b3)|;

:: EUCLID_2:th 28
theorem
for b1, b2, b3 being FinSequence of REAL
      st len b1 = len b2 & len b2 = len b3
   holds |(b1,b2 - b3)| = |(b1,b2)| - |(b1,b3)|;

:: EUCLID_2:th 29
theorem
for b1, b2, b3, b4 being FinSequence of REAL
      st len b1 = len b2 & len b2 = len b3 & len b3 = len b4
   holds |(b1 + b2,b3 + b4)| = ((|(b1,b3)| + |(b1,b4)|) + |(b2,b3)|) + |(b2,b4)|;

:: EUCLID_2:th 30
theorem
for b1, b2, b3, b4 being FinSequence of REAL
      st len b1 = len b2 & len b2 = len b3 & len b3 = len b4
   holds |(b1 - b2,b3 - b4)| = ((|(b1,b3)| - |(b1,b4)|) - |(b2,b3)|) + |(b2,b4)|;

:: EUCLID_2:th 31
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2
   holds |(b1 + b2,b1 + b2)| = (|(b1,b1)| + (2 * |(b1,b2)|)) + |(b2,b2)|;

:: EUCLID_2:th 32
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2
   holds |(b1 - b2,b1 - b2)| = (|(b1,b1)| - (2 * |(b1,b2)|)) + |(b2,b2)|;

:: EUCLID_2:th 33
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2
   holds |.b1 + b2.| ^2 = (|.b1.| ^2 + (2 * |(b2,b1)|)) + (|.b2.| ^2);

:: EUCLID_2:th 34
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2
   holds |.b1 - b2.| ^2 = (|.b1.| ^2 - (2 * |(b2,b1)|)) + (|.b2.| ^2);

:: EUCLID_2:th 35
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2
   holds |.b1 + b2.| ^2 + (|.b1 - b2.| ^2) = 2 * (|.b1.| ^2 + (|.b2.| ^2));

:: EUCLID_2:th 36
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2
   holds |.b1 + b2.| ^2 - (|.b1 - b2.| ^2) = 4 * |(b1,b2)|;

:: EUCLID_2:th 37
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2
   holds abs |(b1,b2)| <= |.b1.| * |.b2.|;

:: EUCLID_2:th 38
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2
   holds |.b1 + b2.| <= |.b1.| + |.b2.|;

:: EUCLID_2:th 39
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|(b2,b3)| = (1 / 4) * (|.b2 + b3.| ^2 - (|.b2 - b3.| ^2));

:: EUCLID_2:th 40
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1 holds
|(b2 + b3,b4)| = |(b2,b4)| + |(b3,b4)|;

:: EUCLID_2:th 41
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being real set holds
   |(b4 * b2,b3)| = b4 * |(b2,b3)|;

:: EUCLID_2:th 42
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being real set holds
   |(b2,b4 * b3)| = b4 * |(b2,b3)|;

:: EUCLID_2:th 43
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|(- b2,b3)| = - |(b2,b3)|;

:: EUCLID_2:th 44
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|(b2,- b3)| = - |(b2,b3)|;

:: EUCLID_2:th 45
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|(- b2,- b3)| = |(b2,b3)|;

:: EUCLID_2:th 46
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1 holds
|(b2 - b3,b4)| = |(b2,b4)| - |(b3,b4)|;

:: EUCLID_2:th 47
theorem
for b1 being Element of NAT
for b2, b3 being real set
for b4, b5, b6 being Element of the carrier of TOP-REAL b1 holds
|((b2 * b4) + (b3 * b5),b6)| = (b2 * |(b4,b6)|) + (b3 * |(b5,b6)|);

:: EUCLID_2:th 48
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1 holds
|(b2,b3 + b4)| = |(b2,b3)| + |(b2,b4)|;

:: EUCLID_2:th 49
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1 holds
|(b2,b3 - b4)| = |(b2,b3)| - |(b2,b4)|;

:: EUCLID_2:th 50
theorem
for b1 being Element of NAT
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL b1 holds
|(b2 + b3,b4 + b5)| = ((|(b2,b4)| + |(b2,b5)|) + |(b3,b4)|) + |(b3,b5)|;

:: EUCLID_2:th 51
theorem
for b1 being Element of NAT
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL b1 holds
|(b2 - b3,b4 - b5)| = ((|(b2,b4)| - |(b2,b5)|) - |(b3,b4)|) + |(b3,b5)|;

:: EUCLID_2:th 52
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|(b2 + b3,b2 + b3)| = (|(b2,b2)| + (2 * |(b2,b3)|)) + |(b3,b3)|;

:: EUCLID_2:th 53
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|(b2 - b3,b2 - b3)| = (|(b2,b2)| - (2 * |(b2,b3)|)) + |(b3,b3)|;

:: EUCLID_2:th 54
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
   |(b2,0.REAL b1)| = 0;

:: EUCLID_2:th 55
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
   |(0.REAL b1,b2)| = 0;

:: EUCLID_2:th 56
theorem
for b1 being Element of NAT holds
   |(0.REAL b1,0.REAL b1)| = 0;

:: EUCLID_2:th 57
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
   0 <= |(b2,b2)|;

:: EUCLID_2:th 58
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
   |(b2,b2)| = |.b2.| ^2;

:: EUCLID_2:th 59
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
   |.b2.| = sqrt |(b2,b2)|;

:: EUCLID_2:th 60
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
   0 <= |.b2.|;

:: EUCLID_2:th 61
theorem
for b1 being Element of NAT holds
   |.0.REAL b1.| = 0;

:: EUCLID_2:th 62
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
      |(b2,b2)| = 0
   iff
      |.b2.| = 0;

:: EUCLID_2:th 63
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
      |(b2,b2)| = 0
   iff
      b2 = 0.REAL b1;

:: EUCLID_2:th 64
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
      |.b2.| = 0
   iff
      b2 = 0.REAL b1;

:: EUCLID_2:th 65
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
      b2 <> 0.REAL b1
   iff
      0 < |(b2,b2)|;

:: EUCLID_2:th 66
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
      b2 <> 0.REAL b1
   iff
      0 < |.b2.|;

:: EUCLID_2:th 67
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2 + b3.| ^2 = (|.b2.| ^2 + (2 * |(b3,b2)|)) + (|.b3.| ^2);

:: EUCLID_2:th 68
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2 - b3.| ^2 = (|.b2.| ^2 - (2 * |(b3,b2)|)) + (|.b3.| ^2);

:: EUCLID_2:th 69
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2 + b3.| ^2 + (|.b2 - b3.| ^2) = 2 * (|.b2.| ^2 + (|.b3.| ^2));

:: EUCLID_2:th 70
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2 + b3.| ^2 - (|.b2 - b3.| ^2) = 4 * |(b2,b3)|;

:: EUCLID_2:th 71
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|(b2,b3)| = (1 / 4) * (|.b2 + b3.| ^2 - (|.b2 - b3.| ^2));

:: EUCLID_2:th 72
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|(b2,b3)| <= |(b2,b2)| + |(b3,b3)|;

:: EUCLID_2:th 73
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
abs |(b2,b3)| <= |.b2.| * |.b3.|;

:: EUCLID_2:th 74
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
|.b2 + b3.| <= |.b2.| + |.b3.|;

:: EUCLID_2:prednot 1 => EUCLID_2:pred 1
definition
  let a1 be Element of NAT;
  let a2, a3 be Element of the carrier of TOP-REAL a1;
  pred A2,A3 are_orthogonal means
    |(a2,a3)| = 0;
  symmetry;
::  for a1 being Element of NAT
::  for a2, a3 being Element of the carrier of TOP-REAL a1
::        st a2,a3 are_orthogonal
::     holds a3,a2 are_orthogonal;
end;

:: EUCLID_2:dfs 2
definiens
  let a1 be Element of NAT;
  let a2, a3 be Element of the carrier of TOP-REAL a1;
To prove
     a2,a3 are_orthogonal
it is sufficient to prove
  thus |(a2,a3)| = 0;

:: EUCLID_2:def 3
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
   b2,b3 are_orthogonal
iff
   |(b2,b3)| = 0;

:: EUCLID_2:th 75
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
   b2,0.REAL b1 are_orthogonal;

:: EUCLID_2:th 76
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
   0.REAL b1,b2 are_orthogonal;

:: EUCLID_2:th 77
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
      b2,b2 are_orthogonal
   iff
      b2 = 0.REAL b1;

:: EUCLID_2:th 78
theorem
for b1 being Element of NAT
for b2 being real set
for b3, b4 being Element of the carrier of TOP-REAL b1
      st b3,b4 are_orthogonal
   holds b2 * b3,b4 are_orthogonal;

:: EUCLID_2:th 79
theorem
for b1 being Element of NAT
for b2 being real set
for b3, b4 being Element of the carrier of TOP-REAL b1
      st b3,b4 are_orthogonal
   holds b3,b2 * b4 are_orthogonal;

:: EUCLID_2:th 80
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
      st for b3 being Element of the carrier of TOP-REAL b1 holds
           b2,b3 are_orthogonal
   holds b2 = 0.REAL b1;