Article BHSP_1, MML version 4.99.1005

:: BHSP_1:structnot 1 => BHSP_1:struct 1
definition
  struct(RLSStruct) UNITSTR(#
    carrier -> set,
    ZeroF -> Element of the carrier of it,
    addF -> Function-like quasi_total Relation of [:the carrier of it,the carrier of it:],the carrier of it,
    Mult -> Function-like quasi_total Relation of [:REAL,the carrier of it:],the carrier of it,
    scalar -> Function-like quasi_total Relation of [:the carrier of it,the carrier of it:],REAL
  #);
end;

:: BHSP_1:attrnot 1 => BHSP_1:attr 1
definition
  let a1 be UNITSTR;
  attr a1 is strict;
end;

:: BHSP_1:exreg 1
registration
  cluster strict UNITSTR;
end;

:: BHSP_1:aggrnot 1 => BHSP_1:aggr 1
definition
  let a1 be set;
  let a2 be Element of a1;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
  let a4 be Function-like quasi_total Relation of [:REAL,a1:],a1;
  let a5 be Function-like quasi_total Relation of [:a1,a1:],REAL;
  aggr UNITSTR(#a1,a2,a3,a4,a5#) -> strict UNITSTR;
end;

:: BHSP_1:selnot 1 => BHSP_1:sel 1
definition
  let a1 be UNITSTR;
  sel the scalar of a1 -> Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],REAL;
end;

:: BHSP_1:exreg 2
registration
  cluster non empty strict UNITSTR;
end;

:: BHSP_1:funcreg 1
registration
  let a1 be non empty set;
  let a2 be Element of a1;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
  let a4 be Function-like quasi_total Relation of [:REAL,a1:],a1;
  let a5 be Function-like quasi_total Relation of [:a1,a1:],REAL;
  cluster UNITSTR(#a1,a2,a3,a4,a5#) -> non empty strict;
end;

:: BHSP_1:funcnot 1 => BHSP_1:func 1
definition
  let a1 be non empty UNITSTR;
  let a2, a3 be Element of the carrier of a1;
  func A2 .|. A3 -> Element of REAL equals
    (the scalar of a1) . [a2,a3];
end;

:: BHSP_1:def 1
theorem
for b1 being non empty UNITSTR
for b2, b3 being Element of the carrier of b1 holds
b2 .|. b3 = (the scalar of b1) . [b2,b3];

:: BHSP_1:attrnot 2 => BHSP_1:attr 2
definition
  let a1 be non empty UNITSTR;
  attr a1 is RealUnitarySpace-like means
    for b1, b2, b3 being Element of the carrier of a1
    for b4 being Element of REAL holds
       (b1 .|. b1 = 0 implies b1 = 0. a1) &
        (b1 = 0. a1 implies b1 .|. b1 = 0) &
        0 <= b1 .|. b1 &
        b1 .|. b2 = b2 .|. b1 &
        (b1 + b2) .|. b3 = (b1 .|. b3) + (b2 .|. b3) &
        (b4 * b1) .|. b2 = b4 * (b1 .|. b2);
end;

:: BHSP_1:dfs 2
definiens
  let a1 be non empty UNITSTR;
To prove
     a1 is RealUnitarySpace-like
it is sufficient to prove
  thus for b1, b2, b3 being Element of the carrier of a1
    for b4 being Element of REAL holds
       (b1 .|. b1 = 0 implies b1 = 0. a1) &
        (b1 = 0. a1 implies b1 .|. b1 = 0) &
        0 <= b1 .|. b1 &
        b1 .|. b2 = b2 .|. b1 &
        (b1 + b2) .|. b3 = (b1 .|. b3) + (b2 .|. b3) &
        (b4 * b1) .|. b2 = b4 * (b1 .|. b2);

:: BHSP_1:def 2
theorem
for b1 being non empty UNITSTR holds
      b1 is RealUnitarySpace-like
   iff
      for b2, b3, b4 being Element of the carrier of b1
      for b5 being Element of REAL holds
         (b2 .|. b2 = 0 implies b2 = 0. b1) &
          (b2 = 0. b1 implies b2 .|. b2 = 0) &
          0 <= b2 .|. b2 &
          b2 .|. b3 = b3 .|. b2 &
          (b2 + b3) .|. b4 = (b2 .|. b4) + (b3 .|. b4) &
          (b5 * b2) .|. b3 = b5 * (b2 .|. b3);

:: BHSP_1:exreg 3
registration
  cluster non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealUnitarySpace-like UNITSTR;
end;

:: BHSP_1:modenot 1
definition
  mode RealUnitarySpace is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
end;

:: BHSP_1:funcnot 2 => BHSP_1:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2, a3 be Element of the carrier of a1;
  redefine func a2 .|. a3 -> Element of REAL;
  commutativity;
::  for a1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
::  for a2, a3 being Element of the carrier of a1 holds
::  a2 .|. a3 = a3 .|. a2;
end;

:: BHSP_1:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
   (0. b1) .|. 0. b1 = 0;

:: BHSP_1:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
b2 .|. (b3 + b4) = (b2 .|. b3) + (b2 .|. b4);

:: BHSP_1:th 8
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3, b4 being Element of the carrier of b2 holds
b3 .|. (b1 * b4) = b1 * (b3 .|. b4);

:: BHSP_1:th 9
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3, b4 being Element of the carrier of b2 holds
(b1 * b3) .|. b4 = b3 .|. (b1 * b4);

:: BHSP_1:th 10
theorem
for b1, b2 being Element of REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b4, b5, b6 being Element of the carrier of b3 holds
((b1 * b4) + (b2 * b5)) .|. b6 = (b1 * (b4 .|. b6)) + (b2 * (b5 .|. b6));

:: BHSP_1:th 11
theorem
for b1, b2 being Element of REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b4, b5, b6 being Element of the carrier of b3 holds
b4 .|. ((b1 * b5) + (b2 * b6)) = (b1 * (b4 .|. b5)) + (b2 * (b4 .|. b6));

:: BHSP_1:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
(- b2) .|. b3 = b2 .|. - b3;

:: BHSP_1:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
(- b2) .|. b3 = - (b2 .|. b3);

:: BHSP_1:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
b2 .|. - b3 = - (b2 .|. b3);

:: BHSP_1:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
(- b2) .|. - b3 = b2 .|. b3;

:: BHSP_1:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 - b3) .|. b4 = (b2 .|. b4) - (b3 .|. b4);

:: BHSP_1:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
b2 .|. (b3 - b4) = (b2 .|. b3) - (b2 .|. b4);

:: BHSP_1:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(b2 - b3) .|. (b4 - b5) = (((b2 .|. b4) - (b2 .|. b5)) - (b3 .|. b4)) + (b3 .|. b5);

:: BHSP_1:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1 holds
   (0. b1) .|. b2 = 0;

:: BHSP_1:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1 holds
   b2 .|. 0. b1 = 0;

:: BHSP_1:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
(b2 + b3) .|. (b2 + b3) = ((b2 .|. b2) + (2 * (b2 .|. b3))) + (b3 .|. b3);

:: BHSP_1:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
(b2 + b3) .|. (b2 - b3) = (b2 .|. b2) - (b3 .|. b3);

:: BHSP_1:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
(b2 - b3) .|. (b2 - b3) = ((b2 .|. b2) - (2 * (b2 .|. b3))) + (b3 .|. b3);

:: BHSP_1:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
abs (b2 .|. b3) <= (sqrt (b2 .|. b2)) * sqrt (b3 .|. b3);

:: BHSP_1:prednot 1 => BHSP_1:pred 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2, a3 be Element of the carrier of a1;
  pred A2,A3 are_orthogonal means
    a2 .|. a3 = 0;
  symmetry;
::  for a1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
::  for a2, a3 being Element of the carrier of a1
::        st a2,a3 are_orthogonal
::     holds a3,a2 are_orthogonal;
end;

:: BHSP_1:dfs 3
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2, a3 be Element of the carrier of a1;
To prove
     a2,a3 are_orthogonal
it is sufficient to prove
  thus a2 .|. a3 = 0;

:: BHSP_1:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
   b2,b3 are_orthogonal
iff
   b2 .|. b3 = 0;

:: BHSP_1:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
      st b2,b3 are_orthogonal
   holds b2,- b3 are_orthogonal;

:: BHSP_1:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
      st b2,b3 are_orthogonal
   holds - b2,b3 are_orthogonal;

:: BHSP_1:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
      st b2,b3 are_orthogonal
   holds - b2,- b3 are_orthogonal;

:: BHSP_1:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1 holds
   b2,0. b1 are_orthogonal;

:: BHSP_1:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
      st b2,b3 are_orthogonal
   holds (b2 + b3) .|. (b2 + b3) = (b2 .|. b2) + (b3 .|. b3);

:: BHSP_1:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
      st b2,b3 are_orthogonal
   holds (b2 - b3) .|. (b2 - b3) = (b2 .|. b2) + (b3 .|. b3);

:: BHSP_1:funcnot 3 => BHSP_1:func 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Element of the carrier of a1;
  func ||.A2.|| -> Element of REAL equals
    sqrt (a2 .|. a2);
end;

:: BHSP_1:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1 holds
   ||.b2.|| = sqrt (b2 .|. b2);

:: BHSP_1:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1 holds
      ||.b2.|| = 0
   iff
      b2 = 0. b1;

:: BHSP_1:th 33
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3 being Element of the carrier of b2 holds
   ||.b1 * b3.|| = (abs b1) * ||.b3.||;

:: BHSP_1:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1 holds
   0 <= ||.b2.||;

:: BHSP_1:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
abs (b2 .|. b3) <= ||.b2.|| * ||.b3.||;

:: BHSP_1:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2 + b3.|| <= ||.b2.|| + ||.b3.||;

:: BHSP_1:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1 holds
   ||.- b2.|| = ||.b2.||;

:: BHSP_1:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2.|| - ||.b3.|| <= ||.b2 - b3.||;

:: BHSP_1:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
abs (||.b2.|| - ||.b3.||) <= ||.b2 - b3.||;

:: BHSP_1:funcnot 4 => BHSP_1:func 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2, a3 be Element of the carrier of a1;
  func dist(A2,A3) -> Element of REAL equals
    ||.a2 - a3.||;
end;

:: BHSP_1:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
dist(b2,b3) = ||.b2 - b3.||;

:: BHSP_1:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
dist(b2,b3) = dist(b3,b2);

:: BHSP_1:funcnot 5 => BHSP_1:func 5
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2, a3 be Element of the carrier of a1;
  redefine func dist(a2,a3) -> Element of REAL;
  commutativity;
::  for a1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
::  for a2, a3 being Element of the carrier of a1 holds
::  dist(a2,a3) = dist(a3,a2);
end;

:: BHSP_1:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1 holds
   dist(b2,b2) = 0;

:: BHSP_1:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
dist(b2,b3) <= (dist(b2,b4)) + dist(b4,b3);

:: BHSP_1:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
   b2 <> b3
iff
   dist(b2,b3) <> 0;

:: BHSP_1:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
0 <= dist(b2,b3);

:: BHSP_1:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
   b2 <> b3
iff
   0 < dist(b2,b3);

:: BHSP_1:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
dist(b2,b3) = sqrt ((b2 - b3) .|. (b2 - b3));

:: BHSP_1:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4, b5 being Element of the carrier of b1 holds
dist(b2 + b3,b4 + b5) <= (dist(b2,b4)) + dist(b3,b5);

:: BHSP_1:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4, b5 being Element of the carrier of b1 holds
dist(b2 - b3,b4 - b5) <= (dist(b2,b4)) + dist(b3,b5);

:: BHSP_1:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
dist(b2 - b3,b4 - b3) = dist(b2,b4);

:: BHSP_1:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
dist(b2 - b3,b4 - b3) <= (dist(b3,b2)) + dist(b3,b4);

:: BHSP_1:funcnot 6 => BHSP_1:func 6
definition
  let a1 be non empty addLoopStr;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  func - A2 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    for b1 being Element of NAT holds
       it . b1 = - (a2 . b1);
end;

:: BHSP_1:def 10
theorem
for b1 being non empty addLoopStr
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b3 = - b2
iff
   for b4 being Element of NAT holds
      b3 . b4 = - (b2 . b4);

:: BHSP_1:funcnot 7 => BHSP_1:func 7
definition
  let a1 be non empty addLoopStr;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  let a3 be Element of the carrier of a1;
  func A2 + A3 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    for b1 being Element of NAT holds
       it . b1 = (a2 . b1) + a3;
end;

:: BHSP_1:def 12
theorem
for b1 being non empty addLoopStr
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 Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b4 = b2 + b3
   iff
      for b5 being Element of NAT holds
         b4 . b5 = (b2 . b5) + b3;

:: BHSP_1:th 55
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 + b3 = b3 + b2;

:: BHSP_1:funcnot 8 => BHSP_1:func 8
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2, a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
  redefine func a2 + a3 -> Function-like quasi_total Relation of NAT,the carrier of a1;
  commutativity;
::  for a1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
::  for a2, a3 being Function-like quasi_total Relation of NAT,the carrier of a1 holds
::  a2 + a3 = a3 + a2;
end;

:: BHSP_1:th 56
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 + (b3 + b4) = (b2 + b3) + b4;

:: BHSP_1:th 57
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is constant & b3 is constant & b4 = b2 + b3
   holds b4 is constant;

:: BHSP_1:th 58
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is constant & b3 is constant & b4 = b2 - b3
   holds b4 is constant;

:: BHSP_1:th 59
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b2
      st b3 is constant & b4 = b1 * b3
   holds b4 is constant;

:: BHSP_1:th 68
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is constant
   iff
      for b3 being Element of NAT holds
         b2 . b3 = b2 . (b3 + 1);

:: BHSP_1:th 69
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is constant
   iff
      for b3, b4 being Element of NAT holds
      b2 . b3 = b2 . (b3 + b4);

:: BHSP_1:th 70
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is constant
   iff
      for b3, b4 being Element of NAT holds
      b2 . b3 = b2 . b4;

:: BHSP_1:th 71
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 - b3 = b2 + - b3;

:: BHSP_1:th 72
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b2 = b2 + 0. b1;

:: BHSP_1:th 73
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
b1 * (b3 + b4) = (b1 * b3) + (b1 * b4);

:: BHSP_1:th 74
theorem
for b1, b2 being Element of REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b4 being Function-like quasi_total Relation of NAT,the carrier of b3 holds
   (b1 + b2) * b4 = (b1 * b4) + (b2 * b4);

:: BHSP_1:th 75
theorem
for b1, b2 being Element of REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b4 being Function-like quasi_total Relation of NAT,the carrier of b3 holds
   (b1 * b2) * b4 = b1 * (b2 * b4);

:: BHSP_1:th 76
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   1 * b2 = b2;

:: BHSP_1:th 77
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   (- 1) * b2 = - b2;

:: BHSP_1:th 78
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b3 - b2 = b3 + - b2;

:: BHSP_1:th 79
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 - b3 = - (b3 - b2);

:: BHSP_1:th 80
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b2 = b2 - 0. b1;

:: BHSP_1:th 81
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b2 = - - b2;

:: BHSP_1:th 82
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 - (b3 + b4) = (b2 - b3) - b4;

:: BHSP_1:th 83
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
(b2 + b3) - b4 = b2 + (b3 - b4);

:: BHSP_1:th 84
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 - (b3 - b4) = (b2 - b3) + b4;

:: BHSP_1:th 85
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
b1 * (b3 - b4) = (b1 * b3) - (b1 * b4);