Article CSSPACE, MML version 4.99.1005

:: CSSPACE:funcnot 1 => CSSPACE:func 1
definition
  func the_set_of_ComplexSequences -> non empty set means
    for b1 being set holds
          b1 in it
       iff
          b1 is Function-like quasi_total Relation of NAT,COMPLEX;
end;

:: CSSPACE:def 1
theorem
for b1 being non empty set holds
      b1 = the_set_of_ComplexSequences
   iff
      for b2 being set holds
            b2 in b1
         iff
            b2 is Function-like quasi_total Relation of NAT,COMPLEX;

:: CSSPACE:funcnot 2 => CSSPACE:func 2
definition
  let a1 be set;
  assume a1 in the_set_of_ComplexSequences;
  func seq_id A1 -> Function-like quasi_total Relation of NAT,COMPLEX equals
    a1;
end;

:: CSSPACE:def 2
theorem
for b1 being set
      st b1 in the_set_of_ComplexSequences
   holds seq_id b1 = b1;

:: CSSPACE:funcnot 3 => CSSPACE:func 3
definition
  let a1 be set;
  assume a1 in COMPLEX;
  func C_id A1 -> Element of COMPLEX equals
    a1;
end;

:: CSSPACE:def 3
theorem
for b1 being set
      st b1 in COMPLEX
   holds C_id b1 = b1;

:: CSSPACE:th 1
theorem
ex b1 being Function-like quasi_total Relation of [:the_set_of_ComplexSequences,the_set_of_ComplexSequences:],the_set_of_ComplexSequences st
   (for b2, b3 being Element of the_set_of_ComplexSequences holds
    b1 .(b2,b3) = (seq_id b2) + seq_id b3) &
    b1 is commutative(the_set_of_ComplexSequences) &
    b1 is associative(the_set_of_ComplexSequences);

:: CSSPACE:th 2
theorem
ex b1 being Function-like quasi_total Relation of [:COMPLEX,the_set_of_ComplexSequences:],the_set_of_ComplexSequences st
   for b2, b3 being set
         st b2 in COMPLEX & b3 in the_set_of_ComplexSequences
      holds b1 .(b2,b3) = (C_id b2) (#) seq_id b3;

:: CSSPACE:funcnot 4 => CSSPACE:func 4
definition
  func l_add -> Function-like quasi_total Relation of [:the_set_of_ComplexSequences,the_set_of_ComplexSequences:],the_set_of_ComplexSequences means
    for b1, b2 being Element of the_set_of_ComplexSequences holds
    it .(b1,b2) = (seq_id b1) + seq_id b2;
end;

:: CSSPACE:def 4
theorem
for b1 being Function-like quasi_total Relation of [:the_set_of_ComplexSequences,the_set_of_ComplexSequences:],the_set_of_ComplexSequences holds
      b1 = l_add
   iff
      for b2, b3 being Element of the_set_of_ComplexSequences holds
      b1 .(b2,b3) = (seq_id b2) + seq_id b3;

:: CSSPACE:funcnot 5 => CSSPACE:func 5
definition
  func l_mult -> Function-like quasi_total Relation of [:COMPLEX,the_set_of_ComplexSequences:],the_set_of_ComplexSequences means
    for b1, b2 being set
          st b1 in COMPLEX & b2 in the_set_of_ComplexSequences
       holds it .(b1,b2) = (C_id b1) (#) seq_id b2;
end;

:: CSSPACE:def 5
theorem
for b1 being Function-like quasi_total Relation of [:COMPLEX,the_set_of_ComplexSequences:],the_set_of_ComplexSequences holds
      b1 = l_mult
   iff
      for b2, b3 being set
            st b2 in COMPLEX & b3 in the_set_of_ComplexSequences
         holds b1 .(b2,b3) = (C_id b2) (#) seq_id b3;

:: CSSPACE:funcnot 6 => CSSPACE:func 6
definition
  func CZeroseq -> Element of the_set_of_ComplexSequences means
    for b1 being Element of NAT holds
       (seq_id it) . b1 = 0c;
end;

:: CSSPACE:def 6
theorem
for b1 being Element of the_set_of_ComplexSequences holds
      b1 = CZeroseq
   iff
      for b2 being Element of NAT holds
         (seq_id b1) . b2 = 0c;

:: CSSPACE:th 3
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX holds
   seq_id b1 = b1;

:: CSSPACE:th 4
theorem
for b1, b2 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
b1 + b2 = (seq_id b1) + seq_id b2;

:: CSSPACE:th 5
theorem
for b1 being Element of COMPLEX
for b2 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
   b1 * b2 = b1 (#) seq_id b2;

:: CSSPACE:funcreg 1
registration
  cluster CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) -> Abelian strict;
end;

:: CSSPACE:th 6
theorem
for b1, b2, b3 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
(b1 + b2) + b3 = b1 + (b2 + b3);

:: CSSPACE:th 7
theorem
for b1 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
   b1 + 0. CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) = b1;

:: CSSPACE:th 8
theorem
for b1 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
   ex b2 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) st
      b1 + b2 = 0. CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#);

:: CSSPACE:th 9
theorem
for b1 being Element of COMPLEX
for b2, b3 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
b1 * (b2 + b3) = (b1 * b2) + (b1 * b3);

:: CSSPACE:th 10
theorem
for b1, b2 being Element of COMPLEX
for b3 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
   (b1 + b2) * b3 = (b1 * b3) + (b2 * b3);

:: CSSPACE:th 11
theorem
for b1, b2 being Element of COMPLEX
for b3 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
   (b1 * b2) * b3 = b1 * (b2 * b3);

:: CSSPACE:th 12
theorem
for b1 being Element of the carrier of CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#) holds
   1r * b1 = b1;

:: CSSPACE:funcnot 7 => CSSPACE:func 7
definition
  func Linear_Space_of_ComplexSequences -> non empty strict CLSStruct equals
    CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#);
end;

:: CSSPACE:def 7
theorem
Linear_Space_of_ComplexSequences = CLSStruct(#the_set_of_ComplexSequences,CZeroseq,l_add,l_mult#);

:: CSSPACE:funcreg 2
registration
  cluster Linear_Space_of_ComplexSequences -> non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like;
end;

:: CSSPACE:funcnot 8 => CSSPACE:func 8
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
  let a2 be Element of bool the carrier of a1;
  assume a2 is linearly-closed(a1) & a2 is not empty;
  func Add_(A2,A1) -> Function-like quasi_total Relation of [:a2,a2:],a2 equals
    (the addF of a1) || a2;
end;

:: CSSPACE:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
      st b2 is linearly-closed(b1) & b2 is not empty
   holds Add_(b2,b1) = (the addF of b1) || b2;

:: CSSPACE:funcnot 9 => CSSPACE:func 9
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
  let a2 be Element of bool the carrier of a1;
  assume a2 is linearly-closed(a1) & a2 is not empty;
  func Mult_(A2,A1) -> Function-like quasi_total Relation of [:COMPLEX,a2:],a2 equals
    (the Mult of a1) | [:COMPLEX,a2:];
end;

:: CSSPACE:def 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
      st b2 is linearly-closed(b1) & b2 is not empty
   holds Mult_(b2,b1) = (the Mult of b1) | [:COMPLEX,b2:];

:: CSSPACE:funcnot 10 => CSSPACE:func 10
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
  let a2 be Element of bool the carrier of a1;
  assume a2 is linearly-closed(a1) & a2 is not empty;
  func Zero_(A2,A1) -> Element of a2 equals
    0. a1;
end;

:: CSSPACE:def 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
      st b2 is linearly-closed(b1) & b2 is not empty
   holds Zero_(b2,b1) = 0. b1;

:: CSSPACE:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
      st b2 is linearly-closed(b1) & b2 is not empty
   holds CLSStruct(#b2,Zero_(b2,b1),Add_(b2,b1),Mult_(b2,b1)#) is Subspace of b1;

:: CSSPACE:funcnot 11 => CSSPACE:func 11
definition
  func the_set_of_l2ComplexSequences -> Element of bool the carrier of Linear_Space_of_ComplexSequences means
    it is not empty &
     (for b1 being set holds
           b1 in it
        iff
           b1 in the_set_of_ComplexSequences &
            |.seq_id b1.| (#) |.seq_id b1.| is summable);
end;

:: CSSPACE:def 11
theorem
for b1 being Element of bool the carrier of Linear_Space_of_ComplexSequences holds
      b1 = the_set_of_l2ComplexSequences
   iff
      b1 is not empty &
       (for b2 being set holds
             b2 in b1
          iff
             b2 in the_set_of_ComplexSequences &
              |.seq_id b2.| (#) |.seq_id b2.| is summable);

:: CSSPACE:th 14
theorem
the_set_of_l2ComplexSequences is linearly-closed(Linear_Space_of_ComplexSequences) & the_set_of_l2ComplexSequences is not empty;

:: CSSPACE:th 15
theorem
CLSStruct(#the_set_of_l2ComplexSequences,Zero_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Add_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Mult_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences)#) is Subspace of Linear_Space_of_ComplexSequences;

:: CSSPACE:th 16
theorem
CLSStruct(#the_set_of_l2ComplexSequences,Zero_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Add_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Mult_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences)#) is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;

:: CSSPACE:th 17
theorem
the carrier of Linear_Space_of_ComplexSequences = the_set_of_ComplexSequences &
 (for b1 being set holds
       b1 is Element of the carrier of Linear_Space_of_ComplexSequences
    iff
       b1 is Function-like quasi_total Relation of NAT,COMPLEX) &
 (for b1 being set holds
       b1 is Element of the carrier of Linear_Space_of_ComplexSequences
    iff
       b1 is Function-like quasi_total Relation of NAT,COMPLEX) &
 (for b1 being Element of the carrier of Linear_Space_of_ComplexSequences holds
    b1 = seq_id b1) &
 (for b1, b2 being Element of the carrier of Linear_Space_of_ComplexSequences holds
 b1 + b2 = (seq_id b1) + seq_id b2) &
 (for b1 being Element of COMPLEX
 for b2 being Element of the carrier of Linear_Space_of_ComplexSequences holds
    b1 * b2 = b1 (#) seq_id b2);

:: CSSPACE:structnot 1 => CSSPACE:struct 1
definition
  struct(CLSStruct) CUNITSTR(#
    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 [:COMPLEX,the carrier of it:],the carrier of it,
    scalar -> Function-like quasi_total Relation of [:the carrier of it,the carrier of it:],COMPLEX
  #);
end;

:: CSSPACE:attrnot 1 => CSSPACE:attr 1
definition
  let a1 be CUNITSTR;
  attr a1 is strict;
end;

:: CSSPACE:exreg 1
registration
  cluster strict CUNITSTR;
end;

:: CSSPACE:aggrnot 1 => CSSPACE: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 [:COMPLEX,a1:],a1;
  let a5 be Function-like quasi_total Relation of [:a1,a1:],COMPLEX;
  aggr CUNITSTR(#a1,a2,a3,a4,a5#) -> strict CUNITSTR;
end;

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

:: CSSPACE:exreg 2
registration
  cluster non empty strict CUNITSTR;
end;

:: CSSPACE:funcreg 3
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 [:COMPLEX,a1:],a1;
  let a5 be Function-like quasi_total Relation of [:a1,a1:],COMPLEX;
  cluster CUNITSTR(#a1,a2,a3,a4,a5#) -> non empty strict;
end;

:: CSSPACE:funcnot 12 => CSSPACE:func 12
definition
  let a1 be non empty CUNITSTR;
  let a2, a3 be Element of the carrier of a1;
  func A2 .|. A3 -> Element of COMPLEX equals
    (the scalar of a1) .(a2,a3);
end;

:: CSSPACE:def 12
theorem
for b1 being non empty CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
b2 .|. b3 = (the scalar of b1) .(b2,b3);

:: CSSPACE:attrnot 2 => CSSPACE:attr 2
definition
  let a1 be non empty CUNITSTR;
  attr a1 is ComplexUnitarySpace-like means
    for b1, b2, b3 being Element of the carrier of a1
    for b4 being Element of COMPLEX holds
       (b1 .|. b1 = 0 implies b1 = 0. a1) &
        (b1 = 0. a1 implies b1 .|. b1 = 0) &
        0 <= Re (b1 .|. b1) &
        0 = Im (b1 .|. b1) &
        b1 .|. b2 = (b2 .|. b1) *' &
        (b1 + b2) .|. b3 = (b1 .|. b3) + (b2 .|. b3) &
        (b4 * b1) .|. b2 = b4 * (b1 .|. b2);
end;

:: CSSPACE:dfs 13
definiens
  let a1 be non empty CUNITSTR;
To prove
     a1 is ComplexUnitarySpace-like
it is sufficient to prove
  thus for b1, b2, b3 being Element of the carrier of a1
    for b4 being Element of COMPLEX holds
       (b1 .|. b1 = 0 implies b1 = 0. a1) &
        (b1 = 0. a1 implies b1 .|. b1 = 0) &
        0 <= Re (b1 .|. b1) &
        0 = Im (b1 .|. b1) &
        b1 .|. b2 = (b2 .|. b1) *' &
        (b1 + b2) .|. b3 = (b1 .|. b3) + (b2 .|. b3) &
        (b4 * b1) .|. b2 = b4 * (b1 .|. b2);

:: CSSPACE:def 13
theorem
for b1 being non empty CUNITSTR holds
      b1 is ComplexUnitarySpace-like
   iff
      for b2, b3, b4 being Element of the carrier of b1
      for b5 being Element of COMPLEX holds
         (b2 .|. b2 = 0 implies b2 = 0. b1) &
          (b2 = 0. b1 implies b2 .|. b2 = 0) &
          0 <= Re (b2 .|. b2) &
          0 = Im (b2 .|. b2) &
          b2 .|. b3 = (b3 .|. b2) *' &
          (b2 + b3) .|. b4 = (b2 .|. b4) + (b3 .|. b4) &
          (b5 * b2) .|. b3 = b5 * (b2 .|. b3);

:: CSSPACE:exreg 3
registration
  cluster non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like strict ComplexUnitarySpace-like CUNITSTR;
end;

:: CSSPACE:modenot 1
definition
  mode ComplexUnitarySpace is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
end;

:: CSSPACE:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR holds
   (0. b1) .|. 0. b1 = 0;

:: CSSPACE:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
b2 .|. (b3 + b4) = (b2 .|. b3) + (b2 .|. b4);

:: CSSPACE:th 20
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b3, b4 being Element of the carrier of b2 holds
b3 .|. (b1 * b4) = b1 *' * (b3 .|. b4);

:: CSSPACE:th 21
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b3, b4 being Element of the carrier of b2 holds
(b1 * b3) .|. b4 = b3 .|. (b1 *' * b4);

:: CSSPACE:th 22
theorem
for b1, b2 being Element of COMPLEX
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b4, b5, b6 being Element of the carrier of b3 holds
((b1 * b4) + (b2 * b5)) .|. b6 = (b1 * (b4 .|. b6)) + (b2 * (b5 .|. b6));

:: CSSPACE:th 23
theorem
for b1, b2 being Element of COMPLEX
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b4, b5, b6 being Element of the carrier of b3 holds
b4 .|. ((b1 * b5) + (b2 * b6)) = (b1 *' * (b4 .|. b5)) + (b2 *' * (b4 .|. b6));

:: CSSPACE:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
(- b2) .|. b3 = b2 .|. - b3;

:: CSSPACE:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
(- b2) .|. b3 = - (b2 .|. b3);

:: CSSPACE:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
b2 .|. - b3 = - (b2 .|. b3);

:: CSSPACE:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
(- b2) .|. - b3 = b2 .|. b3;

:: CSSPACE:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 - b3) .|. b4 = (b2 .|. b4) - (b3 .|. b4);

:: CSSPACE:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
b2 .|. (b3 - b4) = (b2 .|. b3) - (b2 .|. b4);

:: CSSPACE:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CSSPACE:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
   (0. b1) .|. b2 = 0;

:: CSSPACE:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
   b2 .|. 0. b1 = 0;

:: CSSPACE:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
(b2 + b3) .|. (b2 + b3) = (((b2 .|. b2) + (b2 .|. b3)) + (b3 .|. b2)) + (b3 .|. b3);

:: CSSPACE:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
(b2 + b3) .|. (b2 - b3) = (((b2 .|. b2) - (b2 .|. b3)) + (b3 .|. b2)) - (b3 .|. b3);

:: CSSPACE:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
(b2 - b3) .|. (b2 - b3) = (((b2 .|. b2) - (b2 .|. b3)) - (b3 .|. b2)) + (b3 .|. b3);

:: CSSPACE:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
   |.b2 .|. b2.| = Re (b2 .|. b2);

:: CSSPACE:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
|.b2 .|. b3.| <= (sqrt |.b2 .|. b2.|) * sqrt |.b3 .|. b3.|;

:: CSSPACE:prednot 1 => CSSPACE:pred 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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 ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
::  for a2, a3 being Element of the carrier of a1
::        st a2,a3 are_orthogonal
::     holds a3,a2 are_orthogonal;
end;

:: CSSPACE:dfs 14
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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;

:: CSSPACE:def 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
   b2,b3 are_orthogonal
iff
   b2 .|. b3 = 0;

:: CSSPACE:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1
      st b2,b3 are_orthogonal
   holds b2,- b3 are_orthogonal;

:: CSSPACE:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1
      st b2,b3 are_orthogonal
   holds - b2,b3 are_orthogonal;

:: CSSPACE:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1
      st b2,b3 are_orthogonal
   holds - b2,- b3 are_orthogonal;

:: CSSPACE:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
   b2,0. b1 are_orthogonal;

:: CSSPACE:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1
      st b2,b3 are_orthogonal
   holds (b2 + b3) .|. (b2 + b3) = (b2 .|. b2) + (b3 .|. b3);

:: CSSPACE:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1
      st b2,b3 are_orthogonal
   holds (b2 - b3) .|. (b2 - b3) = (b2 .|. b2) + (b3 .|. b3);

:: CSSPACE:funcnot 13 => CSSPACE:func 13
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  let a2 be Element of the carrier of a1;
  func ||.A2.|| -> Element of REAL equals
    sqrt |.a2 .|. a2.|;
end;

:: CSSPACE:def 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
   ||.b2.|| = sqrt |.b2 .|. b2.|;

:: CSSPACE:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
      ||.b2.|| = 0
   iff
      b2 = 0. b1;

:: CSSPACE:th 45
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b3 being Element of the carrier of b2 holds
   ||.b1 * b3.|| = |.b1.| * ||.b3.||;

:: CSSPACE:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
   0 <= ||.b2.||;

:: CSSPACE:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
|.b2 .|. b3.| <= ||.b2.|| * ||.b3.||;

:: CSSPACE:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2 + b3.|| <= ||.b2.|| + ||.b3.||;

:: CSSPACE:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
   ||.- b2.|| = ||.b2.||;

:: CSSPACE:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2.|| - ||.b3.|| <= ||.b2 - b3.||;

:: CSSPACE:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
abs (||.b2.|| - ||.b3.||) <= ||.b2 - b3.||;

:: CSSPACE:funcnot 14 => CSSPACE:func 14
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  let a2, a3 be Element of the carrier of a1;
  func dist(A2,A3) -> Element of REAL equals
    ||.a2 - a3.||;
end;

:: CSSPACE:def 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
dist(b2,b3) = ||.b2 - b3.||;

:: CSSPACE:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
dist(b2,b3) = dist(b3,b2);

:: CSSPACE:funcnot 15 => CSSPACE:func 15
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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 ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
::  for a2, a3 being Element of the carrier of a1 holds
::  dist(a2,a3) = dist(a3,a2);
end;

:: CSSPACE:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Element of the carrier of b1 holds
   dist(b2,b2) = 0;

:: CSSPACE:th 54
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
dist(b2,b3) <= (dist(b2,b4)) + dist(b4,b3);

:: CSSPACE:th 55
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
   b2 <> b3
iff
   dist(b2,b3) <> 0;

:: CSSPACE:th 56
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
0 <= dist(b2,b3);

:: CSSPACE:th 57
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
   b2 <> b3
iff
   0 < dist(b2,b3);

:: CSSPACE:th 58
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Element of the carrier of b1 holds
dist(b2,b3) = sqrt |.(b2 - b3) .|. (b2 - b3).|;

:: CSSPACE:th 59
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4, b5 being Element of the carrier of b1 holds
dist(b2 + b3,b4 + b5) <= (dist(b2,b4)) + dist(b3,b5);

:: CSSPACE:th 60
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4, b5 being Element of the carrier of b1 holds
dist(b2 - b3,b4 - b5) <= (dist(b2,b4)) + dist(b3,b5);

:: CSSPACE:th 61
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
dist(b2 - b3,b4 - b3) = dist(b2,b4);

:: CSSPACE:th 62
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Element of the carrier of b1 holds
dist(b2 - b3,b4 - b3) <= (dist(b3,b2)) + dist(b3,b4);

:: CSSPACE:th 63
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 + b3 = b3 + b2;

:: CSSPACE:funcnot 16 => CSSPACE:func 16
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR;
  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 ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
::  for a2, a3 being Function-like quasi_total Relation of NAT,the carrier of a1 holds
::  a2 + a3 = a3 + a2;
end;

:: CSSPACE:th 64
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 + (b3 + b4) = (b2 + b3) + b4;

:: CSSPACE:th 65
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CSSPACE:th 66
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CSSPACE:th 67
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CSSPACE:th 68
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CSSPACE:th 69
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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);

:: CSSPACE:th 70
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CSSPACE:th 71
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 - b3 = b2 + - b3;

:: CSSPACE:th 72
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b2 = b2 + 0. b1;

:: CSSPACE:th 73
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
b1 * (b3 + b4) = (b1 * b3) + (b1 * b4);

:: CSSPACE:th 74
theorem
for b1, b2 being Element of COMPLEX
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b4 being Function-like quasi_total Relation of NAT,the carrier of b3 holds
   (b1 + b2) * b4 = (b1 * b4) + (b2 * b4);

:: CSSPACE:th 75
theorem
for b1, b2 being Element of COMPLEX
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b4 being Function-like quasi_total Relation of NAT,the carrier of b3 holds
   (b1 * b2) * b4 = b1 * (b2 * b4);

:: CSSPACE:th 76
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   1r * b2 = b2;

:: CSSPACE:th 77
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   (- 1r) * b2 = - b2;

:: CSSPACE:th 78
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
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;

:: CSSPACE:th 79
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 - b3 = - (b3 - b2);

:: CSSPACE:th 80
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b2 = b2 - 0. b1;

:: CSSPACE:th 81
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   b2 = - - b2;

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

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

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

:: CSSPACE:th 85
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexUnitarySpace-like CUNITSTR
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b2 holds
b1 * (b3 - b4) = (b1 * b3) - (b1 * b4);

:: CSSPACE:th 86
theorem
ex b1 being Function-like quasi_total Relation of [:the_set_of_l2ComplexSequences,the_set_of_l2ComplexSequences:],COMPLEX st
   for b2, b3 being set
         st b2 in the_set_of_l2ComplexSequences & b3 in the_set_of_l2ComplexSequences
      holds b1 .(b2,b3) = Sum ((seq_id b2) (#) ((seq_id b3) *'));

:: CSSPACE:funcnot 17 => CSSPACE:func 17
definition
  func cl_scalar -> Function-like quasi_total Relation of [:the_set_of_l2ComplexSequences,the_set_of_l2ComplexSequences:],COMPLEX means
    for b1, b2 being set
          st b1 in the_set_of_l2ComplexSequences & b2 in the_set_of_l2ComplexSequences
       holds it .(b1,b2) = Sum ((seq_id b1) (#) ((seq_id b2) *'));
end;

:: CSSPACE:def 19
theorem
for b1 being Function-like quasi_total Relation of [:the_set_of_l2ComplexSequences,the_set_of_l2ComplexSequences:],COMPLEX holds
      b1 = cl_scalar
   iff
      for b2, b3 being set
            st b2 in the_set_of_l2ComplexSequences & b3 in the_set_of_l2ComplexSequences
         holds b1 .(b2,b3) = Sum ((seq_id b2) (#) ((seq_id b3) *'));

:: CSSPACE:funcreg 4
registration
  cluster CUNITSTR(#the_set_of_l2ComplexSequences,Zero_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Add_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Mult_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),cl_scalar#) -> non empty strict;
end;

:: CSSPACE:funcnot 18 => CSSPACE:func 18
definition
  func Complex_l2_Space -> non empty CUNITSTR equals
    CUNITSTR(#the_set_of_l2ComplexSequences,Zero_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Add_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Mult_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),cl_scalar#);
end;

:: CSSPACE:def 20
theorem
Complex_l2_Space = CUNITSTR(#the_set_of_l2ComplexSequences,Zero_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Add_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),Mult_(the_set_of_l2ComplexSequences,Linear_Space_of_ComplexSequences),cl_scalar#);

:: CSSPACE:th 87
theorem
for b1 being CLSStruct
      st CLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#) is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
   holds b1 is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;

:: CSSPACE:th 88
theorem
for b1 being Function-like quasi_total Relation of NAT,COMPLEX
      st for b2 being Element of NAT holds
           b1 . b2 = 0c
   holds b1 is summable & Sum b1 = 0c;

:: CSSPACE:funcreg 5
registration
  cluster Complex_l2_Space -> non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like;
end;