Article CLVECT_1, MML version 4.99.1005

:: CLVECT_1:structnot 1 => CLVECT_1:struct 1
definition
  struct(addLoopStr) CLSStruct(#
    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
  #);
end;

:: CLVECT_1:attrnot 1 => CLVECT_1:attr 1
definition
  let a1 be CLSStruct;
  attr a1 is strict;
end;

:: CLVECT_1:exreg 1
registration
  cluster strict CLSStruct;
end;

:: CLVECT_1:aggrnot 1 => CLVECT_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 [:COMPLEX,a1:],a1;
  aggr CLSStruct(#a1,a2,a3,a4#) -> strict CLSStruct;
end;

:: CLVECT_1:selnot 1 => CLVECT_1:sel 1
definition
  let a1 be CLSStruct;
  sel the Mult of a1 -> Function-like quasi_total Relation of [:COMPLEX,the carrier of a1:],the carrier of a1;
end;

:: CLVECT_1:exreg 2
registration
  cluster non empty CLSStruct;
end;

:: CLVECT_1:modenot 1
definition
  let a1 be CLSStruct;
  mode VECTOR of a1 is Element of the carrier of a1;
end;

:: CLVECT_1:funcnot 1 => CLVECT_1:func 1
definition
  let a1 be non empty CLSStruct;
  let a2 be Element of the carrier of a1;
  let a3 be Element of COMPLEX;
  func A3 * A2 -> Element of the carrier of a1 equals
    (the Mult of a1) . [a3,a2];
end;

:: CLVECT_1:def 1
theorem
for b1 being non empty CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX holds
   b3 * b2 = (the Mult of b1) . [b3,b2];

:: CLVECT_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 [:COMPLEX,a1:],a1;
  cluster CLSStruct(#a1,a2,a3,a4#) -> non empty strict;
end;

:: CLVECT_1:attrnot 2 => CLVECT_1:attr 2
definition
  let a1 be non empty CLSStruct;
  attr a1 is ComplexLinearSpace-like means
    (for b1 being Element of COMPLEX
     for b2, b3 being Element of the carrier of a1 holds
     b1 * (b2 + b3) = (b1 * b2) + (b1 * b3)) &
     (for b1, b2 being Element of COMPLEX
     for b3 being Element of the carrier of a1 holds
        (b1 + b2) * b3 = (b1 * b3) + (b2 * b3)) &
     (for b1, b2 being Element of COMPLEX
     for b3 being Element of the carrier of a1 holds
        (b1 * b2) * b3 = b1 * (b2 * b3)) &
     (for b1 being Element of the carrier of a1 holds
        1r * b1 = b1);
end;

:: CLVECT_1:dfs 2
definiens
  let a1 be non empty CLSStruct;
To prove
     a1 is ComplexLinearSpace-like
it is sufficient to prove
  thus (for b1 being Element of COMPLEX
     for b2, b3 being Element of the carrier of a1 holds
     b1 * (b2 + b3) = (b1 * b2) + (b1 * b3)) &
     (for b1, b2 being Element of COMPLEX
     for b3 being Element of the carrier of a1 holds
        (b1 + b2) * b3 = (b1 * b3) + (b2 * b3)) &
     (for b1, b2 being Element of COMPLEX
     for b3 being Element of the carrier of a1 holds
        (b1 * b2) * b3 = b1 * (b2 * b3)) &
     (for b1 being Element of the carrier of a1 holds
        1r * b1 = b1);

:: CLVECT_1:def 2
theorem
for b1 being non empty CLSStruct holds
      b1 is ComplexLinearSpace-like
   iff
      (for b2 being Element of COMPLEX
       for b3, b4 being Element of the carrier of b1 holds
       b2 * (b3 + b4) = (b2 * b3) + (b2 * b4)) &
       (for b2, b3 being Element of COMPLEX
       for b4 being Element of the carrier of b1 holds
          (b2 + b3) * b4 = (b2 * b4) + (b3 * b4)) &
       (for b2, b3 being Element of COMPLEX
       for b4 being Element of the carrier of b1 holds
          (b2 * b3) * b4 = b2 * (b3 * b4)) &
       (for b2 being Element of the carrier of b1 holds
          1r * b2 = b2);

:: CLVECT_1:funcnot 2 => CLVECT_1:func 2
definition
  func Trivial-CLSStruct -> strict CLSStruct equals
    CLSStruct(#1,op0,op2,pr2(COMPLEX,1)#);
end;

:: CLVECT_1:def 3
theorem
Trivial-CLSStruct = CLSStruct(#1,op0,op2,pr2(COMPLEX,1)#);

:: CLVECT_1:funcreg 2
registration
  cluster Trivial-CLSStruct -> non empty trivial strict;
end;

:: CLVECT_1:exreg 3
registration
  cluster non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like CLSStruct;
end;

:: CLVECT_1:modenot 2
definition
  mode ComplexLinearSpace is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
end;

:: CLVECT_1:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
      st (b3 = 0 or b2 = 0. b1)
   holds b3 * b2 = 0. b1;

:: CLVECT_1:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
      st b3 * b2 = 0. b1 & b3 <> 0
   holds b2 = 0. b1;

:: CLVECT_1:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1 holds
   - b2 = (- 1r) * b2;

:: CLVECT_1:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
      st b2 = - b2
   holds b2 = 0. b1;

:: CLVECT_1:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
      st b2 + b2 = 0. b1
   holds b2 = 0. b1;

:: CLVECT_1:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX holds
   b3 * - b2 = (- b3) * b2;

:: CLVECT_1:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX holds
   b3 * - b2 = - (b3 * b2);

:: CLVECT_1:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX holds
   (- b3) * - b2 = b3 * b2;

:: CLVECT_1:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of COMPLEX holds
   b4 * (b2 - b3) = (b4 * b2) - (b4 * b3);

:: CLVECT_1:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of COMPLEX holds
(b3 - b4) * b2 = (b3 * b2) - (b4 * b2);

:: CLVECT_1:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of COMPLEX
      st b4 <> 0 & b4 * b2 = b4 * b3
   holds b2 = b3;

:: CLVECT_1:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of COMPLEX
      st b2 <> 0. b1 & b3 * b2 = b4 * b2
   holds b3 = b4;

:: CLVECT_1:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of COMPLEX
for b3, b4 being FinSequence of the carrier of b1
      st len b3 = len b4 &
         (for b5 being Element of NAT
         for b6 being Element of the carrier of b1
               st b5 in dom b3 & b6 = b4 . b5
            holds b3 . b5 = b2 * b6)
   holds Sum b3 = b2 * Sum b4;

:: CLVECT_1:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of COMPLEX holds
   b2 * Sum <*> the carrier of b1 = 0. b1;

:: CLVECT_1:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of COMPLEX holds
   b4 * Sum <*b2,b3*> = (b4 * b2) + (b4 * b3);

:: CLVECT_1:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Element of COMPLEX holds
   b5 * Sum <*b2,b3,b4*> = ((b5 * b2) + (b5 * b3)) + (b5 * b4);

:: CLVECT_1:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1 holds
   Sum <*b2,b2*> = [*2,0*] * b2;

:: CLVECT_1:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1 holds
   Sum <*- b2,- b2*> = [*- 2,0*] * b2;

:: CLVECT_1:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1 holds
   Sum <*b2,b2,b2*> = [*3,0*] * b2;

:: CLVECT_1:attrnot 3 => CLVECT_1:attr 3
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;
  attr a2 is linearly-closed means
    (for b1, b2 being Element of the carrier of a1
           st b1 in a2 & b2 in a2
        holds b1 + b2 in a2) &
     (for b1 being Element of COMPLEX
     for b2 being Element of the carrier of a1
           st b2 in a2
        holds b1 * b2 in a2);
end;

:: CLVECT_1:dfs 4
definiens
  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;
To prove
     a2 is linearly-closed
it is sufficient to prove
  thus (for b1, b2 being Element of the carrier of a1
           st b1 in a2 & b2 in a2
        holds b1 + b2 in a2) &
     (for b1 being Element of COMPLEX
     for b2 being Element of the carrier of a1
           st b2 in a2
        holds b1 * b2 in a2);

:: CLVECT_1:def 4
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 holds
      b2 is linearly-closed(b1)
   iff
      (for b3, b4 being Element of the carrier of b1
             st b3 in b2 & b4 in b2
          holds b3 + b4 in b2) &
       (for b3 being Element of COMPLEX
       for b4 being Element of the carrier of b1
             st b4 in b2
          holds b3 * b4 in b2);

:: CLVECT_1:th 21
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 <> {} & b2 is linearly-closed(b1)
   holds 0. b1 in b2;

:: CLVECT_1:th 22
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)
for b3 being Element of the carrier of b1
      st b3 in b2
   holds - b3 in b2;

:: CLVECT_1:th 23
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)
for b3, b4 being Element of the carrier of b1
      st b3 in b2 & b4 in b2
   holds b3 - b4 in b2;

:: CLVECT_1:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
   {0. b1} is linearly-closed(b1);

:: CLVECT_1:th 25
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 the carrier of b1 = b2
   holds b2 is linearly-closed(b1);

:: CLVECT_1:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st b2 is linearly-closed(b1) &
         b3 is linearly-closed(b1) &
         b4 = {b5 + b6 where b5 is Element of the carrier of b1, b6 is Element of the carrier of b1: b5 in b2 & b6 in b3}
   holds b4 is linearly-closed(b1);

:: CLVECT_1:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is linearly-closed(b1) & b3 is linearly-closed(b1)
   holds b2 /\ b3 is linearly-closed(b1);

:: CLVECT_1:modenot 3 => CLVECT_1:mode 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
  mode Subspace of A1 -> non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct means
    the carrier of it c= the carrier of a1 &
     0. it = 0. a1 &
     the addF of it = (the addF of a1) || the carrier of it &
     the Mult of it = (the Mult of a1) | [:COMPLEX,the carrier of it:];
end;

:: CLVECT_1:dfs 5
definiens
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
To prove
     a2 is Subspace of a1
it is sufficient to prove
  thus the carrier of a2 c= the carrier of a1 &
     0. a2 = 0. a1 &
     the addF of a2 = (the addF of a1) || the carrier of a2 &
     the Mult of a2 = (the Mult of a1) | [:COMPLEX,the carrier of a2:];

:: CLVECT_1:def 5
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
   b2 is Subspace of b1
iff
   the carrier of b2 c= the carrier of b1 &
    0. b2 = 0. b1 &
    the addF of b2 = (the addF of b1) || the carrier of b2 &
    the Mult of b2 = (the Mult of b1) | [:COMPLEX,the carrier of b2:];

:: CLVECT_1:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Subspace of b1
for b4 being set
      st b4 in b2 & b2 is Subspace of b3
   holds b4 in b3;

:: CLVECT_1:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1
for b3 being set
      st b3 in b2
   holds b3 in b1;

:: CLVECT_1:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1
for b3 being Element of the carrier of b2 holds
   b3 is Element of the carrier of b1;

:: CLVECT_1:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
   0. b2 = 0. b1;

:: CLVECT_1:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Subspace of b1 holds
0. b2 = 0. b3;

:: CLVECT_1:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
for b5, b6 being Element of the carrier of b4
      st b5 = b2 & b6 = b3
   holds b5 + b6 = b2 + b3;

:: CLVECT_1:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
for b4 being Subspace of b1
for b5 being Element of the carrier of b4
      st b5 = b2
   holds b3 * b5 = b3 * b2;

:: CLVECT_1:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1
for b4 being Element of the carrier of b3
      st b4 = b2
   holds - b2 = - b4;

:: CLVECT_1:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
for b5, b6 being Element of the carrier of b4
      st b5 = b2 & b6 = b3
   holds b5 - b6 = b2 - b3;

:: CLVECT_1:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
   0. b1 in b2;

:: CLVECT_1:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Subspace of b1 holds
0. b2 in b3;

:: CLVECT_1:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
   0. b2 in b1;

:: CLVECT_1:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
      st b2 in b4 & b3 in b4
   holds b2 + b3 in b4;

:: CLVECT_1:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
for b4 being Subspace of b1
      st b2 in b4
   holds b3 * b2 in b4;

:: CLVECT_1:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1
      st b2 in b3
   holds - b2 in b3;

:: CLVECT_1:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
      st b2 in b4 & b3 in b4
   holds b2 - b3 in b4;

:: CLVECT_1:th 44
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
for b3 being non empty set
for b4 being Element of b3
for b5 being Function-like quasi_total Relation of [:b3,b3:],b3
for b6 being Function-like quasi_total Relation of [:COMPLEX,b3:],b3
      st b2 = b3 &
         b4 = 0. b1 &
         b5 = (the addF of b1) || b2 &
         b6 = (the Mult of b1) | [:COMPLEX,b2:]
   holds CLSStruct(#b3,b4,b5,b6#) is Subspace of b1;

:: CLVECT_1:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
   b1 is Subspace of b1;

:: CLVECT_1:th 46
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like CLSStruct
      st b1 is Subspace of b2 & b2 is Subspace of b1
   holds b1 = b2;

:: CLVECT_1:th 47
theorem
for b1, b2, b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
      st b1 is Subspace of b2 & b2 is Subspace of b3
   holds b1 is Subspace of b3;

:: CLVECT_1:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Subspace of b1
      st the carrier of b2 c= the carrier of b3
   holds b2 is Subspace of b3;

:: CLVECT_1:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Subspace of b1
      st for b4 being Element of the carrier of b1
              st b4 in b2
           holds b4 in b3
   holds b2 is Subspace of b3;

:: CLVECT_1:exreg 4
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
  cluster non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like Subspace of a1;
end;

:: CLVECT_1:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being strict Subspace of b1
      st the carrier of b2 = the carrier of b3
   holds b2 = b3;

:: CLVECT_1:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being strict Subspace of b1
      st for b4 being Element of the carrier of b1 holds
              b4 in b2
           iff
              b4 in b3
   holds b2 = b3;

:: CLVECT_1:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like CLSStruct
for b2 being strict Subspace of b1
      st the carrier of b2 = the carrier of b1
   holds b2 = b1;

:: CLVECT_1:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like CLSStruct
for b2 being strict Subspace of b1
      st for b3 being Element of the carrier of b1 holds
              b3 in b2
           iff
              b3 in b1
   holds b2 = b1;

:: CLVECT_1:th 54
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
for b3 being Subspace of b1
      st the carrier of b3 = b2
   holds b2 is linearly-closed(b1);

:: CLVECT_1:th 55
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 <> {} & b2 is linearly-closed(b1)
   holds ex b3 being strict Subspace of b1 st
      b2 = the carrier of b3;

:: CLVECT_1:funcnot 3 => CLVECT_1:func 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
  func (0). A1 -> strict Subspace of a1 means
    the carrier of it = {0. a1};
end;

:: CLVECT_1:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being strict Subspace of b1 holds
      b2 = (0). b1
   iff
      the carrier of b2 = {0. b1};

:: CLVECT_1:funcnot 4 => CLVECT_1:func 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
  func (Omega). A1 -> strict Subspace of a1 equals
    CLSStruct(#the carrier of a1,the ZeroF of a1,the addF of a1,the Mult of a1#);
end;

:: CLVECT_1:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
   (Omega). b1 = CLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#);

:: CLVECT_1:th 56
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
   (0). b2 = (0). b1;

:: CLVECT_1:th 57
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Subspace of b1 holds
(0). b2 = (0). b3;

:: CLVECT_1:th 58
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
   (0). b2 is Subspace of b1;

:: CLVECT_1:th 59
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
   (0). b1 is Subspace of b2;

:: CLVECT_1:th 60
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Subspace of b1 holds
(0). b2 is Subspace of b3;

:: CLVECT_1:th 61
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like CLSStruct holds
   b1 is Subspace of (Omega). b1;

:: CLVECT_1:funcnot 5 => CLVECT_1:func 5
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
  let a2 be Element of the carrier of a1;
  let a3 be Subspace of a1;
  func A2 + A3 -> Element of bool the carrier of a1 equals
    {a2 + b1 where b1 is Element of the carrier of a1: b1 in a3};
end;

:: CLVECT_1:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
   b2 + b3 = {b2 + b4 where b4 is Element of the carrier of b1: b4 in b3};

:: CLVECT_1:modenot 4 => CLVECT_1:mode 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
  let a2 be Subspace of a1;
  mode Coset of A2 -> Element of bool the carrier of a1 means
    ex b1 being Element of the carrier of a1 st
       it = b1 + a2;
end;

:: CLVECT_1:dfs 9
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
  let a2 be Subspace of a1;
  let a3 be Element of bool the carrier of a1;
To prove
     a3 is Coset of a2
it is sufficient to prove
  thus ex b1 being Element of the carrier of a1 st
       a3 = b1 + a2;

:: CLVECT_1:def 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1
for b3 being Element of bool the carrier of b1 holds
      b3 is Coset of b2
   iff
      ex b4 being Element of the carrier of b1 st
         b3 = b4 + b2;

:: CLVECT_1:th 62
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
      0. b1 in b2 + b3
   iff
      b2 in b3;

:: CLVECT_1:th 63
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
   b2 in b2 + b3;

:: CLVECT_1:th 64
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
   (0. b1) + b2 = the carrier of b2;

:: CLVECT_1:th 65
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1 holds
   b2 + (0). b1 = {b2};

:: CLVECT_1:th 66
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1 holds
   b2 + (Omega). b1 = the carrier of b1;

:: CLVECT_1:th 67
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
      0. b1 in b2 + b3
   iff
      b2 + b3 = the carrier of b3;

:: CLVECT_1:th 68
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
      b2 in b3
   iff
      b2 + b3 = the carrier of b3;

:: CLVECT_1:th 69
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
for b4 being Subspace of b1
      st b2 in b4
   holds (b3 * b2) + b4 = the carrier of b4;

:: CLVECT_1:th 70
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
for b4 being Subspace of b1
      st b3 <> 0 & (b3 * b2) + b4 = the carrier of b4
   holds b2 in b4;

:: CLVECT_1:th 71
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
      b2 in b3
   iff
      (- b2) + b3 = the carrier of b3;

:: CLVECT_1:th 72
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
      b2 in b4
   iff
      b3 + b4 = (b3 + b2) + b4;

:: CLVECT_1:th 73
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
      b2 in b4
   iff
      b3 + b4 = (b3 - b2) + b4;

:: CLVECT_1:th 74
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
      b2 in b3 + b4
   iff
      b3 + b4 = b2 + b4;

:: CLVECT_1:th 75
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
      b2 + b3 = (- b2) + b3
   iff
      b2 in b3;

:: CLVECT_1:th 76
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Subspace of b1
      st b2 in b3 + b5 & b2 in b4 + b5
   holds b3 + b5 = b4 + b5;

:: CLVECT_1:th 77
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
      st b2 in b3 + b4 & b2 in (- b3) + b4
   holds b3 in b4;

:: CLVECT_1:th 78
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
for b4 being Subspace of b1
      st b3 <> 1r & b3 * b2 in b2 + b4
   holds b2 in b4;

:: CLVECT_1:th 79
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Element of COMPLEX
for b4 being Subspace of b1
      st b2 in b4
   holds b3 * b2 in b2 + b4;

:: CLVECT_1:th 80
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1 holds
      - b2 in b2 + b3
   iff
      b2 in b3;

:: CLVECT_1:th 81
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
      b2 + b3 in b3 + b4
   iff
      b2 in b4;

:: CLVECT_1:th 82
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
      b2 - b3 in b2 + b4
   iff
      b3 in b4;

:: CLVECT_1:th 83
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
      b2 in b3 + b4
   iff
      ex b5 being Element of the carrier of b1 st
         b5 in b4 & b2 = b3 + b5;

:: CLVECT_1:th 84
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
      b2 in b3 + b4
   iff
      ex b5 being Element of the carrier of b1 st
         b5 in b4 & b2 = b3 - b5;

:: CLVECT_1:th 85
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
      ex b5 being Element of the carrier of b1 st
         b2 in b5 + b4 & b3 in b5 + b4
   iff
      b2 - b3 in b4;

:: CLVECT_1:th 86
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
      st b2 + b4 = b3 + b4
   holds ex b5 being Element of the carrier of b1 st
      b5 in b4 & b2 + b5 = b3;

:: CLVECT_1:th 87
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
      st b2 + b4 = b3 + b4
   holds ex b5 being Element of the carrier of b1 st
      b5 in b4 & b2 - b5 = b3;

:: CLVECT_1:th 88
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3, b4 being strict Subspace of b1 holds
   b2 + b3 = b2 + b4
iff
   b3 = b4;

:: CLVECT_1:th 89
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being strict Subspace of b1
      st b2 + b4 = b3 + b5
   holds b4 = b5;

:: CLVECT_1:th 90
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1
for b3 being Coset of b2 holds
      b3 is linearly-closed(b1)
   iff
      b3 = the carrier of b2;

:: CLVECT_1:th 91
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being strict Subspace of b1
for b4 being Coset of b2
for b5 being Coset of b3
      st b4 = b5
   holds b2 = b3;

:: CLVECT_1:th 92
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1 holds
   {b2} is Coset of (0). b1;

:: CLVECT_1:th 93
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 Coset of (0). b1
   holds ex b3 being Element of the carrier of b1 st
      b2 = {b3};

:: CLVECT_1:th 94
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
   the carrier of b2 is Coset of b2;

:: CLVECT_1:th 95
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
   the carrier of b1 is Coset of (Omega). b1;

:: CLVECT_1:th 96
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 Coset of (Omega). b1
   holds b2 = the carrier of b1;

:: CLVECT_1:th 97
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1
for b3 being Coset of b2 holds
      0. b1 in b3
   iff
      b3 = the carrier of b2;

:: CLVECT_1:th 98
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1
for b4 being Coset of b3 holds
      b2 in b4
   iff
      b4 = b2 + b3;

:: CLVECT_1:th 99
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
for b5 being Coset of b4
      st b2 in b5 & b3 in b5
   holds ex b6 being Element of the carrier of b1 st
      b6 in b4 & b2 + b6 = b3;

:: CLVECT_1:th 100
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1
for b5 being Coset of b4
      st b2 in b5 & b3 in b5
   holds ex b6 being Element of the carrier of b1 st
      b6 in b4 & b2 - b6 = b3;

:: CLVECT_1:th 101
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Subspace of b1 holds
      ex b5 being Coset of b4 st
         b2 in b5 & b3 in b5
   iff
      b2 - b3 in b4;

:: CLVECT_1:th 102
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Element of the carrier of b1
for b3 being Subspace of b1
for b4, b5 being Coset of b3
      st b2 in b4 & b2 in b5
   holds b4 = b5;

:: CLVECT_1:structnot 2 => CLVECT_1:struct 2
definition
  struct(CLSStruct) CNORMSTR(#
    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,
    norm -> Function-like quasi_total Relation of the carrier of it,REAL
  #);
end;

:: CLVECT_1:attrnot 4 => CLVECT_1:attr 4
definition
  let a1 be CNORMSTR;
  attr a1 is strict;
end;

:: CLVECT_1:exreg 5
registration
  cluster strict CNORMSTR;
end;

:: CLVECT_1:aggrnot 2 => CLVECT_1:aggr 2
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,REAL;
  aggr CNORMSTR(#a1,a2,a3,a4,a5#) -> strict CNORMSTR;
end;

:: CLVECT_1:selnot 2 => CLVECT_1:sel 2
definition
  let a1 be CNORMSTR;
  sel the norm of a1 -> Function-like quasi_total Relation of the carrier of a1,REAL;
end;

:: CLVECT_1:exreg 6
registration
  cluster non empty CNORMSTR;
end;

:: CLVECT_1:funcnot 6 => CLVECT_1:func 6
definition
  let a1 be non empty CNORMSTR;
  let a2 be Element of the carrier of a1;
  func ||.A2.|| -> Element of REAL equals
    (the norm of a1) . a2;
end;

:: CLVECT_1:def 10
theorem
for b1 being non empty CNORMSTR
for b2 being Element of the carrier of b1 holds
   ||.b2.|| = (the norm of b1) . b2;

:: CLVECT_1:attrnot 5 => CLVECT_1:attr 5
definition
  let a1 be non empty CNORMSTR;
  attr a1 is ComplexNormSpace-like means
    for b1, b2 being Element of the carrier of a1
    for b3 being Element of COMPLEX holds
       (||.b1.|| = 0 implies b1 = 0. a1) &
        (b1 = 0. a1 implies ||.b1.|| = 0) &
        ||.b3 * b1.|| = |.b3.| * ||.b1.|| &
        ||.b1 + b2.|| <= ||.b1.|| + ||.b2.||;
end;

:: CLVECT_1:dfs 11
definiens
  let a1 be non empty CNORMSTR;
To prove
     a1 is ComplexNormSpace-like
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
    for b3 being Element of COMPLEX holds
       (||.b1.|| = 0 implies b1 = 0. a1) &
        (b1 = 0. a1 implies ||.b1.|| = 0) &
        ||.b3 * b1.|| = |.b3.| * ||.b1.|| &
        ||.b1 + b2.|| <= ||.b1.|| + ||.b2.||;

:: CLVECT_1:def 11
theorem
for b1 being non empty CNORMSTR holds
      b1 is ComplexNormSpace-like
   iff
      for b2, b3 being Element of the carrier of b1
      for b4 being Element of COMPLEX holds
         (||.b2.|| = 0 implies b2 = 0. b1) &
          (b2 = 0. b1 implies ||.b2.|| = 0) &
          ||.b4 * b2.|| = |.b4.| * ||.b2.|| &
          ||.b2 + b3.|| <= ||.b2.|| + ||.b3.||;

:: CLVECT_1:exreg 7
registration
  cluster non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like strict ComplexNormSpace-like CNORMSTR;
end;

:: CLVECT_1:modenot 5
definition
  mode ComplexNormSpace is non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
end;

:: CLVECT_1:th 103
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR holds
   ||.0. b1.|| = 0;

:: CLVECT_1:th 104
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of the carrier of b1 holds
   ||.- b2.|| = ||.b2.||;

:: CLVECT_1:th 105
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2 - b3.|| <= ||.b2.|| + ||.b3.||;

:: CLVECT_1:th 106
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of the carrier of b1 holds
   0 <= ||.b2.||;

:: CLVECT_1:th 107
theorem
for b1, b2 being Element of COMPLEX
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4, b5 being Element of the carrier of b3 holds
||.(b1 * b4) + (b2 * b5).|| <= (|.b1.| * ||.b4.||) + (|.b2.| * ||.b5.||);

:: CLVECT_1:th 108
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Element of the carrier of b1 holds
   ||.b2 - b3.|| = 0
iff
   b2 = b3;

:: CLVECT_1:th 109
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2 - b3.|| = ||.b3 - b2.||;

:: CLVECT_1:th 110
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2.|| - ||.b3.|| <= ||.b2 - b3.||;

:: CLVECT_1:th 111
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Element of the carrier of b1 holds
abs (||.b2.|| - ||.b3.||) <= ||.b2 - b3.||;

:: CLVECT_1:th 112
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3, b4 being Element of the carrier of b1 holds
||.b2 - b3.|| <= ||.b2 - b4.|| + ||.b4 - b3.||;

:: CLVECT_1:th 113
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Element of the carrier of b1
      st b2 <> b3
   holds ||.b2 - b3.|| <> 0;

:: CLVECT_1:funcnot 7 => CLVECT_1:func 7
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  let a3 be Element of COMPLEX;
  func A3 * A2 -> Function-like quasi_total Relation of NAT,the carrier of a1 means
    for b1 being Element of NAT holds
       it . b1 = a3 * (a2 . b1);
end;

:: CLVECT_1:def 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of COMPLEX
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b4 = b3 * b2
   iff
      for b5 being Element of NAT holds
         b4 . b5 = b3 * (b2 . b5);

:: CLVECT_1:attrnot 6 => CLVECT_1:attr 6
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  attr a2 is convergent means
    ex b1 being Element of the carrier of a1 st
       for b2 being Element of REAL
             st 0 < b2
          holds ex b3 being Element of NAT st
             for b4 being Element of NAT
                   st b3 <= b4
                holds ||.(a2 . b4) - b1.|| < b2;
end;

:: CLVECT_1:dfs 13
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
     a2 is convergent
it is sufficient to prove
  thus ex b1 being Element of the carrier of a1 st
       for b2 being Element of REAL
             st 0 < b2
          holds ex b3 being Element of NAT st
             for b4 being Element of NAT
                   st b3 <= b4
                holds ||.(a2 . b4) - b1.|| < b2;

:: CLVECT_1:def 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is convergent(b1)
   iff
      ex b3 being Element of the carrier of b1 st
         for b4 being Element of REAL
               st 0 < b4
            holds ex b5 being Element of NAT st
               for b6 being Element of NAT
                     st b5 <= b6
                  holds ||.(b2 . b6) - b3.|| < b4;

:: CLVECT_1:th 115
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1) & b3 is convergent(b1)
   holds b2 + b3 is convergent(b1);

:: CLVECT_1:th 116
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1) & b3 is convergent(b1)
   holds b2 - b3 is convergent(b1);

:: CLVECT_1:th 117
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is convergent(b1)
   holds b3 - b2 is convergent(b1);

:: CLVECT_1:th 118
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st b3 is convergent(b2)
   holds b1 * b3 is convergent(b2);

:: CLVECT_1:funcnot 8 => CLVECT_1:func 8
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  func ||.A2.|| -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = ||.a2 . b1.||;
end;

:: CLVECT_1:def 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,REAL holds
      b3 = ||.b2.||
   iff
      for b4 being Element of NAT holds
         b3 . b4 = ||.b2 . b4.||;

:: CLVECT_1:th 119
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1)
   holds ||.b2.|| is convergent;

:: CLVECT_1:funcnot 9 => CLVECT_1:func 9
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  assume a2 is convergent(a1);
  func lim A2 -> Element of the carrier of a1 means
    for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being Element of NAT st
          for b3 being Element of NAT
                st b2 <= b3
             holds ||.(a2 . b3) - it.|| < b1;
end;

:: CLVECT_1:def 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
   st b2 is convergent(b1)
for b3 being Element of the carrier of b1 holds
      b3 = lim b2
   iff
      for b4 being Element of REAL
            st 0 < b4
         holds ex b5 being Element of NAT st
            for b6 being Element of NAT
                  st b5 <= b6
               holds ||.(b2 . b6) - b3.|| < b4;

:: CLVECT_1:th 120
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is convergent(b1) & lim b3 = b2
   holds ||.b3 - b2.|| is convergent & lim ||.b3 - b2.|| = 0;

:: CLVECT_1:th 121
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1) & b3 is convergent(b1)
   holds lim (b2 + b3) = (lim b2) + lim b3;

:: CLVECT_1:th 122
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b2 is convergent(b1) & b3 is convergent(b1)
   holds lim (b2 - b3) = (lim b2) - lim b3;

:: CLVECT_1:th 123
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is convergent(b1)
   holds lim (b3 - b2) = (lim b3) - b2;

:: CLVECT_1:th 124
theorem
for b1 being Element of COMPLEX
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st b3 is convergent(b2)
   holds lim (b1 * b3) = b1 * lim b3;