Article CONVEX4, MML version 4.99.1005

:: CONVEX4:modenot 1 => CONVEX4:mode 1
definition
  let a1 be non empty ZeroStr;
  mode C_Linear_Combination of A1 -> Element of Funcs(the carrier of a1,COMPLEX) means
    ex b1 being finite Element of bool the carrier of a1 st
       for b2 being Element of the carrier of a1
             st not b2 in b1
          holds it . b2 = {};
end;

:: CONVEX4:dfs 1
definiens
  let a1 be non empty ZeroStr;
  let a2 be Element of Funcs(the carrier of a1,COMPLEX);
To prove
     a2 is C_Linear_Combination of a1
it is sufficient to prove
  thus ex b1 being finite Element of bool the carrier of a1 st
       for b2 being Element of the carrier of a1
             st not b2 in b1
          holds a2 . b2 = {};

:: CONVEX4:def 1
theorem
for b1 being non empty ZeroStr
for b2 being Element of Funcs(the carrier of b1,COMPLEX) holds
      b2 is C_Linear_Combination of b1
   iff
      ex b3 being finite Element of bool the carrier of b1 st
         for b4 being Element of the carrier of b1
               st not b4 in b3
            holds b2 . b4 = {};

:: CONVEX4:funcnot 1 => CONVEX4:func 1
definition
  let a1 be non empty addLoopStr;
  let a2 be Element of Funcs(the carrier of a1,COMPLEX);
  func Carrier A2 -> Element of bool the carrier of a1 equals
    {b1 where b1 is Element of the carrier of a1: a2 . b1 <> 0c};
end;

:: CONVEX4:def 2
theorem
for b1 being non empty addLoopStr
for b2 being Element of Funcs(the carrier of b1,COMPLEX) holds
   Carrier b2 = {b3 where b3 is Element of the carrier of b1: b2 . b3 <> 0c};

:: CONVEX4:funcreg 1
registration
  let a1 be non empty addLoopStr;
  let a2 be C_Linear_Combination of a1;
  cluster Carrier a2 -> finite;
end;

:: CONVEX4:th 1
theorem
for b1 being non empty addLoopStr
for b2 being C_Linear_Combination of b1
for b3 being Element of the carrier of b1 holds
      b2 . b3 = 0c
   iff
      not b3 in Carrier b2;

:: CONVEX4:funcnot 2 => CONVEX4:func 2
definition
  let a1 be non empty addLoopStr;
  func ZeroCLC A1 -> C_Linear_Combination of a1 means
    Carrier it = {};
end;

:: CONVEX4:def 3
theorem
for b1 being non empty addLoopStr
for b2 being C_Linear_Combination of b1 holds
      b2 = ZeroCLC b1
   iff
      Carrier b2 = {};

:: CONVEX4:funcreg 2
registration
  let a1 be non empty addLoopStr;
  cluster Carrier ZeroCLC a1 -> empty;
end;

:: CONVEX4:th 2
theorem
for b1 being non empty addLoopStr
for b2 being Element of the carrier of b1 holds
   (ZeroCLC b1) . b2 = 0c;

:: CONVEX4:modenot 2 => CONVEX4:mode 2
definition
  let a1 be non empty addLoopStr;
  let a2 be Element of bool the carrier of a1;
  mode C_Linear_Combination of A2 -> C_Linear_Combination of a1 means
    Carrier it c= a2;
end;

:: CONVEX4:dfs 4
definiens
  let a1 be non empty addLoopStr;
  let a2 be Element of bool the carrier of a1;
  let a3 be C_Linear_Combination of a1;
To prove
     a3 is C_Linear_Combination of a2
it is sufficient to prove
  thus Carrier a3 c= a2;

:: CONVEX4:def 4
theorem
for b1 being non empty addLoopStr
for b2 being Element of bool the carrier of b1
for b3 being C_Linear_Combination of b1 holds
      b3 is C_Linear_Combination of b2
   iff
      Carrier b3 c= b2;

:: CONVEX4:th 3
theorem
for b1 being non empty addLoopStr
for b2, b3 being Element of bool the carrier of b1
for b4 being C_Linear_Combination of b2
      st b2 c= b3
   holds b4 is C_Linear_Combination of b3;

:: CONVEX4:th 4
theorem
for b1 being non empty addLoopStr
for b2 being Element of bool the carrier of b1 holds
   ZeroCLC b1 is C_Linear_Combination of b2;

:: CONVEX4:th 5
theorem
for b1 being non empty addLoopStr
for b2 being C_Linear_Combination of {} the carrier of b1 holds
   b2 = ZeroCLC b1;

:: CONVEX4:funcnot 3 => CONVEX4:func 3
definition
  let a1 be non empty CLSStruct;
  let a2 be FinSequence of the carrier of a1;
  let a3 be Function-like quasi_total Relation of the carrier of a1,COMPLEX;
  func A3 (#) A2 -> FinSequence of the carrier of a1 means
    len it = len a2 &
     (for b1 being natural set
           st b1 in dom it
        holds it . b1 = (a3 . (a2 /. b1)) * (a2 /. b1));
end;

:: CONVEX4:def 5
theorem
for b1 being non empty CLSStruct
for b2 being FinSequence of the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,COMPLEX
for b4 being FinSequence of the carrier of b1 holds
      b4 = b3 (#) b2
   iff
      len b4 = len b2 &
       (for b5 being natural set
             st b5 in dom b4
          holds b4 . b5 = (b3 . (b2 /. b5)) * (b2 /. b5));

:: CONVEX4:th 6
theorem
for b1 being non empty CLSStruct
for b2 being Element of the carrier of b1
for b3 being set
for b4 being FinSequence of the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,COMPLEX
      st b3 in dom b4 & b2 = b4 . b3
   holds (b5 (#) b4) . b3 = (b5 . b2) * b2;

:: CONVEX4:th 7
theorem
for b1 being non empty CLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,COMPLEX holds
   b2 (#) <*> the carrier of b1 = <*> the carrier of b1;

:: CONVEX4:th 8
theorem
for b1 being non empty CLSStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,COMPLEX holds
   b3 (#) <*b2*> = <*(b3 . b2) * b2*>;

:: CONVEX4:th 9
theorem
for b1 being non empty CLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,COMPLEX holds
   b4 (#) <*b2,b3*> = <*(b4 . b2) * b2,(b4 . b3) * b3*>;

:: CONVEX4:th 10
theorem
for b1 being non empty CLSStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,COMPLEX holds
   b5 (#) <*b2,b3,b4*> = <*(b5 . b2) * b2,(b5 . b3) * b3,(b5 . b4) * b4*>;

:: CONVEX4:funcnot 4 => CONVEX4:func 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed CLSStruct;
  let a2 be C_Linear_Combination of a1;
  func Sum A2 -> Element of the carrier of a1 means
    ex b1 being FinSequence of the carrier of a1 st
       b1 is one-to-one & rng b1 = Carrier a2 & it = Sum (a2 (#) b1);
end;

:: CONVEX4:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed CLSStruct
for b2 being C_Linear_Combination of b1
for b3 being Element of the carrier of b1 holds
      b3 = Sum b2
   iff
      ex b4 being FinSequence of the carrier of b1 st
         b4 is one-to-one & rng b4 = Carrier b2 & b3 = Sum (b2 (#) b4);

:: CONVEX4:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed CLSStruct holds
   Sum ZeroCLC b1 = 0. b1;

:: CONVEX4:th 12
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 <> {}
   holds    b2 is linearly-closed(b1)
   iff
      for b3 being C_Linear_Combination of b2 holds
         Sum b3 in b2;

:: CONVEX4:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed CLSStruct
for b2 being C_Linear_Combination of {} the carrier of b1 holds
   Sum b2 = 0. b1;

:: CONVEX4:th 14
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 C_Linear_Combination of {b2} holds
   Sum b3 = (b3 . b2) * b2;

:: CONVEX4:th 15
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
   st b2 <> b3
for b4 being C_Linear_Combination of {b2,b3} holds
   Sum b4 = ((b4 . b2) * b2) + ((b4 . b3) * b3);

:: CONVEX4:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed CLSStruct
for b2 being C_Linear_Combination of b1
      st Carrier b2 = {}
   holds Sum b2 = 0. b1;

:: CONVEX4:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being C_Linear_Combination of b1
for b3 being Element of the carrier of b1
      st Carrier b2 = {b3}
   holds Sum b2 = (b2 . b3) * b3;

:: CONVEX4:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being C_Linear_Combination of b1
for b3, b4 being Element of the carrier of b1
      st Carrier b2 = {b3,b4} & b3 <> b4
   holds Sum b2 = ((b2 . b3) * b3) + ((b2 . b4) * b4);

:: CONVEX4:prednot 1 => CONVEX4:pred 1
definition
  let a1 be non empty addLoopStr;
  let a2, a3 be C_Linear_Combination of a1;
  redefine pred A2 = A3 means
    for b1 being Element of the carrier of a1 holds
       a2 . b1 = a3 . b1;
  symmetry;
::  for a1 being non empty addLoopStr
::  for a2, a3 being C_Linear_Combination of a1
::        st a2 = a3
::     holds a3 = a2;
  reflexivity;
::  for a1 being non empty addLoopStr
::  for a2 being C_Linear_Combination of a1 holds
::     a2 = a2;
end;

:: CONVEX4:dfs 7
definiens
  let a1 be non empty addLoopStr;
  let a2, a3 be C_Linear_Combination of a1;
To prove
     a2 = a3
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       a2 . b1 = a3 . b1;

:: CONVEX4:def 7
theorem
for b1 being non empty addLoopStr
for b2, b3 being C_Linear_Combination of b1 holds
   b2 = b3
iff
   for b4 being Element of the carrier of b1 holds
      b2 . b4 = b3 . b4;

:: CONVEX4:funcnot 5 => CONVEX4:func 5
definition
  let a1 be non empty addLoopStr;
  let a2, a3 be C_Linear_Combination of a1;
  redefine func A2 + A3 -> C_Linear_Combination of a1 means
    for b1 being Element of the carrier of a1 holds
       it . b1 = (a2 . b1) + (a3 . b1);
  commutativity;
::  for a1 being non empty addLoopStr
::  for a2, a3 being C_Linear_Combination of a1 holds
::  a2 + a3 = a3 + a2;
end;

:: CONVEX4:def 8
theorem
for b1 being non empty addLoopStr
for b2, b3, b4 being C_Linear_Combination of b1 holds
   b4 = b2 + b3
iff
   for b5 being Element of the carrier of b1 holds
      b4 . b5 = (b2 . b5) + (b3 . b5);

:: CONVEX4:th 19
theorem
for b1 being non empty CLSStruct
for b2, b3 being C_Linear_Combination of b1 holds
Carrier (b2 + b3) c= (Carrier b2) \/ Carrier b3;

:: CONVEX4:th 20
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
for b3, b4 being C_Linear_Combination of b1
      st b3 is C_Linear_Combination of b2 & b4 is C_Linear_Combination of b2
   holds b3 + b4 is C_Linear_Combination of b2;

:: CONVEX4:funcnot 6 => CONVEX4:func 6
definition
  let a1 be non empty CLSStruct;
  let a2 be Element of bool the carrier of a1;
  let a3, a4 be C_Linear_Combination of a2;
  redefine func a3 + a4 -> C_Linear_Combination of a2;
  commutativity;
::  for a1 being non empty CLSStruct
::  for a2 being Element of bool the carrier of a1
::  for a3, a4 being C_Linear_Combination of a2 holds
::  a3 + a4 = a4 + a3;
end;

:: CONVEX4:th 21
theorem
for b1 being non empty addLoopStr
for b2, b3 being C_Linear_Combination of b1 holds
b2 + b3 = b3 + b2;

:: CONVEX4:th 22
theorem
for b1 being non empty CLSStruct
for b2, b3, b4 being C_Linear_Combination of b1 holds
b2 + (b3 + b4) = (b2 + b3) + b4;

:: CONVEX4:th 23
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
   b2 + ZeroCLC b1 = b2;

:: CONVEX4:funcnot 7 => CONVEX4:func 7
definition
  let a1 be non empty CLSStruct;
  let a2 be Element of COMPLEX;
  let a3 be C_Linear_Combination of a1;
  func A2 * A3 -> C_Linear_Combination of a1 means
    for b1 being Element of the carrier of a1 holds
       it . b1 = a2 * (a3 . b1);
end;

:: CONVEX4:def 9
theorem
for b1 being non empty CLSStruct
for b2 being Element of COMPLEX
for b3, b4 being C_Linear_Combination of b1 holds
   b4 = b2 * b3
iff
   for b5 being Element of the carrier of b1 holds
      b4 . b5 = b2 * (b3 . b5);

:: CONVEX4:th 24
theorem
for b1 being non empty CLSStruct
for b2 being Element of COMPLEX
for b3 being C_Linear_Combination of b1
      st b2 <> 0c
   holds Carrier (b2 * b3) = Carrier b3;

:: CONVEX4:th 25
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
   0c * b2 = ZeroCLC b1;

:: CONVEX4:th 26
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of COMPLEX
for b4 being C_Linear_Combination of b1
      st b4 is C_Linear_Combination of b2
   holds b3 * b4 is C_Linear_Combination of b2;

:: CONVEX4:th 27
theorem
for b1 being non empty CLSStruct
for b2, b3 being Element of COMPLEX
for b4 being C_Linear_Combination of b1 holds
   (b2 + b3) * b4 = (b2 * b4) + (b3 * b4);

:: CONVEX4:th 28
theorem
for b1 being non empty CLSStruct
for b2 being Element of COMPLEX
for b3, b4 being C_Linear_Combination of b1 holds
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4);

:: CONVEX4:th 29
theorem
for b1 being non empty CLSStruct
for b2, b3 being Element of COMPLEX
for b4 being C_Linear_Combination of b1 holds
   b2 * (b3 * b4) = (b2 * b3) * b4;

:: CONVEX4:th 30
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
   1r * b2 = b2;

:: CONVEX4:funcnot 8 => CONVEX4:func 8
definition
  let a1 be non empty CLSStruct;
  let a2 be C_Linear_Combination of a1;
  func - A2 -> C_Linear_Combination of a1 equals
    (- 1r) * a2;
end;

:: CONVEX4:def 10
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
   - b2 = (- 1r) * b2;

:: CONVEX4:th 31
theorem
for b1 being non empty CLSStruct
for b2 being Element of the carrier of b1
for b3 being C_Linear_Combination of b1 holds
   (- b3) . b2 = - (b3 . b2);

:: CONVEX4:th 32
theorem
for b1 being non empty CLSStruct
for b2, b3 being C_Linear_Combination of b1
      st b2 + b3 = ZeroCLC b1
   holds b3 = - b2;

:: CONVEX4:th 33
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
   - - b2 = b2;

:: CONVEX4:funcnot 9 => CONVEX4:func 9
definition
  let a1 be non empty CLSStruct;
  let a2, a3 be C_Linear_Combination of a1;
  func A2 - A3 -> C_Linear_Combination of a1 equals
    a2 + - a3;
end;

:: CONVEX4:def 11
theorem
for b1 being non empty CLSStruct
for b2, b3 being C_Linear_Combination of b1 holds
b2 - b3 = b2 + - b3;

:: CONVEX4:th 34
theorem
for b1 being non empty CLSStruct
for b2 being Element of the carrier of b1
for b3, b4 being C_Linear_Combination of b1 holds
(b3 - b4) . b2 = (b3 . b2) - (b4 . b2);

:: CONVEX4:th 35
theorem
for b1 being non empty CLSStruct
for b2, b3 being C_Linear_Combination of b1 holds
Carrier (b2 - b3) c= (Carrier b2) \/ Carrier b3;

:: CONVEX4:th 36
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
for b3, b4 being C_Linear_Combination of b1
      st b3 is C_Linear_Combination of b2 & b4 is C_Linear_Combination of b2
   holds b3 - b4 is C_Linear_Combination of b2;

:: CONVEX4:th 37
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
   b2 - b2 = ZeroCLC b1;

:: CONVEX4:funcnot 10 => CONVEX4:func 10
definition
  let a1 be non empty CLSStruct;
  func C_LinComb A1 -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 is C_Linear_Combination of a1;
end;

:: CONVEX4:def 12
theorem
for b1 being non empty CLSStruct
for b2 being set holds
      b2 = C_LinComb b1
   iff
      for b3 being set holds
            b3 in b2
         iff
            b3 is C_Linear_Combination of b1;

:: CONVEX4:funcreg 3
registration
  let a1 be non empty CLSStruct;
  cluster C_LinComb a1 -> non empty;
end;

:: CONVEX4:funcnot 11 => CONVEX4:func 11
definition
  let a1 be non empty CLSStruct;
  let a2 be Element of C_LinComb a1;
  func @ A2 -> C_Linear_Combination of a1 equals
    a2;
end;

:: CONVEX4:def 13
theorem
for b1 being non empty CLSStruct
for b2 being Element of C_LinComb b1 holds
   @ b2 = b2;

:: CONVEX4:funcnot 12 => CONVEX4:func 12
definition
  let a1 be non empty CLSStruct;
  let a2 be C_Linear_Combination of a1;
  func @ A2 -> Element of C_LinComb a1 equals
    a2;
end;

:: CONVEX4:def 14
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
   @ b2 = b2;

:: CONVEX4:funcnot 13 => CONVEX4:func 13
definition
  let a1 be non empty CLSStruct;
  func C_LCAdd A1 -> Function-like quasi_total Relation of [:C_LinComb a1,C_LinComb a1:],C_LinComb a1 means
    for b1, b2 being Element of C_LinComb a1 holds
    it .(b1,b2) = (@ b1) + @ b2;
end;

:: CONVEX4:def 15
theorem
for b1 being non empty CLSStruct
for b2 being Function-like quasi_total Relation of [:C_LinComb b1,C_LinComb b1:],C_LinComb b1 holds
      b2 = C_LCAdd b1
   iff
      for b3, b4 being Element of C_LinComb b1 holds
      b2 .(b3,b4) = (@ b3) + @ b4;

:: CONVEX4:funcnot 14 => CONVEX4:func 14
definition
  let a1 be non empty CLSStruct;
  func C_LCMult A1 -> Function-like quasi_total Relation of [:COMPLEX,C_LinComb a1:],C_LinComb a1 means
    for b1 being Element of COMPLEX
    for b2 being Element of C_LinComb a1 holds
       it . [b1,b2] = b1 * @ b2;
end;

:: CONVEX4:def 16
theorem
for b1 being non empty CLSStruct
for b2 being Function-like quasi_total Relation of [:COMPLEX,C_LinComb b1:],C_LinComb b1 holds
      b2 = C_LCMult b1
   iff
      for b3 being Element of COMPLEX
      for b4 being Element of C_LinComb b1 holds
         b2 . [b3,b4] = b3 * @ b4;

:: CONVEX4:funcnot 15 => CONVEX4:func 15
definition
  let a1 be non empty CLSStruct;
  func LC_CLSpace A1 -> non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct equals
    CLSStruct(#C_LinComb a1,@ ZeroCLC a1,C_LCAdd a1,C_LCMult a1#);
end;

:: CONVEX4:def 17
theorem
for b1 being non empty CLSStruct holds
   LC_CLSpace b1 = CLSStruct(#C_LinComb b1,@ ZeroCLC b1,C_LCAdd b1,C_LCMult b1#);

:: CONVEX4:funcreg 4
registration
  let a1 be non empty CLSStruct;
  cluster LC_CLSpace a1 -> non empty right_complementable Abelian add-associative right_zeroed strict ComplexLinearSpace-like;
end;

:: CONVEX4:th 38
theorem
for b1 being non empty CLSStruct
for b2, b3 being C_Linear_Combination of b1 holds
(vector(LC_CLSpace b1,b2)) + vector(LC_CLSpace b1,b3) = b2 + b3;

:: CONVEX4:th 39
theorem
for b1 being non empty CLSStruct
for b2 being Element of COMPLEX
for b3 being C_Linear_Combination of b1 holds
   b2 * vector(LC_CLSpace b1,b3) = b2 * b3;

:: CONVEX4:th 40
theorem
for b1 being non empty CLSStruct
for b2 being C_Linear_Combination of b1 holds
   - vector(LC_CLSpace b1,b2) = - b2;

:: CONVEX4:th 41
theorem
for b1 being non empty CLSStruct
for b2, b3 being C_Linear_Combination of b1 holds
(vector(LC_CLSpace b1,b2)) - vector(LC_CLSpace b1,b3) = b2 - b3;

:: CONVEX4:funcnot 16 => CONVEX4:func 16
definition
  let a1 be non empty CLSStruct;
  let a2 be Element of bool the carrier of a1;
  func LC_CLSpace A2 -> strict Subspace of LC_CLSpace a1 means
    the carrier of it = {b1 where b1 is C_Linear_Combination of a2: TRUE};
end;

:: CONVEX4:def 18
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
for b3 being strict Subspace of LC_CLSpace b1 holds
      b3 = LC_CLSpace b2
   iff
      the carrier of b3 = {b4 where b4 is C_Linear_Combination of b2: TRUE};

:: CONVEX4:funcnot 17 => CONVEX4:func 17
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
  let a2 be Subspace of a1;
  func Up A2 -> Element of bool the carrier of a1 equals
    the carrier of a2;
end;

:: CONVEX4:def 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being Subspace of b1 holds
   Up b2 = the carrier of b2;

:: CONVEX4:funcreg 5
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
  let a2 be Subspace of a1;
  cluster Up a2 -> non empty;
end;

:: CONVEX4:attrnot 1 => CONVEX4:attr 1
definition
  let a1 be non empty CLSStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is Affine means
    for b1, b2 being Element of the carrier of a1
    for b3 being Element of COMPLEX
          st (ex b4 being Element of REAL st
                b4 = b3) &
             b1 in a2 &
             b2 in a2
       holds ((1r - b3) * b1) + (b3 * b2) in a2;
end;

:: CONVEX4:dfs 20
definiens
  let a1 be non empty CLSStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is Affine
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
    for b3 being Element of COMPLEX
          st (ex b4 being Element of REAL st
                b4 = b3) &
             b1 in a2 &
             b2 in a2
       holds ((1r - b3) * b1) + (b3 * b2) in a2;

:: CONVEX4:def 20
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is Affine(b1)
   iff
      for b3, b4 being Element of the carrier of b1
      for b5 being Element of COMPLEX
            st (ex b6 being Element of REAL st
                  b6 = b5) &
               b3 in b2 &
               b4 in b2
         holds ((1r - b5) * b3) + (b5 * b4) in b2;

:: CONVEX4:funcnot 18 => CONVEX4:func 18
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;

:: CONVEX4:def 21
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#);

:: CONVEX4:funcreg 6
registration
  let a1 be non empty CLSStruct;
  cluster [#] a1 -> Affine;
end;

:: CONVEX4:funcreg 7
registration
  let a1 be non empty CLSStruct;
  cluster {} a1 -> Affine;
end;

:: CONVEX4:exreg 1
registration
  let a1 be non empty CLSStruct;
  cluster non empty Affine Element of bool the carrier of a1;
end;

:: CONVEX4:exreg 2
registration
  let a1 be non empty CLSStruct;
  cluster empty Affine Element of bool the carrier of a1;
end;

:: CONVEX4:th 42
theorem
for b1 being real set
for b2 being complex set holds
   Re (b1 * b2) = b1 * Re b2;

:: CONVEX4:th 43
theorem
for b1 being real set
for b2 being complex set holds
   Im (b1 * b2) = b1 * Im b2;

:: CONVEX4:th 44
theorem
for b1 being real set
for b2 being complex set
      st {} <= b1 & b1 <= 1
   holds |.b1 * b2.| = b1 * |.b2.| &
    |.(1r - b1) * b2.| = (1r - b1) * |.b2.|;

:: CONVEX4:funcnot 19 => CONVEX4:func 19
definition
  let a1 be non empty CLSStruct;
  let a2 be Element of bool the carrier of a1;
  let a3 be Element of COMPLEX;
  func A3 * A2 -> Element of bool the carrier of a1 equals
    {a3 * b1 where b1 is Element of the carrier of a1: b1 in a2};
end;

:: CONVEX4:def 22
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of COMPLEX holds
   b3 * b2 = {b3 * b4 where b4 is Element of the carrier of b1: b4 in b2};

:: CONVEX4:attrnot 2 => CONVEX4:attr 2
definition
  let a1 be non empty CLSStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is convex means
    for b1, b2 being Element of the carrier of a1
    for b3 being Element of COMPLEX
          st (ex b4 being Element of REAL st
                b3 = b4 & {} < b4 & b4 < 1) &
             b1 in a2 &
             b2 in a2
       holds (b3 * b1) + ((1r - b3) * b2) in a2;
end;

:: CONVEX4:dfs 23
definiens
  let a1 be non empty CLSStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is convex
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
    for b3 being Element of COMPLEX
          st (ex b4 being Element of REAL st
                b3 = b4 & {} < b4 & b4 < 1) &
             b1 in a2 &
             b2 in a2
       holds (b3 * b1) + ((1r - b3) * b2) in a2;

:: CONVEX4:def 23
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is convex(b1)
   iff
      for b3, b4 being Element of the carrier of b1
      for b5 being Element of COMPLEX
            st (ex b6 being Element of REAL st
                  b5 = b6 & {} < b6 & b6 < 1) &
               b3 in b2 &
               b4 in b2
         holds (b5 * b3) + ((1r - b5) * b4) in b2;

:: CONVEX4:th 45
theorem
for b1 being non empty ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of COMPLEX
      st b2 is convex(b1)
   holds b3 * b2 is convex(b1);

:: CONVEX4:th 46
theorem
for b1 being non empty Abelian add-associative ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is convex(b1) & b3 is convex(b1)
   holds b2 + b3 is convex(b1);

:: CONVEX4:th 47
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 convex(b1) & b3 is convex(b1)
   holds b2 - b3 is convex(b1);

:: CONVEX4:th 48
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is convex(b1)
   iff
      for b3 being Element of COMPLEX
            st ex b4 being Element of REAL st
                 b3 = b4 & {} < b4 & b4 < 1
         holds (b3 * b2) + ((1r - b3) * b2) c= b2;

:: CONVEX4:th 49
theorem
for b1 being non empty Abelian CLSStruct
for b2 being Element of bool the carrier of b1
   st b2 is convex(b1)
for b3 being Element of COMPLEX
      st ex b4 being Element of REAL st
           b3 = b4 & {} < b4 & b4 < 1
   holds ((1r - b3) * b2) + (b3 * b2) c= b2;

:: CONVEX4:th 50
theorem
for b1 being non empty Abelian add-associative ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of bool the carrier of b1
   st b2 is convex(b1) & b3 is convex(b1)
for b4 being Element of COMPLEX
      st ex b5 being Element of REAL st
           b4 = b5
   holds (b4 * b2) + ((1r - b4) * b3) is convex(b1);

:: CONVEX4:th 51
theorem
for b1 being non empty ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1 holds
   1r * b2 = b2;

:: CONVEX4:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being non empty Element of bool the carrier of b1 holds
   0c * b2 = {0. b1};

:: CONVEX4:th 53
theorem
for b1 being non empty add-associative addLoopStr
for b2, b3, b4 being Element of bool the carrier of b1 holds
(b2 + b3) + b4 = b2 + (b3 + b4);

:: CONVEX4:th 54
theorem
for b1 being non empty ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of COMPLEX holds
b3 * (b4 * b2) = (b3 * b4) * b2;

:: CONVEX4:th 55
theorem
for b1 being non empty ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of COMPLEX holds
   b4 * (b2 + b3) = (b4 * b2) + (b4 * b3);

:: CONVEX4:th 56
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 Element of the carrier of b1 holds
      b2 is convex(b1)
   iff
      b3 + b2 is convex(b1);

:: CONVEX4:th 57
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
   Up (0). b1 is convex(b1);

:: CONVEX4:th 58
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct holds
   Up (Omega). b1 is convex(b1);

:: CONVEX4:th 59
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
      st b2 = {}
   holds b2 is convex(b1);

:: CONVEX4:th 60
theorem
for b1 being non empty Abelian add-associative ComplexLinearSpace-like CLSStruct
for b2, b3 being Element of bool the carrier of b1
for b4, b5 being Element of COMPLEX
      st b2 is convex(b1) & b3 is convex(b1)
   holds (b4 * b2) + (b5 * b3) is convex(b1);

:: CONVEX4:th 61
theorem
for b1 being non empty ComplexLinearSpace-like CLSStruct
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of COMPLEX holds
(b3 + b4) * b2 c= (b3 * b2) + (b4 * b2);

:: CONVEX4:th 62
theorem
for b1 being non empty CLSStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of COMPLEX
      st b2 c= b3
   holds b4 * b2 c= b4 * b3;

:: CONVEX4:th 63
theorem
for b1 being non empty CLSStruct
for b2 being empty Element of bool the carrier of b1
for b3 being Element of COMPLEX holds
   b3 * b2 = {};

:: CONVEX4:th 64
theorem
for b1 being non empty addLoopStr
for b2 being empty Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1 holds
   b2 + b3 = {};

:: CONVEX4:th 65
theorem
for b1 being non empty right_zeroed addLoopStr
for b2 being Element of bool the carrier of b1 holds
   b2 + {0. b1} = b2;

:: CONVEX4:th 66
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, b4 being Element of COMPLEX
      st (ex b5, b6 being Element of REAL st
            b3 = b5 & b4 = b6 & {} <= b5 & {} <= b6) &
         b2 is convex(b1)
   holds (b3 * b2) + (b4 * b2) = (b3 + b4) * b2;

:: CONVEX4:th 67
theorem
for b1 being non empty Abelian add-associative ComplexLinearSpace-like CLSStruct
for b2, b3, b4 being Element of bool the carrier of b1
for b5, b6, b7 being Element of COMPLEX
      st b2 is convex(b1) & b3 is convex(b1) & b4 is convex(b1)
   holds ((b5 * b2) + (b6 * b3)) + (b7 * b4) is convex(b1);

:: CONVEX4:th 68
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool bool the carrier of b1
      st for b3 being Element of bool the carrier of b1
              st b3 in b2
           holds b3 is convex(b1)
   holds meet b2 is convex(b1);

:: CONVEX4:th 69
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
      st b2 is Affine(b1)
   holds b2 is convex(b1);

:: CONVEX4:exreg 3
registration
  let a1 be non empty CLSStruct;
  cluster non empty convex Element of bool the carrier of a1;
end;

:: CONVEX4:exreg 4
registration
  let a1 be non empty CLSStruct;
  cluster empty convex Element of bool the carrier of a1;
end;

:: CONVEX4:th 70
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: b4 <= Re (b5 .|. b3)}
   holds b2 is convex(b1);

:: CONVEX4:th 71
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: b4 < Re (b5 .|. b3)}
   holds b2 is convex(b1);

:: CONVEX4:th 72
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: Re (b5 .|. b3) <= b4}
   holds b2 is convex(b1);

:: CONVEX4:th 73
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: Re (b5 .|. b3) < b4}
   holds b2 is convex(b1);

:: CONVEX4:th 74
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: b4 <= Im (b5 .|. b3)}
   holds b2 is convex(b1);

:: CONVEX4:th 75
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: b4 < Im (b5 .|. b3)}
   holds b2 is convex(b1);

:: CONVEX4:th 76
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: Im (b5 .|. b3) <= b4}
   holds b2 is convex(b1);

:: CONVEX4:th 77
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: Im (b5 .|. b3) < b4}
   holds b2 is convex(b1);

:: CONVEX4:th 78
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: |.b5 .|. b3.| <= b4}
   holds b2 is convex(b1);

:: CONVEX4:th 79
theorem
for b1 being non empty ComplexUnitarySpace-like CUNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: |.b5 .|. b3.| < b4}
   holds b2 is convex(b1);

:: CONVEX4:attrnot 3 => CONVEX4:attr 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
  let a2 be C_Linear_Combination of a1;
  attr a2 is convex means
    ex b1 being FinSequence of the carrier of a1 st
       b1 is one-to-one &
        rng b1 = Carrier a2 &
        (ex b2 being FinSequence of REAL st
           len b2 = len b1 &
            Sum b2 = 1 &
            (for b3 being natural set
                  st b3 in dom b2
               holds b2 . b3 = a2 . (b1 . b3) & {} <= b2 . b3));
end;

:: CONVEX4:dfs 24
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct;
  let a2 be C_Linear_Combination of a1;
To prove
     a2 is convex
it is sufficient to prove
  thus ex b1 being FinSequence of the carrier of a1 st
       b1 is one-to-one &
        rng b1 = Carrier a2 &
        (ex b2 being FinSequence of REAL st
           len b2 = len b1 &
            Sum b2 = 1 &
            (for b3 being natural set
                  st b3 in dom b2
               holds b2 . b3 = a2 . (b1 . b3) & {} <= b2 . b3));

:: CONVEX4:def 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being C_Linear_Combination of b1 holds
      b2 is convex(b1)
   iff
      ex b3 being FinSequence of the carrier of b1 st
         b3 is one-to-one &
          rng b3 = Carrier b2 &
          (ex b4 being FinSequence of REAL st
             len b4 = len b3 &
              Sum b4 = 1 &
              (for b5 being natural set
                    st b5 in dom b4
                 holds b4 . b5 = b2 . (b3 . b5) & {} <= b4 . b5));

:: CONVEX4:th 80
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being C_Linear_Combination of b1
      st b2 is convex(b1)
   holds Carrier b2 <> {};

:: CONVEX4:th 81
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being C_Linear_Combination of b1
for b3 being Element of the carrier of b1
      st b2 is convex(b1) &
         (ex b4 being Element of REAL st
            b4 = b2 . b3 & b4 <= {})
   holds not b3 in Carrier b2;

:: CONVEX4:th 82
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like CLSStruct
for b2 being C_Linear_Combination of b1
      st b2 is convex(b1)
   holds b2 <> ZeroCLC b1;

:: CONVEX4:th 83
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 C_Linear_Combination of b1
      st b3 is convex(b1) & Carrier b3 = {b2}
   holds (ex b4 being Element of REAL st
       b4 = b3 . b2 & b4 = 1) &
    Sum b3 = (b3 . b2) * b2;

:: CONVEX4: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 C_Linear_Combination of b1
      st b4 is convex(b1) & Carrier b4 = {b2,b3} & b2 <> b3
   holds (ex b5, b6 being Element of REAL st
       b5 = b4 . b2 & b6 = b4 . b3 & b5 + b6 = 1 & {} <= b5 & {} <= b6) &
    Sum b4 = ((b4 . b2) * b2) + ((b4 . b3) * b3);

:: CONVEX4:th 85
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 C_Linear_Combination of b1
      st b5 is convex(b1) & Carrier b5 = {b2,b3,b4} & b2 <> b3 & b3 <> b4 & b4 <> b2
   holds (ex b6, b7, b8 being real set st
       b6 = b5 . b2 & b7 = b5 . b3 & b8 = b5 . b4 & (b6 + b7) + b8 = 1 & {} <= b6 & {} <= b7 & {} <= b8) &
    Sum b5 = (((b5 . b2) * b2) + ((b5 . b3) * b3)) + ((b5 . b4) * b4);

:: CONVEX4:th 86
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 C_Linear_Combination of {b2}
      st b3 is convex(b1)
   holds (ex b4 being Element of REAL st
       b4 = b3 . b2 & b4 = 1) &
    Sum b3 = (b3 . b2) * b2;

:: CONVEX4: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 C_Linear_Combination of {b2,b3}
      st b2 <> b3 & b4 is convex(b1)
   holds (ex b5, b6 being real set st
       b5 = b4 . b2 & b6 = b4 . b3 & {} <= b5 & {} <= b6) &
    Sum b4 = ((b4 . b2) * b2) + ((b4 . b3) * b3);

:: CONVEX4:th 88
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 C_Linear_Combination of {b2,b3,b4}
      st b2 <> b3 & b3 <> b4 & b4 <> b2 & b5 is convex(b1)
   holds (ex b6, b7, b8 being real set st
       b6 = b5 . b2 & b7 = b5 . b3 & b8 = b5 . b4 & (b6 + b7) + b8 = 1 & {} <= b6 & {} <= b7 & {} <= b8) &
    Sum b5 = (((b5 . b2) * b2) + ((b5 . b3) * b3)) + ((b5 . b4) * b4);

:: CONVEX4:funcnot 20 => CONVEX4:func 20
definition
  let a1 be non empty CLSStruct;
  let a2 be Element of bool the carrier of a1;
  func Convex-Family A2 -> Element of bool bool the carrier of a1 means
    for b1 being Element of bool the carrier of a1 holds
          b1 in it
       iff
          b1 is convex(a1) & a2 c= b1;
end;

:: CONVEX4:def 25
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1 holds
      b3 = Convex-Family b2
   iff
      for b4 being Element of bool the carrier of b1 holds
            b4 in b3
         iff
            b4 is convex(b1) & b2 c= b4;

:: CONVEX4:funcnot 21 => CONVEX4:func 21
definition
  let a1 be non empty CLSStruct;
  let a2 be Element of bool the carrier of a1;
  func conv A2 -> convex Element of bool the carrier of a1 equals
    meet Convex-Family a2;
end;

:: CONVEX4:def 26
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1 holds
   conv b2 = meet Convex-Family b2;

:: CONVEX4:th 89
theorem
for b1 being non empty CLSStruct
for b2 being Element of bool the carrier of b1
for b3 being convex Element of bool the carrier of b1
      st b2 c= b3
   holds conv b2 c= b3;