Article HERMITAN, MML version 4.99.1005

:: HERMITAN:th 1
theorem
for b1 being complex set
      st b1 = b1 *'
   holds Im b1 = 0;

:: HERMITAN:th 2
theorem
for b1 being Element of COMPLEX
      st b1 <> 0c
   holds |.[*(Re b1) / |.b1.|,(- Im b1) / |.b1.|*].| = 1 &
    Re ([*(Re b1) / |.b1.|,(- Im b1) / |.b1.|*] * b1) = |.b1.| &
    Im ([*(Re b1) / |.b1.|,(- Im b1) / |.b1.|*] * b1) = 0;

:: HERMITAN:th 3
theorem
for b1 being Element of COMPLEX holds
   ex b2 being Element of COMPLEX st
      |.b2.| = 1 & Re (b2 * b1) = |.b1.| & Im (b2 * b1) = 0;

:: HERMITAN:th 4
theorem
for b1 being Element of COMPLEX holds
   b1 * (b1 *') = |.b1.| ^2 + (0 * <i>);

:: HERMITAN:th 5
theorem
for b1 being Element of the carrier of F_Complex
      st b1 = b1 *'
   holds Im b1 = 0;

:: HERMITAN:th 7
theorem
i_FC * (i_FC *') = 1_ F_Complex;

:: HERMITAN:th 8
theorem
for b1 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds |.[**(Re b1) / |.b1.|,(- Im b1) / |.b1.|**].| = 1 &
    Re ([**(Re b1) / |.b1.|,(- Im b1) / |.b1.|**] * b1) = |.b1.| &
    Im ([**(Re b1) / |.b1.|,(- Im b1) / |.b1.|**] * b1) = 0;

:: HERMITAN:th 9
theorem
for b1 being Element of the carrier of F_Complex holds
   ex b2 being Element of the carrier of F_Complex st
      |.b2.| = 1 & Re (b2 * b1) = |.b1.| & Im (b2 * b1) = 0;

:: HERMITAN:th 10
theorem
for b1, b2 being Element of the carrier of F_Complex holds
Re (b1 - b2) = (Re b1) - Re b2 &
 Im (b1 - b2) = (Im b1) - Im b2;

:: HERMITAN:th 11
theorem
for b1, b2 being Element of the carrier of F_Complex
      st Im b1 = 0
   holds Re (b1 * b2) = (Re b1) * Re b2 &
    Im (b1 * b2) = (Re b1) * Im b2;

:: HERMITAN:th 12
theorem
for b1, b2 being Element of the carrier of F_Complex
      st Im b1 = 0 & Im b2 = 0
   holds Im (b1 * b2) = 0;

:: HERMITAN:th 14
theorem
for b1 being Element of the carrier of F_Complex
      st Im b1 = 0
   holds b1 = b1 *';

:: HERMITAN:th 16
theorem
for b1 being Element of the carrier of F_Complex holds
   b1 * (b1 *') = |.b1.| ^2;

:: HERMITAN:th 17
theorem
for b1 being Element of the carrier of F_Complex
      st 0 <= Re b1 & Im b1 = 0
   holds |.b1.| = Re b1;

:: HERMITAN:th 18
theorem
for b1 being Element of the carrier of F_Complex holds
   (Re b1) + Re (b1 *') = 2 * Re b1;

:: HERMITAN:attrnot 1 => HERMITAN:attr 1
definition
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of F_Complex;
  attr a2 is cmplxhomogeneous means
    for b1 being Element of the carrier of a1
    for b2 being Element of the carrier of F_Complex holds
       a2 . (b2 * b1) = b2 *' * (a2 . b1);
end;

:: HERMITAN:dfs 1
definiens
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of F_Complex;
To prove
     a2 is cmplxhomogeneous
it is sufficient to prove
  thus for b1 being Element of the carrier of a1
    for b2 being Element of the carrier of F_Complex holds
       a2 . (b2 * b1) = b2 *' * (a2 . b1);

:: HERMITAN:def 1
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of F_Complex holds
      b2 is cmplxhomogeneous(b1)
   iff
      for b3 being Element of the carrier of b1
      for b4 being Element of the carrier of F_Complex holds
         b2 . (b4 * b3) = b4 *' * (b2 . b3);

:: HERMITAN:funcreg 1
registration
  let a1 be non empty VectSpStr over F_Complex;
  cluster 0Functional a1 -> Function-like quasi_total cmplxhomogeneous;
end;

:: HERMITAN:condreg 1
registration
  let a1 be non empty right_complementable add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  cluster Function-like quasi_total cmplxhomogeneous -> 0-preserving (Relation of the carrier of a1,the carrier of F_Complex);
end;

:: HERMITAN:exreg 1
registration
  let a1 be non empty VectSpStr over F_Complex;
  cluster Relation-like Function-like quasi_total additive 0-preserving cmplxhomogeneous Relation of the carrier of a1,the carrier of F_Complex;
end;

:: HERMITAN:modenot 1
definition
  let a1 be non empty VectSpStr over F_Complex;
  mode antilinear-Functional of a1 is Function-like quasi_total additive cmplxhomogeneous Relation of the carrier of a1,the carrier of F_Complex;
end;

:: HERMITAN:funcreg 2
registration
  let a1 be non empty VectSpStr over F_Complex;
  let a2, a3 be Function-like quasi_total cmplxhomogeneous Relation of the carrier of a1,the carrier of F_Complex;
  cluster a2 + a3 -> Function-like quasi_total cmplxhomogeneous;
end;

:: HERMITAN:funcreg 3
registration
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total cmplxhomogeneous Relation of the carrier of a1,the carrier of F_Complex;
  cluster - a2 -> Function-like quasi_total cmplxhomogeneous;
end;

:: HERMITAN:funcreg 4
registration
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Element of the carrier of F_Complex;
  let a3 be Function-like quasi_total cmplxhomogeneous Relation of the carrier of a1,the carrier of F_Complex;
  cluster a2 * a3 -> Function-like quasi_total cmplxhomogeneous;
end;

:: HERMITAN:funcreg 5
registration
  let a1 be non empty VectSpStr over F_Complex;
  let a2, a3 be Function-like quasi_total cmplxhomogeneous Relation of the carrier of a1,the carrier of F_Complex;
  cluster a2 - a3 -> Function-like quasi_total cmplxhomogeneous;
end;

:: HERMITAN:funcnot 1 => HERMITAN:func 1
definition
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of F_Complex;
  func A2 *' -> Function-like quasi_total Relation of the carrier of a1,the carrier of F_Complex means
    for b1 being Element of the carrier of a1 holds
       it . b1 = (a2 . b1) *';
end;

:: HERMITAN:def 2
theorem
for b1 being non empty VectSpStr over F_Complex
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of F_Complex holds
   b3 = b2 *'
iff
   for b4 being Element of the carrier of b1 holds
      b3 . b4 = (b2 . b4) *';

:: HERMITAN:funcreg 6
registration
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total additive Relation of the carrier of a1,the carrier of F_Complex;
  cluster a2 *' -> Function-like quasi_total additive;
end;

:: HERMITAN:funcreg 7
registration
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total homogeneous Relation of the carrier of a1,the carrier of F_Complex;
  cluster a2 *' -> Function-like quasi_total cmplxhomogeneous;
end;

:: HERMITAN:funcreg 8
registration
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total cmplxhomogeneous Relation of the carrier of a1,the carrier of F_Complex;
  cluster a2 *' -> Function-like quasi_total homogeneous;
end;

:: HERMITAN:funcreg 9
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a2 be Function-like non constant quasi_total Relation of the carrier of a1,the carrier of F_Complex;
  cluster a2 *' -> Function-like non constant quasi_total;
end;

:: HERMITAN:exreg 2
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  cluster Relation-like Function-like non constant quasi_total non trivial additive cmplxhomogeneous Relation of the carrier of a1,the carrier of F_Complex;
end;

:: HERMITAN:th 19
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of F_Complex holds
   b2 *' *' = b2;

:: HERMITAN:th 20
theorem
for b1 being non empty VectSpStr over F_Complex holds
   (0Functional b1) *' = 0Functional b1;

:: HERMITAN:th 21
theorem
for b1 being non empty VectSpStr over F_Complex
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of F_Complex holds
(b2 + b3) *' = b2 *' + (b3 *');

:: HERMITAN:th 22
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of F_Complex holds
   (- b2) *' = - (b2 *');

:: HERMITAN:th 23
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of F_Complex
for b3 being Element of the carrier of F_Complex holds
   (b3 * b2) *' = b3 *' * (b2 *');

:: HERMITAN:th 24
theorem
for b1 being non empty VectSpStr over F_Complex
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of F_Complex holds
(b2 - b3) *' = b2 *' - (b3 *');

:: HERMITAN:th 25
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of F_Complex
for b3 being Element of the carrier of b1 holds
      b2 . b3 = 0. F_Complex
   iff
      b2 *' . b3 = 0. F_Complex;

:: HERMITAN:th 26
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of F_Complex holds
   ker b2 = ker (b2 *');

:: HERMITAN:th 27
theorem
for b1 being non empty right_complementable add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additive cmplxhomogeneous Relation of the carrier of b1,the carrier of F_Complex holds
   ker b2 is linearly-closed(F_Complex, b1);

:: HERMITAN:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Subspace of b1
for b3 being Function-like quasi_total additive cmplxhomogeneous Relation of the carrier of b1,the carrier of F_Complex
      st the carrier of b2 c= ker (b3 *')
   holds QFunctional(b3,b2) is cmplxhomogeneous(VectQuot(b1,b2));

:: HERMITAN:funcnot 2 => HERMITAN:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a2 be Function-like quasi_total additive cmplxhomogeneous Relation of the carrier of a1,the carrier of F_Complex;
  func QcFunctional A2 -> Function-like quasi_total additive cmplxhomogeneous Relation of the carrier of VectQuot(a1,Ker (a2 *')),the carrier of F_Complex equals
    QFunctional(a2,Ker (a2 *'));
end;

:: HERMITAN:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additive cmplxhomogeneous Relation of the carrier of b1,the carrier of F_Complex holds
   QcFunctional b2 = QFunctional(b2,Ker (b2 *'));

:: HERMITAN:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additive cmplxhomogeneous Relation of the carrier of b1,the carrier of F_Complex
for b3 being Element of the carrier of VectQuot(b1,Ker (b2 *'))
for b4 being Element of the carrier of b1
      st b3 = b4 + Ker (b2 *')
   holds (QcFunctional b2) . b3 = b2 . b4;

:: HERMITAN:funcreg 10
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a2 be Function-like non constant quasi_total additive cmplxhomogeneous Relation of the carrier of a1,the carrier of F_Complex;
  cluster QcFunctional a2 -> Function-like non constant quasi_total additive cmplxhomogeneous;
end;

:: HERMITAN:funcreg 11
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a2 be Function-like quasi_total additive cmplxhomogeneous Relation of the carrier of a1,the carrier of F_Complex;
  cluster QcFunctional a2 -> Function-like quasi_total additive non degenerated cmplxhomogeneous;
end;

:: HERMITAN:attrnot 2 => HERMITAN:attr 2
definition
  let a1, a2 be non empty VectSpStr over F_Complex;
  let a3 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  attr a3 is cmplxhomogeneousFAF means
    for b1 being Element of the carrier of a1 holds
       FunctionalFAF(a3,b1) is cmplxhomogeneous(a2);
end;

:: HERMITAN:dfs 4
definiens
  let a1, a2 be non empty VectSpStr over F_Complex;
  let a3 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
To prove
     a3 is cmplxhomogeneousFAF
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       FunctionalFAF(a3,b1) is cmplxhomogeneous(a2);

:: HERMITAN:def 4
theorem
for b1, b2 being non empty VectSpStr over F_Complex
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
      b3 is cmplxhomogeneousFAF(b1, b2)
   iff
      for b4 being Element of the carrier of b1 holds
         FunctionalFAF(b3,b4) is cmplxhomogeneous(b2);

:: HERMITAN:th 30
theorem
for b1, b2 being non empty VectSpStr over F_Complex
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Element of the carrier of F_Complex
for b6 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex
      st b6 is cmplxhomogeneousFAF(b1, b2)
   holds b6 .(b3,b5 * b4) = b5 *' * (b6 .(b3,b4));

:: HERMITAN:attrnot 3 => HERMITAN:attr 3
definition
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
  attr a2 is hermitan means
    for b1, b2 being Element of the carrier of a1 holds
    a2 .(b1,b2) = (a2 .(b2,b1)) *';
end;

:: HERMITAN:dfs 5
definiens
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
To prove
     a2 is hermitan
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1 holds
    a2 .(b1,b2) = (a2 .(b2,b1)) *';

:: HERMITAN:def 5
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex holds
      b2 is hermitan(b1)
   iff
      for b3, b4 being Element of the carrier of b1 holds
      b2 .(b3,b4) = (b2 .(b4,b3)) *';

:: HERMITAN:attrnot 4 => HERMITAN:attr 4
definition
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
  attr a2 is diagRvalued means
    for b1 being Element of the carrier of a1 holds
       Im (a2 .(b1,b1)) = 0;
end;

:: HERMITAN:dfs 6
definiens
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
To prove
     a2 is diagRvalued
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       Im (a2 .(b1,b1)) = 0;

:: HERMITAN:def 6
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex holds
      b2 is diagRvalued(b1)
   iff
      for b3 being Element of the carrier of b1 holds
         Im (b2 .(b3,b3)) = 0;

:: HERMITAN:attrnot 5 => HERMITAN:attr 5
definition
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
  attr a2 is diagReR+0valued means
    for b1 being Element of the carrier of a1 holds
       0 <= Re (a2 .(b1,b1));
end;

:: HERMITAN:dfs 7
definiens
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
To prove
     a2 is diagReR+0valued
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       0 <= Re (a2 .(b1,b1));

:: HERMITAN:def 7
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex holds
      b2 is diagReR+0valued(b1)
   iff
      for b3 being Element of the carrier of b1 holds
         0 <= Re (b2 .(b3,b3));

:: HERMITAN:funcreg 12
registration
  let a1, a2 be non empty VectSpStr over F_Complex;
  cluster NulForm(a1,a2) -> Function-like quasi_total cmplxhomogeneousFAF;
end;

:: HERMITAN:funcreg 13
registration
  let a1 be non empty VectSpStr over F_Complex;
  cluster NulForm(a1,a1) -> Function-like quasi_total hermitan;
end;

:: HERMITAN:funcreg 14
registration
  let a1 be non empty VectSpStr over F_Complex;
  cluster NulForm(a1,a1) -> Function-like quasi_total diagReR+0valued;
end;

:: HERMITAN:condreg 2
registration
  let a1 be non empty VectSpStr over F_Complex;
  cluster Function-like quasi_total hermitan -> diagRvalued (Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex);
end;

:: HERMITAN:exreg 3
registration
  let a1 be non empty VectSpStr over F_Complex;
  cluster Relation-like Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF hermitan diagRvalued diagReR+0valued Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
end;

:: HERMITAN:exreg 4
registration
  let a1, a2 be non empty VectSpStr over F_Complex;
  cluster Relation-like Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
end;

:: HERMITAN:modenot 2
definition
  let a1, a2 be non empty VectSpStr over F_Complex;
  mode sesquilinear-Form of a1,a2 is Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
end;

:: HERMITAN:condreg 3
registration
  let a1 be non empty VectSpStr over F_Complex;
  cluster Function-like quasi_total additiveFAF hermitan -> additiveSAF (Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex);
end;

:: HERMITAN:condreg 4
registration
  let a1 be non empty VectSpStr over F_Complex;
  cluster Function-like quasi_total additiveSAF hermitan -> additiveFAF (Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex);
end;

:: HERMITAN:condreg 5
registration
  let a1 be non empty VectSpStr over F_Complex;
  cluster Function-like quasi_total homogeneousSAF hermitan -> cmplxhomogeneousFAF (Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex);
end;

:: HERMITAN:condreg 6
registration
  let a1 be non empty VectSpStr over F_Complex;
  cluster Function-like quasi_total cmplxhomogeneousFAF hermitan -> homogeneousSAF (Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex);
end;

:: HERMITAN:modenot 3
definition
  let a1 be non empty VectSpStr over F_Complex;
  mode hermitan-Form of a1 is Function-like quasi_total additiveSAF homogeneousSAF hermitan Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
end;

:: HERMITAN:funcreg 15
registration
  let a1, a2 be non empty VectSpStr over F_Complex;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of F_Complex;
  let a4 be Function-like quasi_total cmplxhomogeneous Relation of the carrier of a2,the carrier of F_Complex;
  cluster FormFunctional(a3,a4) -> Function-like quasi_total cmplxhomogeneousFAF;
end;

:: HERMITAN:funcreg 16
registration
  let a1, a2 be non empty VectSpStr over F_Complex;
  let a3 be Function-like quasi_total cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  let a4 be Element of the carrier of a1;
  cluster FunctionalFAF(a3,a4) -> Function-like quasi_total cmplxhomogeneous;
end;

:: HERMITAN:funcreg 17
registration
  let a1, a2 be non empty VectSpStr over F_Complex;
  let a3, a4 be Function-like quasi_total cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  cluster a3 + a4 -> Function-like quasi_total cmplxhomogeneousFAF;
end;

:: HERMITAN:funcreg 18
registration
  let a1, a2 be non empty VectSpStr over F_Complex;
  let a3 be Function-like quasi_total cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  let a4 be Element of the carrier of F_Complex;
  cluster a4 * a3 -> Function-like quasi_total cmplxhomogeneousFAF;
end;

:: HERMITAN:funcreg 19
registration
  let a1, a2 be non empty VectSpStr over F_Complex;
  let a3 be Function-like quasi_total cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  cluster - a3 -> Function-like quasi_total cmplxhomogeneousFAF;
end;

:: HERMITAN:funcreg 20
registration
  let a1, a2 be non empty VectSpStr over F_Complex;
  let a3, a4 be Function-like quasi_total cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  cluster a3 - a4 -> Function-like quasi_total cmplxhomogeneousFAF;
end;

:: HERMITAN:exreg 5
registration
  let a1, a2 be non empty non trivial right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  cluster Relation-like Function-like non constant quasi_total non trivial additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
end;

:: HERMITAN:funcnot 3 => HERMITAN:func 3
definition
  let a1, a2 be non empty VectSpStr over F_Complex;
  let a3 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  func A3 *' -> Function-like quasi_total Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex means
    for b1 being Element of the carrier of a1
    for b2 being Element of the carrier of a2 holds
       it .(b1,b2) = (a3 .(b1,b2)) *';
end;

:: HERMITAN:def 8
theorem
for b1, b2 being non empty VectSpStr over F_Complex
for b3, b4 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
   b4 = b3 *'
iff
   for b5 being Element of the carrier of b1
   for b6 being Element of the carrier of b2 holds
      b4 .(b5,b6) = (b3 .(b5,b6)) *';

:: HERMITAN:funcreg 21
registration
  let a1, a2 be non empty VectSpStr over F_Complex;
  let a3 be Function-like quasi_total additiveFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  cluster a3 *' -> Function-like quasi_total additiveFAF;
end;

:: HERMITAN:funcreg 22
registration
  let a1, a2 be non empty VectSpStr over F_Complex;
  let a3 be Function-like quasi_total additiveSAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  cluster a3 *' -> Function-like quasi_total additiveSAF;
end;

:: HERMITAN:funcreg 23
registration
  let a1, a2 be non empty VectSpStr over F_Complex;
  let a3 be Function-like quasi_total homogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  cluster a3 *' -> Function-like quasi_total cmplxhomogeneousFAF;
end;

:: HERMITAN:funcreg 24
registration
  let a1, a2 be non empty VectSpStr over F_Complex;
  let a3 be Function-like quasi_total cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  cluster a3 *' -> Function-like quasi_total homogeneousFAF;
end;

:: HERMITAN:funcreg 25
registration
  let a1, a2 be non empty non trivial right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a3 be Function-like non constant quasi_total Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  cluster a3 *' -> Function-like non constant quasi_total;
end;

:: HERMITAN:th 31
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of F_Complex
for b3 being Element of the carrier of b1 holds
   (FormFunctional(b2,b2 *')) .(b3,b3) = |.b2 . b3.| ^2;

:: HERMITAN:funcreg 26
registration
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of F_Complex;
  cluster FormFunctional(a2,a2 *') -> Function-like quasi_total hermitan diagRvalued diagReR+0valued;
end;

:: HERMITAN:exreg 6
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  cluster Relation-like Function-like non constant quasi_total non trivial additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF hermitan diagRvalued diagReR+0valued Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
end;

:: HERMITAN:th 32
theorem
for b1, b2 being non empty VectSpStr over F_Complex
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
   b3 *' *' = b3;

:: HERMITAN:th 33
theorem
for b1, b2 being non empty VectSpStr over F_Complex holds
(NulForm(b1,b2)) *' = NulForm(b1,b2);

:: HERMITAN:th 34
theorem
for b1, b2 being non empty VectSpStr over F_Complex
for b3, b4 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
(b3 + b4) *' = b3 *' + (b4 *');

:: HERMITAN:th 35
theorem
for b1, b2 being non empty VectSpStr over F_Complex
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
   (- b3) *' = - (b3 *');

:: HERMITAN:th 36
theorem
for b1, b2 being non empty VectSpStr over F_Complex
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex
for b4 being Element of the carrier of F_Complex holds
   (b4 * b3) *' = b4 *' * (b3 *');

:: HERMITAN:th 37
theorem
for b1, b2 being non empty VectSpStr over F_Complex
for b3, b4 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
(b3 - b4) *' = b3 *' - (b4 *');

:: HERMITAN:th 38
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b3 being Element of the carrier of b1
for b4, b5 being Element of the carrier of b2
for b6 being Function-like quasi_total additiveFAF cmplxhomogeneousFAF Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
   b6 .(b3,b4 - b5) = (b6 .(b3,b4)) - (b6 .(b3,b5));

:: HERMITAN:th 39
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
for b7 being Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
   b7 .(b3 - b4,b5 - b6) = ((b7 .(b3,b5)) - (b7 .(b3,b6))) - ((b7 .(b4,b5)) - (b7 .(b4,b6)));

:: HERMITAN:th 40
theorem
for b1, b2 being non empty right_complementable add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
for b7, b8 being Element of the carrier of F_Complex
for b9 being Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
   b9 .(b3 + (b7 * b4),b5 + (b8 * b6)) = ((b9 .(b3,b5)) + (b8 *' * (b9 .(b3,b6)))) + ((b7 * (b9 .(b4,b5))) + (b7 * (b8 *' * (b9 .(b4,b6)))));

:: HERMITAN:th 41
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
for b7, b8 being Element of the carrier of F_Complex
for b9 being Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
   b9 .(b3 - (b7 * b4),b5 - (b8 * b6)) = ((b9 .(b3,b5)) - (b8 *' * (b9 .(b3,b6)))) - ((b7 * (b9 .(b4,b5))) - (b7 * (b8 *' * (b9 .(b4,b6)))));

:: HERMITAN:th 42
theorem
for b1 being non empty right_complementable add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total cmplxhomogeneousFAF Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex
for b3 being Element of the carrier of b1 holds
   b2 .(b3,0. b1) = 0. F_Complex;

:: HERMITAN:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total additiveSAF homogeneousSAF hermitan Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex holds
   (((b4 .(b2,b3)) + (b4 .(b2,b3))) + (b4 .(b2,b3))) + (b4 .(b2,b3)) = (((b4 .(b2 + b3,b2 + b3)) - (b4 .(b2 - b3,b2 - b3))) + (i_FC * (b4 .(b2 + (i_FC * b3),b2 + (i_FC * b3))))) - (i_FC * (b4 .(b2 - (i_FC * b3),b2 - (i_FC * b3))));

:: HERMITAN:funcnot 4 => HERMITAN:func 4
definition
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
  let a3 be Element of the carrier of a1;
  func signnorm(A2,A3) -> real set equals
    Re (a2 .(a3,a3));
end;

:: HERMITAN:def 9
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex
for b3 being Element of the carrier of b1 holds
   signnorm(b2,b3) = Re (b2 .(b3,b3));

:: HERMITAN:th 44
theorem
for b1 being non empty right_complementable add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total diagRvalued diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex
for b3 being Element of the carrier of b1 holds
   |.b2 .(b3,b3).| = Re (b2 .(b3,b3)) &
    signnorm(b2,b3) = |.b2 .(b3,b3).|;

:: HERMITAN:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex
for b5 being Element of REAL
for b6 being Element of the carrier of F_Complex
      st |.b6.| = 1 &
         Re (b6 * (b4 .(b3,b2))) = |.b4 .(b3,b2).| &
         Im (b6 * (b4 .(b3,b2))) = 0
   holds b4 .(b2 - (([**b5,0**] * b6) * b3),b2 - (([**b5,0**] * b6) * b3)) = (((b4 .(b2,b2)) - ([**b5,0**] * (b6 * (b4 .(b3,b2))))) - ([**b5,0**] * (b6 *' * (b4 .(b2,b3))))) + ([**b5 ^2,0**] * (b4 .(b3,b3)));

:: HERMITAN:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex
for b5 being Element of REAL
for b6 being Element of the carrier of F_Complex
      st |.b6.| = 1 &
         Re (b6 * (b4 .(b3,b2))) = |.b4 .(b3,b2).| &
         Im (b6 * (b4 .(b3,b2))) = 0
   holds Re (b4 .(b2 - (([**b5,0**] * b6) * b3),b2 - (([**b5,0**] * b6) * b3))) = ((signnorm(b4,b2)) - ((2 * |.b4 .(b3,b2).|) * b5)) + ((signnorm(b4,b3)) * (b5 ^2)) &
    0 <= ((signnorm(b4,b2)) - ((2 * |.b4 .(b3,b2).|) * b5)) + ((signnorm(b4,b3)) * (b5 ^2));

:: HERMITAN:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex
      st signnorm(b4,b3) = 0
   holds |.b4 .(b3,b2).| = 0;

:: HERMITAN:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex holds
   |.b4 .(b2,b3).| ^2 <= (signnorm(b4,b2)) * signnorm(b4,b3);

:: HERMITAN:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex
for b3, b4 being Element of the carrier of b1 holds
|.b2 .(b3,b4).| ^2 <= |.b2 .(b3,b3).| * |.b2 .(b4,b4).|;

:: HERMITAN:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex
for b3, b4 being Element of the carrier of b1 holds
signnorm(b2,b3 + b4) <= ((sqrt signnorm(b2,b3)) + sqrt signnorm(b2,b4)) ^2;

:: HERMITAN:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex
for b3, b4 being Element of the carrier of b1 holds
|.b2 .(b3 + b4,b3 + b4).| <= ((sqrt |.b2 .(b3,b3).|) + sqrt |.b2 .(b4,b4).|) ^2;

:: HERMITAN:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additiveSAF homogeneousSAF hermitan Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex
for b3, b4 being Element of the carrier of b1 holds
(signnorm(b2,b3 + b4)) + signnorm(b2,b3 - b4) = (2 * signnorm(b2,b3)) + (2 * signnorm(b2,b4));

:: HERMITAN:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex
for b3, b4 being Element of the carrier of b1 holds
|.b2 .(b3 + b4,b3 + b4).| + |.b2 .(b3 - b4,b3 - b4).| = (2 * |.b2 .(b3,b3).|) + (2 * |.b2 .(b4,b4).|);

:: HERMITAN:funcnot 5 => HERMITAN:func 5
definition
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
  func quasinorm A2 -> Function-like quasi_total Relation of the carrier of a1,REAL means
    for b1 being Element of the carrier of a1 holds
       it . b1 = sqrt signnorm(a2,b1);
end;

:: HERMITAN:def 10
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex
for b3 being Function-like quasi_total Relation of the carrier of b1,REAL holds
      b3 = quasinorm b2
   iff
      for b4 being Element of the carrier of b1 holds
         b3 . b4 = sqrt signnorm(b2,b4);

:: HERMITAN:funcnot 6 => HERMITAN:func 6
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a2 be Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
  redefine func quasinorm a2 -> Function-like quasi_total subadditive homogeneous Relation of the carrier of a1,REAL;
end;

:: HERMITAN:funcreg 27
registration
  let a1 be non empty right_complementable add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a2 be Function-like quasi_total cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
  cluster diagker a2 -> non empty;
end;

:: HERMITAN:th 54
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex holds
   diagker b2 is linearly-closed(F_Complex, b1);

:: HERMITAN:th 55
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex holds
   diagker b2 = leftker b2;

:: HERMITAN:th 56
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex holds
   diagker b2 = rightker b2;

:: HERMITAN:th 57
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex holds
   diagker b2 = diagker (b2 *');

:: HERMITAN:th 58
theorem
for b1, b2 being non empty VectSpStr over F_Complex
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
   leftker b3 = leftker (b3 *') & rightker b3 = rightker (b3 *');

:: HERMITAN:th 59
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex holds
   LKer b2 = RKer (b2 *');

:: HERMITAN:th 60
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total diagRvalued diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex
for b3 being Element of the carrier of b1
      st Re (b2 .(b3,b3)) = 0
   holds b2 .(b3,b3) = 0. F_Complex;

:: HERMITAN:th 61
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex
for b3 being Element of the carrier of b1
      st Re (b2 .(b3,b3)) = 0 &
         (b2 is degenerated-on-left(F_Complex, b1, b1) implies b2 is degenerated-on-right(not F_Complex, b1, b1))
   holds b3 = 0. b1;

:: HERMITAN:funcnot 7 => HERMITAN:func 7
definition
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a3 be Function-like quasi_total additiveFAF cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  func RQ*Form A3 -> Function-like quasi_total additiveFAF cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of VectQuot(a2,RKer (a3 *')):],the carrier of F_Complex equals
    (RQForm (a3 *')) *';
end;

:: HERMITAN:def 11
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b3 being Function-like quasi_total additiveFAF cmplxhomogeneousFAF Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
   RQ*Form b3 = (RQForm (b3 *')) *';

:: HERMITAN:th 62
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b3 being Function-like quasi_total additiveFAF cmplxhomogeneousFAF Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex
for b4 being Element of the carrier of b1
for b5 being Element of the carrier of b2 holds
   (RQ*Form b3) .(b4,b5 + RKer (b3 *')) = b3 .(b4,b5);

:: HERMITAN:funcreg 28
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a3 be Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  cluster LQForm a3 -> Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF;
end;

:: HERMITAN:funcreg 29
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a3 be Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  cluster RQ*Form a3 -> Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF;
end;

:: HERMITAN:funcnot 8 => HERMITAN:func 8
definition
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a3 be Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  func Q*Form A3 -> Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of VectQuot(a1,LKer a3),the carrier of VectQuot(a2,RKer (a3 *')):],the carrier of F_Complex means
    for b1 being Element of the carrier of VectQuot(a1,LKer a3)
    for b2 being Element of the carrier of VectQuot(a2,RKer (a3 *'))
    for b3 being Element of the carrier of a1
    for b4 being Element of the carrier of a2
          st b1 = b3 + LKer a3 & b2 = b4 + RKer (a3 *')
       holds it .(b1,b2) = a3 .(b3,b4);
end;

:: HERMITAN:def 12
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b3 being Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex
for b4 being Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of VectQuot(b1,LKer b3),the carrier of VectQuot(b2,RKer (b3 *')):],the carrier of F_Complex holds
      b4 = Q*Form b3
   iff
      for b5 being Element of the carrier of VectQuot(b1,LKer b3)
      for b6 being Element of the carrier of VectQuot(b2,RKer (b3 *'))
      for b7 being Element of the carrier of b1
      for b8 being Element of the carrier of b2
            st b5 = b7 + LKer b3 & b6 = b8 + RKer (b3 *')
         holds b4 .(b5,b6) = b3 .(b7,b8);

:: HERMITAN:funcreg 30
registration
  let a1, a2 be non empty non trivial right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a3 be Function-like non constant quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  cluster Q*Form a3 -> Function-like non constant quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF;
end;

:: HERMITAN:funcreg 31
registration
  let a1 be non empty right_zeroed VectSpStr over F_Complex;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a3 be Function-like quasi_total additiveFAF cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  cluster RQ*Form a3 -> Function-like quasi_total additiveFAF non degenerated-on-right cmplxhomogeneousFAF;
end;

:: HERMITAN:th 63
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b3 being Function-like quasi_total additiveFAF cmplxhomogeneousFAF Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
   leftker b3 = leftker RQ*Form b3;

:: HERMITAN:th 64
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b3 being Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
   RKer (b3 *') = RKer ((LQForm b3) *');

:: HERMITAN:th 65
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b3 being Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
   LKer b3 = LKer RQ*Form b3;

:: HERMITAN:th 66
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b3 being Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
   Q*Form b3 = RQ*Form LQForm b3 & Q*Form b3 = LQForm RQ*Form b3;

:: HERMITAN:th 67
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b3 being Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of b1,the carrier of b2:],the carrier of F_Complex holds
   leftker Q*Form b3 = leftker RQ*Form LQForm b3 &
    rightker Q*Form b3 = rightker RQ*Form LQForm b3 &
    leftker Q*Form b3 = leftker LQForm RQ*Form b3 &
    rightker Q*Form b3 = rightker LQForm RQ*Form b3;

:: HERMITAN:funcreg 32
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a3 be Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  cluster RQ*Form LQForm a3 -> Function-like quasi_total additiveFAF non degenerated-on-left non degenerated-on-right cmplxhomogeneousFAF;
end;

:: HERMITAN:funcreg 33
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a3 be Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  cluster LQForm RQ*Form a3 -> Function-like quasi_total additiveSAF homogeneousSAF non degenerated-on-left non degenerated-on-right;
end;

:: HERMITAN:funcreg 34
registration
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a3 be Function-like quasi_total additiveFAF additiveSAF homogeneousSAF cmplxhomogeneousFAF Relation of [:the carrier of a1,the carrier of a2:],the carrier of F_Complex;
  cluster Q*Form a3 -> Function-like quasi_total additiveFAF additiveSAF homogeneousSAF non degenerated-on-left non degenerated-on-right cmplxhomogeneousFAF;
end;

:: HERMITAN:attrnot 6 => HERMITAN:attr 6
definition
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
  attr a2 is positivediagvalued means
    for b1 being Element of the carrier of a1
          st b1 <> 0. a1
       holds 0 < Re (a2 .(b1,b1));
end;

:: HERMITAN:dfs 13
definiens
  let a1 be non empty VectSpStr over F_Complex;
  let a2 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
To prove
     a2 is positivediagvalued
it is sufficient to prove
  thus for b1 being Element of the carrier of a1
          st b1 <> 0. a1
       holds 0 < Re (a2 .(b1,b1));

:: HERMITAN:def 13
theorem
for b1 being non empty VectSpStr over F_Complex
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex holds
      b2 is positivediagvalued(b1)
   iff
      for b3 being Element of the carrier of b1
            st b3 <> 0. b1
         holds 0 < Re (b2 .(b3,b3));

:: HERMITAN:condreg 7
registration
  let a1 be non empty right_zeroed VectSpStr over F_Complex;
  cluster Function-like quasi_total additiveSAF positivediagvalued -> diagReR+0valued (Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex);
end;

:: HERMITAN:condreg 8
registration
  let a1 be non empty right_zeroed VectSpStr over F_Complex;
  cluster Function-like quasi_total additiveFAF positivediagvalued -> diagReR+0valued (Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex);
end;

:: HERMITAN:funcnot 9 => HERMITAN:func 9
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a2 be Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
  func ScalarForm A2 -> Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of VectQuot(a1,LKer a2),the carrier of VectQuot(a1,LKer a2):],the carrier of F_Complex equals
    Q*Form a2;
end;

:: HERMITAN:def 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex holds
   ScalarForm b2 = Q*Form b2;

:: HERMITAN:th 68
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex
for b3, b4 being Element of the carrier of VectQuot(b1,LKer b2)
for b5, b6 being Element of the carrier of b1
      st b3 = b5 + LKer b2 & b4 = b6 + LKer b2
   holds (ScalarForm b2) .(b3,b4) = b2 .(b5,b6);

:: HERMITAN:th 69
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex holds
   leftker ScalarForm b2 = leftker Q*Form b2;

:: HERMITAN:th 70
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex
for b2 being Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of b1,the carrier of b1:],the carrier of F_Complex holds
   rightker ScalarForm b2 = rightker Q*Form b2;

:: HERMITAN:funcreg 35
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a2 be Function-like quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
  cluster ScalarForm a2 -> Function-like quasi_total additiveSAF homogeneousSAF non degenerated-on-left non degenerated-on-right hermitan diagReR+0valued positivediagvalued;
end;

:: HERMITAN:funcreg 36
registration
  let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over F_Complex;
  let a2 be Function-like non constant quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued Relation of [:the carrier of a1,the carrier of a1:],the carrier of F_Complex;
  cluster ScalarForm a2 -> Function-like non constant quasi_total additiveSAF homogeneousSAF hermitan diagReR+0valued;
end;