Article HURWITZ, MML version 4.99.1005

:: HURWITZ:th 1
theorem
for b1 being non empty right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b2 being Element of the carrier of b1
      st b2 <> 0. b1
   holds - (b2 ") = (- b2) ";

:: HURWITZ:th 2
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b2 being Element of NAT holds
   (power b1) .(- 1_ b1,b2) <> 0. b1;

:: HURWITZ:th 3
theorem
for b1 being non empty associative well-unital multLoopStr
for b2 being Element of the carrier of b1
for b3, b4 being Element of NAT holds
((power b1) .(b2,b3)) * ((power b1) .(b2,b4)) = (power b1) .(b2,b3 + b4);

:: HURWITZ:th 4
theorem
for b1 being non empty right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b2 being Element of NAT holds
   (power b1) .(- 1_ b1,2 * b2) = 1_ b1 &
    (power b1) .(- 1_ b1,(2 * b2) + 1) = - 1_ b1;

:: HURWITZ:th 5
theorem
for b1 being Element of the carrier of F_Complex
for b2 being Element of NAT holds
   ((power F_Complex) .(b1,b2)) *' = (power F_Complex) .(b1 *',b2);

:: HURWITZ:th 6
theorem
for b1, b2 being FinSequence of the carrier of F_Complex
      st len b2 = len b1 &
         (for b3 being Element of NAT
               st b3 in dom b2
            holds b2 /. b3 = (b1 /. b3) *')
   holds Sum b2 = (Sum b1) *';

:: HURWITZ:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being FinSequence of the carrier of b1
      st len b2 = len b3 &
         (for b4 being Element of NAT
               st b4 in dom b2
            holds b2 /. b4 = - (b3 /. b4))
   holds Sum b2 = - Sum b3;

:: HURWITZ:th 8
theorem
for b1 being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for b2 being Element of the carrier of b1
for b3 being FinSequence of the carrier of b1 holds
   b2 * Sum b3 = Sum (b2 * b3);

:: HURWITZ:th 9
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr holds
   - 0_. b1 = 0_. b1;

:: HURWITZ:th 10
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
   - - b2 = b2;

:: HURWITZ:th 11
theorem
for b1 being non empty right_complementable distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
- (b2 + b3) = (- b2) + - b3;

:: HURWITZ:th 12
theorem
for b1 being non empty right_complementable distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
- (b2 *' b3) = (- b2) *' b3 & - (b2 *' b3) = b2 *' - b3;

:: HURWITZ:funcnot 1 => HURWITZ:func 1
definition
  let a1 be non empty right_complementable distributive add-associative right_zeroed doubleLoopStr;
  let a2 be FinSequence of the carrier of Polynom-Ring a1;
  let a3 be Element of NAT;
  func Coeff(A2,A3) -> FinSequence of the carrier of a1 means
    len it = len a2 &
     (for b1 being Element of NAT
           st b1 in dom it
        holds ex b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of a1 st
           b2 = a2 . b1 & it . b1 = b2 . a3);
end;

:: HURWITZ:def 1
theorem
for b1 being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for b2 being FinSequence of the carrier of Polynom-Ring b1
for b3 being Element of NAT
for b4 being FinSequence of the carrier of b1 holds
      b4 = Coeff(b2,b3)
   iff
      len b4 = len b2 &
       (for b5 being Element of NAT
             st b5 in dom b4
          holds ex b6 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 st
             b6 = b2 . b5 & b4 . b5 = b6 . b3);

:: HURWITZ:th 13
theorem
for b1 being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3 being FinSequence of the carrier of Polynom-Ring b1
   st b2 = Sum b3
for b4 being Element of NAT holds
   b2 . b4 = Sum Coeff(b3,b4);

:: HURWITZ:th 14
theorem
for b1 being non empty associative doubleLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3, b4 being Element of the carrier of b1 holds
b3 * (b4 * b2) = (b3 * b4) * b2;

:: HURWITZ:th 15
theorem
for b1 being non empty right_complementable left-distributive add-associative right_zeroed doubleLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1 holds
   - (b3 * b2) = (- b3) * b2;

:: HURWITZ:th 16
theorem
for b1 being non empty right_complementable right-distributive add-associative right_zeroed doubleLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1 holds
   - (b3 * b2) = b3 * - b2;

:: HURWITZ:th 17
theorem
for b1 being non empty left-distributive doubleLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3, b4 being Element of the carrier of b1 holds
(b3 + b4) * b2 = (b3 * b2) + (b4 * b2);

:: HURWITZ:th 18
theorem
for b1 being non empty right-distributive doubleLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b4 being Element of the carrier of b1 holds
   b4 * (b2 + b3) = (b4 * b2) + (b4 * b3);

:: HURWITZ:th 19
theorem
for b1 being non empty right_complementable associative commutative distributive add-associative right_zeroed doubleLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b4 being Element of the carrier of b1 holds
   b2 *' (b4 * b3) = b4 * (b2 *' b3);

:: HURWITZ:funcnot 2 => HURWITZ:func 2
definition
  let a1 be non empty ZeroStr;
  let a2 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
  func degree A2 -> integer set equals
    (len a2) - 1;
end;

:: HURWITZ:def 2
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
   degree b2 = (len b2) - 1;

:: HURWITZ:funcnot 3 => HURWITZ:func 2
notation
  let a1 be non empty ZeroStr;
  let a2 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
  synonym deg a2 for degree a2;
end;

:: HURWITZ:th 20
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
      degree b2 = - 1
   iff
      b2 = 0_. b1;

:: HURWITZ:th 21
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
      st degree b2 <> degree b3
   holds degree (b2 + b3) = max(degree b2,degree b3);

:: HURWITZ:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
degree (b2 + b3) <= max(degree b2,degree b3);

:: HURWITZ:th 23
theorem
for b1 being non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
      st b2 <> 0_. b1 & b3 <> 0_. b1
   holds degree (b2 *' b3) = (degree b2) + degree b3;

:: HURWITZ:th 24
theorem
for b1 being non empty right_complementable unital add-associative right_zeroed doubleLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
      st degree b2 = 0
   holds b2 is not with_roots(b1);

:: HURWITZ:funcnot 4 => HURWITZ:func 3
definition
  let a1 be non empty unital doubleLoopStr;
  let a2 be Element of the carrier of a1;
  let a3 be Element of NAT;
  func rpoly(A3,A2) -> Function-like quasi_total finite-Support Relation of NAT,the carrier of a1 equals
    (0_. a1) +* ((0,a3)-->(- ((power a1) .(a2,a3)),1_ a1));
end;

:: HURWITZ:def 3
theorem
for b1 being non empty unital doubleLoopStr
for b2 being Element of the carrier of b1
for b3 being Element of NAT holds
   rpoly(b3,b2) = (0_. b1) +* ((0,b3)-->(- ((power b1) .(b2,b3)),1_ b1));

:: HURWITZ:th 25
theorem
for b1 being non empty unital doubleLoopStr
for b2 being Element of the carrier of b1
for b3 being Element of NAT
      st b3 <> 0
   holds (rpoly(b3,b2)) . 0 = - ((power b1) .(b2,b3)) &
    (rpoly(b3,b2)) . b3 = 1_ b1;

:: HURWITZ:th 26
theorem
for b1 being non empty unital doubleLoopStr
for b2 being Element of the carrier of b1
for b3, b4 being Element of NAT
      st b3 <> 0 & b3 <> b4
   holds (rpoly(b4,b2)) . b3 = 0. b1;

:: HURWITZ:th 27
theorem
for b1 being non empty non degenerated well-unital doubleLoopStr
for b2 being Element of the carrier of b1
for b3 being Element of NAT holds
   degree rpoly(b3,b2) = b3;

:: HURWITZ:th 28
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
      degree b2 = 1
   iff
      ex b3, b4 being Element of the carrier of b1 st
         b3 <> 0. b1 & b2 = b3 * rpoly(1,b4);

:: HURWITZ:th 29
theorem
for b1 being non empty non degenerated right_complementable well-unital Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being Element of the carrier of b1 holds
eval(rpoly(1,b3),b2) = b2 - b3;

:: HURWITZ:th 30
theorem
for b1 being non empty non degenerated right_complementable well-unital Abelian add-associative right_zeroed doubleLoopStr
for b2 being Element of the carrier of b1 holds
   b2 is_a_root_of rpoly(1,b2);

:: HURWITZ:funcnot 5 => HURWITZ:func 4
definition
  let a1 be non empty well-unital doubleLoopStr;
  let a2 be Element of the carrier of a1;
  let a3 be natural set;
  func qpoly(A3,A2) -> Function-like quasi_total finite-Support Relation of NAT,the carrier of a1 means
    (for b1 being natural set
           st b1 < a3
        holds it . b1 = (power a1) .(a2,(a3 - b1) - 1)) &
     (for b1 being natural set
           st a3 <= b1
        holds it . b1 = 0. a1);
end;

:: HURWITZ:def 4
theorem
for b1 being non empty well-unital doubleLoopStr
for b2 being Element of the carrier of b1
for b3 being natural set
for b4 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
      b4 = qpoly(b3,b2)
   iff
      (for b5 being natural set
             st b5 < b3
          holds b4 . b5 = (power b1) .(b2,(b3 - b5) - 1)) &
       (for b5 being natural set
             st b3 <= b5
          holds b4 . b5 = 0. b1);

:: HURWITZ:th 31
theorem
for b1 being non empty non degenerated well-unital doubleLoopStr
for b2 being Element of the carrier of b1
for b3 being Element of NAT
      st 1 <= b3
   holds degree qpoly(b3,b2) = b3 - 1;

:: HURWITZ:th 32
theorem
for b1 being non empty right_complementable commutative left-distributive well-unital add-associative right_zeroed doubleLoopStr
for b2 being Element of the carrier of b1
for b3 being Element of NAT
      st 1 < b3
   holds (rpoly(1,b2)) *' qpoly(b3,b2) = rpoly(b3,b2);

:: HURWITZ:th 33
theorem
for b1 being non empty right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
      st b3 is_a_root_of b2
   holds ex b4 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 st
      b2 = (rpoly(1,b3)) *' b4;

:: HURWITZ:funcnot 6 => HURWITZ:func 5
definition
  let a1 be non empty right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2, a3 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
  assume a3 <> 0_. a1;
  func A2 div A3 -> Function-like quasi_total finite-Support Relation of NAT,the carrier of a1 means
    ex b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of a1 st
       a2 = (it *' a3) + b1 & degree b1 < degree a3;
end;

:: HURWITZ:def 5
theorem
for b1 being non empty right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
   st b3 <> 0_. b1
for b4 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
      b4 = b2 div b3
   iff
      ex b5 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 st
         b2 = (b4 *' b3) + b5 & degree b5 < degree b3;

:: HURWITZ:funcnot 7 => HURWITZ:func 6
definition
  let a1 be non empty right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2, a3 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
  func A2 mod A3 -> Function-like quasi_total finite-Support Relation of NAT,the carrier of a1 equals
    a2 - ((a2 div a3) *' a3);
end;

:: HURWITZ:def 6
theorem
for b1 being non empty right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
b2 mod b3 = b2 - ((b2 div b3) *' b3);

:: HURWITZ:prednot 1 => HURWITZ:pred 1
definition
  let a1 be non empty right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2, a3 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
  pred A3 divides A2 means
    a2 mod a3 = 0_. a1;
end;

:: HURWITZ:dfs 7
definiens
  let a1 be non empty right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2, a3 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
To prove
     a3 divides a2
it is sufficient to prove
  thus a2 mod a3 = 0_. a1;

:: HURWITZ:def 7
theorem
for b1 being non empty right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
   b3 divides b2
iff
   b2 mod b3 = 0_. b1;

:: HURWITZ:th 34
theorem
for b1 being non empty right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
      st b3 <> 0_. b1
   holds    b3 divides b2
   iff
      ex b4 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 st
         b4 *' b3 = b2;

:: HURWITZ:th 35
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
      st b3 is_a_root_of b2
   holds rpoly(1,b3) divides b2;

:: HURWITZ:th 36
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1
      st b2 <> 0_. b1 & b3 is_a_root_of b2
   holds degree (b2 div rpoly(1,b3)) = (degree b2) - 1;

:: HURWITZ:attrnot 1 => HURWITZ:attr 1
definition
  let a1 be Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex;
  attr a1 is Hurwitz means
    for b1 being Element of the carrier of F_Complex
          st b1 is_a_root_of a1
       holds Re b1 < 0;
end;

:: HURWITZ:dfs 8
definiens
  let a1 be Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex;
To prove
     a1 is Hurwitz
it is sufficient to prove
  thus for b1 being Element of the carrier of F_Complex
          st b1 is_a_root_of a1
       holds Re b1 < 0;

:: HURWITZ:def 8
theorem
for b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex holds
      b1 is Hurwitz
   iff
      for b2 being Element of the carrier of F_Complex
            st b2 is_a_root_of b1
         holds Re b2 < 0;

:: HURWITZ:th 37
theorem
0_. F_Complex is not Hurwitz;

:: HURWITZ:th 38
theorem
for b1 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds b1 * 1_. F_Complex is Hurwitz;

:: HURWITZ:th 39
theorem
for b1, b2 being Element of the carrier of F_Complex
      st b1 <> 0. F_Complex
   holds    b1 * rpoly(1,b2) is Hurwitz
   iff
      Re b2 < 0;

:: HURWITZ:th 40
theorem
for b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex
for b2 being Element of the carrier of F_Complex
      st b2 <> 0. F_Complex
   holds    b1 is Hurwitz
   iff
      b2 * b1 is Hurwitz;

:: HURWITZ:th 41
theorem
for b1, b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex holds
   b1 *' b2 is Hurwitz
iff
   b1 is Hurwitz & b2 is Hurwitz;

:: HURWITZ:funcnot 8 => HURWITZ:func 7
definition
  let a1 be Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex;
  func A1 *' -> Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex means
    for b1 being Element of NAT holds
       it . b1 = ((power F_Complex) .(- 1_ F_Complex,b1)) * ((a1 . b1) *');
end;

:: HURWITZ:def 9
theorem
for b1, b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex holds
   b2 = b1 *'
iff
   for b3 being Element of NAT holds
      b2 . b3 = ((power F_Complex) .(- 1_ F_Complex,b3)) * ((b1 . b3) *');

:: HURWITZ:th 42
theorem
for b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex holds
   degree (b1 *') = degree b1;

:: HURWITZ:th 43
theorem
for b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex holds
   b1 *' *' = b1;

:: HURWITZ:th 44
theorem
for b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex
for b2 being Element of the carrier of F_Complex holds
   (b2 * b1) *' = b2 *' * (b1 *');

:: HURWITZ:th 45
theorem
for b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex holds
   (- b1) *' = - (b1 *');

:: HURWITZ:th 46
theorem
for b1, b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex holds
(b1 + b2) *' = b1 *' + (b2 *');

:: HURWITZ:th 47
theorem
for b1, b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex holds
(b1 *' b2) *' = b1 *' *' (b2 *');

:: HURWITZ:th 48
theorem
for b1, b2 being Element of the carrier of F_Complex holds
eval((rpoly(1,b2)) *',b1) = (- b1) - (b2 *');

:: HURWITZ:th 49
theorem
for b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex
   st b1 is Hurwitz
for b2 being Element of the carrier of F_Complex
      st 0 <= Re b2
   holds 0 < |.eval(b1,b2).|;

:: HURWITZ:th 50
theorem
for b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex
   st 1 <= degree b1 & b1 is Hurwitz
for b2 being Element of the carrier of F_Complex holds
   (Re b2 < 0 implies |.eval(b1,b2).| < |.eval(b1 *',b2).|) &
    (0 < Re b2 implies |.eval(b1 *',b2).| < |.eval(b1,b2).|) &
    (Re b2 = 0 implies |.eval(b1,b2).| = |.eval(b1 *',b2).|);

:: HURWITZ:funcnot 9 => HURWITZ:func 8
definition
  let a1 be Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex;
  let a2 be Element of the carrier of F_Complex;
  func F*(A1,A2) -> Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex equals
    ((eval(a1 *',a2)) * a1) - ((eval(a1,a2)) * (a1 *'));
end;

:: HURWITZ:def 10
theorem
for b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex
for b2 being Element of the carrier of F_Complex holds
   F*(b1,b2) = ((eval(b1 *',b2)) * b1) - ((eval(b1,b2)) * (b1 *'));

:: HURWITZ:th 51
theorem
for b1, b2 being Element of the carrier of F_Complex
   st |.b2.| < |.b1.|
for b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex
      st 1 <= degree b3
   holds    b3 is Hurwitz
   iff
      (b1 * b3) - (b2 * (b3 *')) is Hurwitz;

:: HURWITZ:th 52
theorem
for b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex
   st 1 <= degree b1
for b2 being Element of the carrier of F_Complex
      st Re b2 < 0 & b1 is Hurwitz
   holds (F*(b1,b2)) div rpoly(1,b2) is Hurwitz;

:: HURWITZ:th 53
theorem
for b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex
      st 1 <= degree b1 &
         (ex b2 being Element of the carrier of F_Complex st
            Re b2 < 0 &
             |.eval(b1 *',b2).| <= |.eval(b1,b2).|)
   holds b1 is not Hurwitz;

:: HURWITZ:th 54
theorem
for b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F_Complex
   st 1 <= degree b1
for b2 being Element of the carrier of F_Complex
      st Re b2 < 0 &
         |.eval(b1,b2).| < |.eval(b1 *',b2).|
   holds    b1 is Hurwitz
   iff
      (F*(b1,b2)) div rpoly(1,b2) is Hurwitz;