Article POLYEQ_1, MML version 4.99.1005

:: POLYEQ_1:funcnot 1 => POLYEQ_1:func 1
definition
  let a1, a2, a3 be complex set;
  func Polynom(A1,A2,A3) -> set equals
    (a1 * a3) + a2;
end;

:: POLYEQ_1:def 1
theorem
for b1, b2, b3 being complex set holds
Polynom(b1,b2,b3) = (b1 * b3) + b2;

:: POLYEQ_1:funcreg 1
registration
  let a1, a2, a3 be complex set;
  cluster Polynom(a1,a2,a3) -> complex;
end;

:: POLYEQ_1:funcreg 2
registration
  let a1, a2, a3 be real set;
  cluster Polynom(a1,a2,a3) -> real;
end;

:: POLYEQ_1:funcnot 2 => POLYEQ_1:func 2
definition
  let a1, a2, a3 be Element of REAL;
  redefine func Polynom(a1,a2,a3) -> Element of REAL;
end;

:: POLYEQ_1:th 1
theorem
for b1, b2, b3 being complex set
      st b1 <> 0 & Polynom(b1,b2,b3) = 0
   holds b3 = - (b2 / b1);

:: POLYEQ_1:th 2
theorem
for b1 being complex set holds
   Polynom(0,0,b1) = 0;

:: POLYEQ_1:th 3
theorem
for b1 being complex set
   st b1 <> 0
for b2 being complex set holds
   Polynom(0,b1,b2) <> 0;

:: POLYEQ_1:funcnot 3 => POLYEQ_1:func 3
definition
  let a1, a2, a3, a4 be complex set;
  func Polynom(A1,A2,A3,A4) -> set equals
    ((a1 * (a4 ^2)) + (a2 * a4)) + a3;
end;

:: POLYEQ_1:def 2
theorem
for b1, b2, b3, b4 being complex set holds
Polynom(b1,b2,b3,b4) = ((b1 * (b4 ^2)) + (b2 * b4)) + b3;

:: POLYEQ_1:funcreg 3
registration
  let a1, a2, a3, a4 be real set;
  cluster Polynom(a1,a2,a3,a4) -> real;
end;

:: POLYEQ_1:funcreg 4
registration
  let a1, a2, a3, a4 be complex set;
  cluster Polynom(a1,a2,a3,a4) -> complex;
end;

:: POLYEQ_1:funcnot 4 => POLYEQ_1:func 4
definition
  let a1, a2, a3, a4 be Element of REAL;
  redefine func Polynom(a1,a2,a3,a4) -> Element of REAL;
end;

:: POLYEQ_1:th 4
theorem
for b1, b2, b3, b4, b5, b6 being complex set
      st for b7 being real set holds
           Polynom(b1,b2,b3,b7) = Polynom(b4,b5,b6,b7)
   holds b1 = b4 & b2 = b5 & b3 = b6;

:: POLYEQ_1:th 5
theorem
for b1, b2, b3 being real set
   st b1 <> 0 & 0 <= delta(b1,b2,b3)
for b4 being real set
      st Polynom(b1,b2,b3,b4) = 0 &
         b4 <> ((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1)
   holds b4 = ((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1);

:: POLYEQ_1:th 6
theorem
for b1, b2, b3, b4 being complex set
      st b1 <> 0 & delta(b1,b2,b3) = 0 & Polynom(b1,b2,b3,b4) = 0
   holds b4 = - (b2 / (2 * b1));

:: POLYEQ_1:th 7
theorem
for b1, b2, b3 being real set
   st b1 <> 0 & delta(b1,b2,b3) < 0
for b4 being real set holds
   Polynom(b1,b2,b3,b4) <> 0;

:: POLYEQ_1:th 8
theorem
for b1 being real set
for b2, b3 being complex set
      st b2 <> 0 &
         (for b4 being real set holds
            Polynom(0,b2,b3,b4) = 0)
   holds b1 = - (b3 / b2);

:: POLYEQ_1:th 9
theorem
for b1 being complex set holds
   Polynom(0,0,0,b1) = 0;

:: POLYEQ_1:th 10
theorem
for b1 being complex set
   st b1 <> 0
for b2 being complex set holds
   Polynom(0,0,b1,b2) <> 0;

:: POLYEQ_1:funcnot 5 => POLYEQ_1:func 5
definition
  let a1, a2, a3, a4 be complex set;
  func Quard(A1,A3,A4,A2) -> set equals
    a1 * ((a2 - a3) * (a2 - a4));
end;

:: POLYEQ_1:def 3
theorem
for b1, b2, b3, b4 being complex set holds
Quard(b1,b3,b4,b2) = b1 * ((b2 - b3) * (b2 - b4));

:: POLYEQ_1:funcreg 5
registration
  let a1, a2, a3, a4 be real set;
  cluster Quard(a1,a3,a4,a2) -> real;
end;

:: POLYEQ_1:funcnot 6 => POLYEQ_1:func 6
definition
  let a1, a2, a3, a4 be Element of REAL;
  redefine func Quard(a1,a3,a4,a2) -> Element of REAL;
end;

:: POLYEQ_1:th 11
theorem
for b1, b2 being real set
for b3, b4, b5 being complex set
      st b3 <> 0 &
         (for b6 being real set holds
            Polynom(b3,b4,b5,b6) = Quard(b3,b1,b2,b6))
   holds b4 / b3 = - (b1 + b2) & b5 / b3 = b1 * b2;

:: POLYEQ_1:funcnot 7 => POLYEQ_1:func 7
definition
  let a1, a2, a3, a4, a5 be complex set;
  func Polynom(A1,A2,A3,A4,A5) -> set equals
    (((a1 * (a5 |^ 3)) + (a2 * (a5 ^2))) + (a3 * a5)) + a4;
end;

:: POLYEQ_1:def 4
theorem
for b1, b2, b3, b4, b5 being complex set holds
Polynom(b1,b2,b3,b4,b5) = (((b1 * (b5 |^ 3)) + (b2 * (b5 ^2))) + (b3 * b5)) + b4;

:: POLYEQ_1:funcreg 6
registration
  let a1, a2, a3, a4, a5 be complex set;
  cluster Polynom(a1,a2,a3,a4,a5) -> complex;
end;

:: POLYEQ_1:funcreg 7
registration
  let a1, a2, a3, a4, a5 be real set;
  cluster Polynom(a1,a2,a3,a4,a5) -> real;
end;

:: POLYEQ_1:funcnot 8 => POLYEQ_1:func 8
definition
  let a1, a2, a3, a4, a5 be Element of REAL;
  redefine func Polynom(a1,a2,a3,a4,a5) -> Element of REAL;
end;

:: POLYEQ_1:th 12
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being real set
      st for b9 being real set holds
           Polynom(b1,b2,b3,b4,b9) = Polynom(b5,b6,b7,b8,b9)
   holds b1 = b5 & b2 = b6 & b3 = b7 & b4 = b8;

:: POLYEQ_1:funcnot 9 => POLYEQ_1:func 9
definition
  let a1, a2, a3, a4, a5 be real set;
  func Tri(A1,A3,A4,A5,A2) -> set equals
    a1 * (((a2 - a3) * (a2 - a4)) * (a2 - a5));
end;

:: POLYEQ_1:def 5
theorem
for b1, b2, b3, b4, b5 being real set holds
Tri(b1,b3,b4,b5,b2) = b1 * (((b2 - b3) * (b2 - b4)) * (b2 - b5));

:: POLYEQ_1:funcreg 8
registration
  let a1, a2, a3, a4, a5 be real set;
  cluster Tri(a1,a3,a4,a5,a2) -> real;
end;

:: POLYEQ_1:funcnot 10 => POLYEQ_1:func 10
definition
  let a1, a2, a3, a4, a5 be Element of REAL;
  redefine func Tri(a1,a3,a4,a5,a2) -> Element of REAL;
end;

:: POLYEQ_1:th 13
theorem
for b1, b2, b3, b4, b5, b6, b7 being real set
      st b1 <> 0 &
         (for b8 being real set holds
            Polynom(b1,b2,b3,b4,b8) = Tri(b1,b5,b6,b7,b8))
   holds b2 / b1 = - ((b5 + b6) + b7) &
    b3 / b1 = ((b5 * b6) + (b6 * b7)) + (b5 * b7) &
    b4 / b1 = - ((b5 * b6) * b7);

:: POLYEQ_1:th 14
theorem
for b1, b2 being real set holds
(b1 + b2) |^ 3 = ((b1 |^ 3) + (((3 * b2) * (b1 ^2)) + ((3 * (b2 ^2)) * b1))) + (b2 |^ 3);

:: POLYEQ_1:th 15
theorem
for b1, b2, b3, b4, b5 being real set
   st b1 <> 0 & Polynom(b1,b2,b3,b4,b5) = 0
for b6, b7, b8, b9, b10 being real set
      st b10 = b5 + (b2 / (3 * b1)) & b9 = - (b2 / (3 * b1)) & b6 = b2 / b1 & b7 = b3 / b1 & b8 = b4 / b1
   holds ((b10 |^ 3) + ((((3 * b9) + b6) * (b10 ^2)) + ((((3 * (b9 ^2)) + (2 * (b6 * b9))) + b7) * b10))) + (((b9 |^ 3) + (b6 * (b9 ^2))) + ((b7 * b9) + b8)) = 0;

:: POLYEQ_1:th 16
theorem
for b1, b2, b3, b4, b5 being real set
   st b1 <> 0 & Polynom(b1,b2,b3,b4,b5) = 0
for b6, b7, b8, b9, b10 being real set
      st b10 = b5 + (b2 / (3 * b1)) & b9 = - (b2 / (3 * b1)) & b6 = b2 / b1 & b7 = b3 / b1 & b8 = b4 / b1
   holds (((b10 |^ 3) + (0 * (b10 ^2))) + (((((3 * b1) * b3) - (b2 ^2)) / (3 * (b1 ^2))) * b10)) + ((2 * ((b2 / (3 * b1)) |^ 3)) + ((((3 * b1) * b4) - (b2 * b3)) / (3 * (b1 ^2)))) = 0;

:: POLYEQ_1:th 17
theorem
for b1, b2, b3, b4, b5 being real set
   st (((b1 |^ 3) + (0 * (b1 ^2))) + (((((3 * b2) * b3) - (b4 ^2)) / (3 * (b2 ^2))) * b1)) + ((2 * ((b4 / (3 * b2)) |^ 3)) + ((((3 * b2) * b5) - (b4 * b3)) / (3 * (b2 ^2)))) = 0
for b6, b7 being real set
      st b6 = (((3 * b2) * b3) - (b4 ^2)) / (3 * (b2 ^2)) &
         b7 = (2 * ((b4 / (3 * b2)) |^ 3)) + ((((3 * b2) * b5) - (b4 * b3)) / (3 * (b2 ^2)))
   holds Polynom(1,0,b6,b7,b1) = 0;

:: POLYEQ_1:th 18
theorem
for b1, b2, b3 being real set
   st Polynom(1,0,b1,b2,b3) = 0
for b4, b5 being real set
      st b3 = b4 + b5 & ((3 * b5) * b4) + b1 = 0
   holds (b4 |^ 3) + (b5 |^ 3) = - b2 &
    (b4 |^ 3) * (b5 |^ 3) = (- (b1 / 3)) |^ 3;

:: POLYEQ_1:th 19
theorem
for b1, b2, b3 being real set
   st Polynom(1,0,b1,b2,b3) = 0
for b4, b5 being real set
      st b3 = b4 + b5 &
         ((3 * b5) * b4) + b1 = 0 &
         b3 <> (3 -root ((- (b2 / 2)) + sqrt ((b2 ^2 / 4) + ((b1 / 3) |^ 3)))) + (3 -root ((- (b2 / 2)) - sqrt ((b2 ^2 / 4) + ((b1 / 3) |^ 3)))) &
         b3 <> (3 -root ((- (b2 / 2)) + sqrt ((b2 ^2 / 4) + ((b1 / 3) |^ 3)))) + (3 -root ((- (b2 / 2)) + sqrt ((b2 ^2 / 4) + ((b1 / 3) |^ 3))))
   holds b3 = (3 -root ((- (b2 / 2)) - sqrt ((b2 ^2 / 4) + ((b1 / 3) |^ 3)))) + (3 -root ((- (b2 / 2)) - sqrt ((b2 ^2 / 4) + ((b1 / 3) |^ 3))));

:: POLYEQ_1:th 20
theorem
for b1, b2, b3, b4 being real set
      st b1 <> 0 &
         0 < delta(b1,b2,b3) &
         Polynom(0,b1,b2,b3,b4) = 0 &
         b4 <> ((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1)
   holds b4 = ((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1);

:: POLYEQ_1:th 21
theorem
for b1, b2, b3, b4, b5, b6 being real set
   st b1 <> 0 & b2 = b3 / b1 & b4 = b5 / b1 & Polynom(b1,0,b3,b5,b6) = 0
for b7, b8 being real set
      st b6 = b7 + b8 &
         ((3 * b8) * b7) + b2 = 0 &
         b6 <> (3 -root ((- (b5 / (2 * b1))) + sqrt ((b5 ^2 / (4 * (b1 ^2))) + ((b3 / (3 * b1)) |^ 3)))) + (3 -root ((- (b5 / (2 * b1))) - sqrt ((b5 ^2 / (4 * (b1 ^2))) + ((b3 / (3 * b1)) |^ 3)))) &
         b6 <> (3 -root ((- (b5 / (2 * b1))) + sqrt ((b5 ^2 / (4 * (b1 ^2))) + ((b3 / (3 * b1)) |^ 3)))) + (3 -root ((- (b5 / (2 * b1))) + sqrt ((b5 ^2 / (4 * (b1 ^2))) + ((b3 / (3 * b1)) |^ 3))))
   holds b6 = (3 -root ((- (b5 / (2 * b1))) - sqrt ((b5 ^2 / (4 * (b1 ^2))) + ((b3 / (3 * b1)) |^ 3)))) + (3 -root ((- (b5 / (2 * b1))) - sqrt ((b5 ^2 / (4 * (b1 ^2))) + ((b3 / (3 * b1)) |^ 3))));

:: POLYEQ_1:th 22
theorem
for b1, b2, b3, b4 being real set
      st b1 <> 0 &
         0 <= delta(b1,b2,b3) &
         Polynom(b1,b2,b3,0,b4) = 0 &
         b4 <> 0 &
         b4 <> ((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1)
   holds b4 = ((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1);

:: POLYEQ_1:th 23
theorem
for b1, b2, b3 being real set
      st b1 <> 0 & b2 / b1 < 0 & Polynom(b1,0,b2,0,b3) = 0 & b3 <> 0 & b3 <> sqrt - (b2 / b1)
   holds b3 = - sqrt - (b2 / b1);