Article POLYEQ_2, MML version 4.99.1005

:: POLYEQ_2:funcnot 1 => POLYEQ_2:func 1
definition
  let a1, a2, a3, a4, a5, a6 be complex set;
  func Polynom(A1,A2,A3,A4,A5,A6) -> set equals
    ((((a1 * (a6 |^ 4)) + (a2 * (a6 |^ 3))) + (a3 * (a6 ^2))) + (a4 * a6)) + a5;
end;

:: POLYEQ_2:def 1
theorem
for b1, b2, b3, b4, b5, b6 being complex set holds
Polynom(b1,b2,b3,b4,b5,b6) = ((((b1 * (b6 |^ 4)) + (b2 * (b6 |^ 3))) + (b3 * (b6 ^2))) + (b4 * b6)) + b5;

:: POLYEQ_2:funcreg 1
registration
  let a1, a2, a3, a4, a5, a6 be complex set;
  cluster Polynom(a1,a2,a3,a4,a5,a6) -> complex;
end;

:: POLYEQ_2:funcreg 2
registration
  let a1, a2, a3, a4, a5, a6 be real set;
  cluster Polynom(a1,a2,a3,a4,a5,a6) -> real;
end;

:: POLYEQ_2:th 1
theorem
for b1, b2, b3, b4 being real set
      st b1 <> 0 &
         b3 <> 0 &
         0 < b2 ^2 - ((4 * b1) * b3) &
         Polynom(b1,0,b2,0,b3,b4) = 0
   holds b4 <> 0 &
    (b4 <> sqrt (((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1)) &
     b4 <> sqrt (((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1)) &
     b4 <> - sqrt (((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1)) implies b4 = - sqrt (((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1)));

:: POLYEQ_2:th 2
theorem
for b1, b2, b3, b4, b5 being real set
      st b1 <> 0 & b5 = b4 + (1 / b4) & Polynom(b1,b2,b3,b2,b1,b4) = 0
   holds b4 <> 0 &
    (((b1 * (b5 ^2)) + (b2 * b5)) + b3) - (2 * b1) = 0;

:: POLYEQ_2:th 3
theorem
for b1, b2, b3, b4, b5 being real set
   st b1 <> 0 &
      0 < (b2 ^2 - ((4 * b1) * b3)) + (8 * (b1 ^2)) &
      b5 = b4 + (1 / b4) &
      Polynom(b1,b2,b3,b2,b1,b4) = 0
for b6, b7 being real set
      st b6 = ((- b2) + sqrt ((b2 ^2 - ((4 * b1) * b3)) + (8 * (b1 ^2)))) / (2 * b1) &
         b7 = ((- b2) - sqrt ((b2 ^2 - ((4 * b1) * b3)) + (8 * (b1 ^2)))) / (2 * b1)
   holds b4 <> 0 &
    (b4 <> (b6 + sqrt delta(1,- b6,1)) / 2 &
     b4 <> (b7 + sqrt delta(1,- b7,1)) / 2 &
     b4 <> (b6 - sqrt delta(1,- b6,1)) / 2 implies b4 = (b7 - sqrt delta(1,- b7,1)) / 2);

:: POLYEQ_2:th 4
theorem
for b1 being real set holds
   b1 |^ 3 = b1 ^2 * b1 & (b1 |^ 3) * b1 = b1 |^ 4 & b1 ^2 * (b1 ^2) = b1 |^ 4;

:: POLYEQ_2:th 5
theorem
for b1, b2 being real set
      st b1 + b2 <> 0
   holds (b1 + b2) |^ 4 = ((((b1 |^ 3) + (((3 * b2) * (b1 ^2)) + ((3 * (b2 ^2)) * b1))) + (b2 |^ 3)) * b1) + ((((b1 |^ 3) + (((3 * b2) * (b1 ^2)) + ((3 * (b2 ^2)) * b1))) + (b2 |^ 3)) * b2);

:: POLYEQ_2:th 6
theorem
for b1, b2 being real set
      st b1 + b2 <> 0
   holds (b1 + b2) |^ 4 = ((b1 |^ 4) + ((((4 * b2) * (b1 |^ 3)) + ((6 * (b2 ^2)) * (b1 ^2))) + ((4 * (b2 |^ 3)) * b1))) + (b2 |^ 4);

:: POLYEQ_2:th 7
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being real set
      st for b11 being real set holds
           Polynom(b1,b2,b3,b4,b5,b11) = Polynom(b6,b7,b8,b9,b10,b11)
   holds b5 = b10 &
    ((b1 - b2) + b3) - b4 = ((b6 - b7) + b8) - b9 &
    ((b1 + b2) + b3) + b4 = ((b6 + b7) + b8) + b9;

:: POLYEQ_2:th 8
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being real set
      st for b11 being real set holds
           Polynom(b1,b2,b3,b4,b5,b11) = Polynom(b6,b7,b8,b9,b10,b11)
   holds b1 - b6 = b8 - b3 & b2 - b7 = b9 - b4;

:: POLYEQ_2:th 9
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being real set
      st for b11 being real set holds
           Polynom(b1,b2,b3,b4,b5,b11) = Polynom(b6,b7,b8,b9,b10,b11)
   holds b1 = b6 & b2 = b7 & b3 = b8 & b4 = b9 & b5 = b10;

:: POLYEQ_2:funcnot 2 => POLYEQ_2:func 2
definition
  let a1, a2, a3, a4, a5, a6 be real set;
  func Four0(A1,A2,A3,A4,A5,A6) -> set equals
    a1 * ((((a6 - a2) * (a6 - a3)) * (a6 - a4)) * (a6 - a5));
end;

:: POLYEQ_2:def 2
theorem
for b1, b2, b3, b4, b5, b6 being real set holds
Four0(b1,b2,b3,b4,b5,b6) = b1 * ((((b6 - b2) * (b6 - b3)) * (b6 - b4)) * (b6 - b5));

:: POLYEQ_2:funcreg 3
registration
  let a1, a2, a3, a4, a5, a6 be real set;
  cluster Four0(a1,a2,a3,a4,a5,a6) -> real;
end;

:: POLYEQ_2:th 10
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being real set
      st b1 <> 0 &
         (for b11 being real set holds
            Polynom(b1,b2,b3,b4,b5,b11) = Four0(b1,b7,b8,b9,b10,b11))
   holds (((((b1 * (b6 |^ 4)) + (b2 * (b6 |^ 3))) + (b3 * (b6 ^2))) + (b4 * b6)) + b5) / b1 = ((((b6 ^2 * (b6 ^2)) - (((b7 + b8) + b9) * (b6 ^2 * b6))) + ((((b7 * b9) + (b8 * b9)) + (b7 * b8)) * (b6 ^2))) - (((b7 * b8) * b9) * b6)) - ((((b6 - b7) * (b6 - b8)) * (b6 - b9)) * b10);

:: POLYEQ_2:th 11
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being real set
      st b1 <> 0 &
         (for b11 being real set holds
            Polynom(b1,b2,b3,b4,b5,b11) = Four0(b1,b7,b8,b9,b10,b11))
   holds (((((b1 * (b6 |^ 4)) + (b2 * (b6 |^ 3))) + (b3 * (b6 ^2))) + (b4 * b6)) + b5) / b1 = ((((b6 |^ 4) - ((((b7 + b8) + b9) + b10) * (b6 |^ 3))) + ((((((b7 * b8) + (b7 * b9)) + (b7 * b10)) + ((b8 * b9) + (b8 * b10))) + (b9 * b10)) * (b6 ^2))) - ((((((b7 * b8) * b9) + ((b7 * b8) * b10)) + ((b7 * b9) * b10)) + ((b8 * b9) * b10)) * b6)) + (((b7 * b8) * b9) * b10);

:: POLYEQ_2:th 12
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being real set
      st b1 <> 0 &
         (for b10 being real set holds
            Polynom(b1,b2,b3,b4,b5,b10) = Four0(b1,b6,b7,b8,b9,b10))
   holds b2 / b1 = - (((b6 + b7) + b8) + b9) &
    b3 / b1 = ((((b6 * b7) + (b6 * b8)) + (b6 * b9)) + ((b7 * b8) + (b7 * b9))) + (b8 * b9) &
    b4 / b1 = - (((((b6 * b7) * b8) + ((b6 * b7) * b9)) + ((b6 * b8) * b9)) + ((b7 * b8) * b9)) &
    b5 / b1 = ((b6 * b7) * b8) * b9;

:: POLYEQ_2:th 13
theorem
for b1, b2, b3 being real set
      st b1 <> 0 &
         (for b4 being real set holds
            (b4 |^ 4) + (b1 |^ 4) = ((b2 * b1) * b4) * (b4 ^2 + (b1 ^2)))
   holds (((b3 |^ 4) - (b2 * (b3 |^ 3))) - (b2 * b3)) + 1 = 0;