Article POLYEQ_4, MML version 4.99.1005

:: POLYEQ_4:th 1
theorem
for b1, b2, b3 being Element of REAL
      st b1 <> 0 & b2 / b1 < 0 & 0 < b3 / b1 & 0 <= delta(b1,b2,b3)
   holds 0 < ((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1) &
    0 < ((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1);

:: POLYEQ_4:th 2
theorem
for b1, b2, b3 being Element of REAL
      st b1 <> 0 & 0 < b2 / b1 & 0 < b3 / b1 & 0 <= delta(b1,b2,b3)
   holds ((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1) < 0 &
    ((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1) < 0;

:: POLYEQ_4:th 3
theorem
for b1, b2, b3 being Element of REAL
      st b1 <> 0 &
         b2 / b1 < 0 &
         (((- b3) + sqrt delta(b1,b3,b2)) / (2 * b1) <= 0 or 0 <= ((- b3) - sqrt delta(b1,b3,b2)) / (2 * b1))
   holds ((- b3) + sqrt delta(b1,b3,b2)) / (2 * b1) < 0 &
    0 < ((- b3) - sqrt delta(b1,b3,b2)) / (2 * b1);

:: POLYEQ_4:th 4
theorem
for b1, b2 being Element of REAL
for b3 being Element of NAT
      st 0 < b1 &
         (ex b4 being Element of NAT st
            b3 = 2 * b4 & 1 <= b4) &
         b2 |^ b3 = b1 &
         b2 <> b3 -root b1
   holds b2 = - (b3 -root b1);

:: POLYEQ_4:th 5
theorem
for b1, b2, b3 being Element of REAL
      st b1 <> 0 & Polynom(b1,b2,0,b3) = 0 & b3 <> 0
   holds b3 = - (b2 / b1);

:: POLYEQ_4:th 6
theorem
for b1, b2 being Element of REAL
      st b1 <> 0 & Polynom(b1,0,0,b2) = 0
   holds b2 = 0;

:: POLYEQ_4:th 7
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Element of NAT
      st b1 <> 0 &
         (ex b6 being Element of NAT st
            b5 = (2 * b6) + 1) &
         0 <= delta(b1,b2,b3) &
         Polynom(b1,b2,b3,b4 |^ b5) = 0 &
         b4 <> b5 -root (((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1))
   holds b4 = b5 -root (((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1));

:: POLYEQ_4:th 8
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Element of NAT
      st b1 <> 0 &
         b2 / b1 < 0 &
         0 < b3 / b1 &
         (ex b6 being Element of NAT st
            b5 = 2 * b6 & 1 <= b6) &
         0 <= delta(b1,b2,b3) &
         Polynom(b1,b2,b3,b4 |^ b5) = 0 &
         b4 <> b5 -root (((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1)) &
         b4 <> - (b5 -root (((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1))) &
         b4 <> b5 -root (((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1))
   holds b4 = - (b5 -root (((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1)));

:: POLYEQ_4:th 9
theorem
for b1, b2, b3 being Element of REAL
for b4 being Element of NAT
      st b1 <> 0 &
         (ex b5 being Element of NAT st
            b4 = (2 * b5) + 1) &
         Polynom(b1,b2,0,b3 |^ b4) = 0 &
         b3 <> 0
   holds b3 = b4 -root - (b2 / b1);

:: POLYEQ_4:th 10
theorem
for b1, b2, b3 being Element of REAL
for b4 being Element of NAT
      st b1 <> 0 &
         b2 / b1 < 0 &
         (ex b5 being Element of NAT st
            b4 = 2 * b5 & 1 <= b5) &
         Polynom(b1,b2,0,b3 |^ b4) = 0 &
         b3 <> 0 &
         b3 <> b4 -root - (b2 / b1)
   holds b3 = - (b4 -root - (b2 / b1));

:: POLYEQ_4:th 11
theorem
for b1, b2 being Element of REAL holds
(b1 |^ 3) + (b2 |^ 3) = (b1 + b2) * ((b1 ^2 - (b1 * b2)) + (b2 ^2)) &
 (b1 |^ 5) + (b2 |^ 5) = (b1 + b2) * (((((b1 |^ 4) - ((b1 |^ 3) * b2)) + ((b1 |^ 2) * (b2 |^ 2))) - (b1 * (b2 |^ 3))) + (b2 |^ 4));

:: POLYEQ_4:th 12
theorem
for b1, b2, b3 being Element of REAL
      st b1 <> 0 &
         0 <= (b2 ^2 - ((2 * b1) * b2)) - (3 * (b1 ^2)) &
         Polynom(b1,b2,b2,b1,b3) = 0 &
         b3 <> - 1 &
         b3 <> ((b1 - b2) + sqrt ((b2 ^2 - ((2 * b1) * b2)) - (3 * (b1 ^2)))) / (2 * b1)
   holds b3 = ((b1 - b2) - sqrt ((b2 ^2 - ((2 * b1) * b2)) - (3 * (b1 ^2)))) / (2 * b1);

:: POLYEQ_4:funcnot 1 => POLYEQ_4:func 1
definition
  let a1, a2, a3, a4, a5, a6, a7 be complex set;
  func Polynom(A1,A2,A3,A4,A5,A6,A7) -> set equals
    (((((a1 * (a7 |^ 5)) + (a2 * (a7 |^ 4))) + (a3 * (a7 |^ 3))) + (a4 * (a7 ^2))) + (a5 * a7)) + a6;
end;

:: POLYEQ_4:def 1
theorem
for b1, b2, b3, b4, b5, b6, b7 being complex set holds
Polynom(b1,b2,b3,b4,b5,b6,b7) = (((((b1 * (b7 |^ 5)) + (b2 * (b7 |^ 4))) + (b3 * (b7 |^ 3))) + (b4 * (b7 ^2))) + (b5 * b7)) + b6;

:: POLYEQ_4:funcreg 1
registration
  let a1, a2, a3, a4, a5, a6, a7 be complex set;
  cluster Polynom(a1,a2,a3,a4,a5,a6,a7) -> complex;
end;

:: POLYEQ_4:funcreg 2
registration
  let a1, a2, a3, a4, a5, a6, a7 be real set;
  cluster Polynom(a1,a2,a3,a4,a5,a6,a7) -> real;
end;

:: POLYEQ_4:th 13
theorem
for b1, b2, b3, b4 being Element of REAL
   st b1 <> 0 &
      0 < ((b2 ^2 + ((2 * b1) * b2)) + (5 * (b1 ^2))) - ((4 * b1) * b3) &
      Polynom(b1,b2,b3,b3,b2,b1,b4) = 0
for b5, b6 being Element of REAL
      st b5 = ((b1 - b2) + sqrt (((b2 ^2 + ((2 * b1) * b2)) + (5 * (b1 ^2))) - ((4 * b1) * b3))) / (2 * b1) &
         b6 = ((b1 - b2) - sqrt (((b2 ^2 + ((2 * b1) * b2)) + (5 * (b1 ^2))) - ((4 * b1) * b3))) / (2 * b1) &
         b4 <> - 1 &
         b4 <> (b5 + sqrt delta(1,- b5,1)) / 2 &
         b4 <> (b6 + sqrt delta(1,- b6,1)) / 2 &
         b4 <> (b5 - sqrt delta(1,- b5,1)) / 2
   holds b4 = (b6 - sqrt delta(1,- b6,1)) / 2;

:: POLYEQ_4:th 14
theorem
for b1, b2, b3, b4 being Element of REAL
      st b1 + b2 = b3 &
         b1 * b2 = b4 &
         0 <= b3 ^2 - (4 * b4) &
         (b1 = (b3 + sqrt (b3 ^2 - (4 * b4))) / 2 implies b2 <> (b3 - sqrt (b3 ^2 - (4 * b4))) / 2)
   holds b1 = (b3 - sqrt (b3 ^2 - (4 * b4))) / 2 &
    b2 = (b3 + sqrt (b3 ^2 - (4 * b4))) / 2;

:: POLYEQ_4:th 15
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Element of NAT
      st (b1 |^ b5) + (b2 |^ b5) = b3 &
         (b1 |^ b5) * (b2 |^ b5) = b4 &
         0 <= b3 ^2 - (4 * b4) &
         (ex b6 being Element of NAT st
            b5 = (2 * b6) + 1) &
         (b1 = b5 -root ((b3 + sqrt (b3 ^2 - (4 * b4))) / 2) implies b2 <> b5 -root ((b3 - sqrt (b3 ^2 - (4 * b4))) / 2))
   holds b1 = b5 -root ((b3 - sqrt (b3 ^2 - (4 * b4))) / 2) &
    b2 = b5 -root ((b3 + sqrt (b3 ^2 - (4 * b4))) / 2);

:: POLYEQ_4:th 16
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Element of NAT
      st (b1 |^ b5) + (b2 |^ b5) = b3 &
         (b1 |^ b5) * (b2 |^ b5) = b4 &
         0 <= b3 ^2 - (4 * b4) &
         0 < b3 &
         0 < b4 &
         (ex b6 being Element of NAT st
            b5 = 2 * b6 & 1 <= b6) &
         (b1 = b5 -root ((b3 + sqrt (b3 ^2 - (4 * b4))) / 2) implies b2 <> b5 -root ((b3 - sqrt (b3 ^2 - (4 * b4))) / 2)) &
         (b1 = - (b5 -root ((b3 + sqrt (b3 ^2 - (4 * b4))) / 2)) implies b2 <> b5 -root ((b3 - sqrt (b3 ^2 - (4 * b4))) / 2)) &
         (b1 = b5 -root ((b3 + sqrt (b3 ^2 - (4 * b4))) / 2) implies b2 <> - (b5 -root ((b3 - sqrt (b3 ^2 - (4 * b4))) / 2))) &
         (b1 = - (b5 -root ((b3 + sqrt (b3 ^2 - (4 * b4))) / 2)) implies b2 <> - (b5 -root ((b3 - sqrt (b3 ^2 - (4 * b4))) / 2))) &
         (b1 = b5 -root ((b3 - sqrt (b3 ^2 - (4 * b4))) / 2) implies b2 <> b5 -root ((b3 + sqrt (b3 ^2 - (4 * b4))) / 2)) &
         (b1 = - (b5 -root ((b3 - sqrt (b3 ^2 - (4 * b4))) / 2)) implies b2 <> b5 -root ((b3 + sqrt (b3 ^2 - (4 * b4))) / 2)) &
         (b1 = b5 -root ((b3 - sqrt (b3 ^2 - (4 * b4))) / 2) implies b2 <> - (b5 -root ((b3 + sqrt (b3 ^2 - (4 * b4))) / 2)))
   holds b1 = - (b5 -root ((b3 - sqrt (b3 ^2 - (4 * b4))) / 2)) &
    b2 = - (b5 -root ((b3 + sqrt (b3 ^2 - (4 * b4))) / 2));

:: POLYEQ_4:th 18
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Element of NAT
      st (b1 |^ b5) + (b2 |^ b5) = b3 &
         (b1 |^ b5) - (b2 |^ b5) = b4 &
         (ex b6 being Element of NAT st
            b5 = 2 * b6 & 1 <= b6) &
         0 < b3 &
         0 < b3 + b4 &
         0 < b3 - b4 &
         (b1 = b5 -root ((b3 + b4) / 2) implies b2 <> b5 -root ((b3 - b4) / 2)) &
         (b1 = b5 -root ((b3 + b4) / 2) implies b2 <> - (b5 -root ((b3 - b4) / 2))) &
         (b1 = - (b5 -root ((b3 + b4) / 2)) implies b2 <> b5 -root ((b3 - b4) / 2))
   holds b1 = - (b5 -root ((b3 + b4) / 2)) &
    b2 = - (b5 -root ((b3 - b4) / 2));

:: POLYEQ_4:th 19
theorem
for b1, b2, b3, b4, b5 being Element of REAL
for b6 being Element of NAT
      st (b1 * (b2 |^ b6)) + (b3 * (b4 |^ b6)) = b5 &
         b2 * b4 = 0 &
         (ex b7 being Element of NAT st
            b6 = (2 * b7) + 1) &
         b1 * b3 <> 0 &
         (b2 = 0 implies b4 <> b6 -root (b5 / b3))
   holds b2 = b6 -root (b5 / b1) & b4 = 0;

:: POLYEQ_4:th 20
theorem
for b1, b2, b3, b4, b5 being Element of REAL
for b6 being Element of NAT
      st (b1 * (b2 |^ b6)) + (b3 * (b4 |^ b6)) = b5 &
         b2 * b4 = 0 &
         (ex b7 being Element of NAT st
            b6 = 2 * b7 & 1 <= b7) &
         0 < b5 / b3 &
         0 < b5 / b1 &
         b1 * b3 <> 0 &
         (b2 = 0 implies b4 <> b6 -root (b5 / b3)) &
         (b2 = 0 implies b4 <> - (b6 -root (b5 / b3))) &
         (b2 = b6 -root (b5 / b1) implies b4 <> 0)
   holds b2 = - (b6 -root (b5 / b1)) & b4 = 0;

:: POLYEQ_4:th 21
theorem
for b1, b2, b3, b4, b5 being Element of REAL
for b6 being Element of NAT
      st b1 * (b2 |^ b6) = b3 &
         b2 * b4 = b5 &
         (ex b7 being Element of NAT st
            b6 = (2 * b7) + 1) &
         b3 * b1 <> 0
   holds b2 = b6 -root (b3 / b1) & b4 = b5 * (b6 -root (b1 / b3));

:: POLYEQ_4:th 22
theorem
for b1, b2, b3, b4, b5 being Element of REAL
for b6 being Element of NAT
      st b1 * (b2 |^ b6) = b3 &
         b2 * b4 = b5 &
         (ex b7 being Element of NAT st
            b6 = 2 * b7 & 1 <= b7) &
         0 < b3 / b1 &
         b1 <> 0 &
         (b2 = b6 -root (b3 / b1) implies b4 <> b5 * (b6 -root (b1 / b3)))
   holds b2 = - (b6 -root (b3 / b1)) & b4 = - (b5 * (b6 -root (b1 / b3)));

:: POLYEQ_4:th 24
theorem
for b1, b2 being Element of REAL
      st 0 < b1 & b1 <> 1 & b1 to_power b2 = 1
   holds b2 = 0;

:: POLYEQ_4:th 25
theorem
for b1, b2 being Element of REAL
      st 0 < b1 & b1 <> 1 & b1 to_power b2 = b1
   holds b2 = 1;

:: POLYEQ_4:th 27
theorem
for b1, b2, b3 being Element of REAL
      st 0 < b1 & b1 <> 1 & 0 < b3 & log(b1,b3) = 0
   holds b3 = 1;

:: POLYEQ_4:th 28
theorem
for b1, b2, b3 being Element of REAL
      st 0 < b1 & b1 <> 1 & 0 < b3 & log(b1,b3) = 1
   holds b3 = b1;