Article QUIN_1, MML version 4.99.1005

:: QUIN_1:funcnot 1 => QUIN_1:func 1
definition
  let a1, a2, a3 be complex set;
  func delta(A1,A2,A3) -> set equals
    a2 ^2 - ((4 * a1) * a3);
end;

:: QUIN_1:def 1
theorem
for b1, b2, b3 being complex set holds
delta(b1,b2,b3) = b2 ^2 - ((4 * b1) * b3);

:: QUIN_1:funcreg 1
registration
  let a1, a2, a3 be complex set;
  cluster delta(a1,a2,a3) -> complex;
end;

:: QUIN_1:funcreg 2
registration
  let a1, a2, a3 be real set;
  cluster delta(a1,a2,a3) -> real;
end;

:: QUIN_1:funcnot 2 => QUIN_1:func 2
definition
  let a1, a2, a3 be Element of REAL;
  redefine func delta(a1,a2,a3) -> Element of REAL;
end;

:: QUIN_1:th 1
theorem
for b1, b2, b3, b4 being complex set
      st b1 <> 0
   holds ((b1 * (b4 ^2)) + (b2 * b4)) + b3 = (b1 * ((b4 + (b2 / (2 * b1))) ^2)) - ((delta(b1,b2,b3)) / (4 * b1));

:: QUIN_1:th 2
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & delta(b1,b2,b3) <= 0
   holds 0 <= ((b1 * (b4 ^2)) + (b2 * b4)) + b3;

:: QUIN_1:th 3
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & delta(b1,b2,b3) < 0
   holds 0 < ((b1 * (b4 ^2)) + (b2 * b4)) + b3;

:: QUIN_1:th 4
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 & delta(b1,b2,b3) <= 0
   holds ((b1 * (b4 ^2)) + (b2 * b4)) + b3 <= 0;

:: QUIN_1:th 5
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 & delta(b1,b2,b3) < 0
   holds ((b1 * (b4 ^2)) + (b2 * b4)) + b3 < 0;

:: QUIN_1:th 6
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 &
         0 <= ((b1 * (b2 ^2)) + (b3 * b2)) + b4
   holds 0 <= (((2 * b1) * b2) + b3) ^2 - delta(b1,b3,b4);

:: QUIN_1:th 7
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 &
         0 < ((b1 * (b2 ^2)) + (b3 * b2)) + b4
   holds 0 < (((2 * b1) * b2) + b3) ^2 - delta(b1,b3,b4);

:: QUIN_1:th 8
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 &
         ((b1 * (b2 ^2)) + (b3 * b2)) + b4 <= 0
   holds 0 <= (((2 * b1) * b2) + b3) ^2 - delta(b1,b3,b4);

:: QUIN_1:th 9
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 &
         ((b1 * (b2 ^2)) + (b3 * b2)) + b4 < 0
   holds 0 < (((2 * b1) * b2) + b3) ^2 - delta(b1,b3,b4);

:: QUIN_1:th 10
theorem
for b1, b2, b3 being real set
      st (for b4 being real set holds
            0 <= ((b1 * (b4 ^2)) + (b2 * b4)) + b3) &
         0 < b1
   holds delta(b1,b2,b3) <= 0;

:: QUIN_1:th 11
theorem
for b1, b2, b3 being real set
      st (for b4 being real set holds
            ((b1 * (b4 ^2)) + (b2 * b4)) + b3 <= 0) &
         b1 < 0
   holds delta(b1,b2,b3) <= 0;

:: QUIN_1:th 12
theorem
for b1, b2, b3 being real set
      st (for b4 being real set holds
            0 < ((b1 * (b4 ^2)) + (b2 * b4)) + b3) &
         0 < b1
   holds delta(b1,b2,b3) < 0;

:: QUIN_1:th 13
theorem
for b1, b2, b3 being real set
      st (for b4 being real set holds
            ((b1 * (b4 ^2)) + (b2 * b4)) + b3 < 0) &
         b1 < 0
   holds delta(b1,b2,b3) < 0;

:: QUIN_1:th 14
theorem
for b1, b2, b3, b4 being complex set
      st b1 <> 0 &
         ((b1 * (b4 ^2)) + (b2 * b4)) + b3 = 0
   holds (((2 * b1) * b4) + b2) ^2 - delta(b1,b2,b3) = 0;

:: QUIN_1:th 15
theorem
for b1, b2, b3, b4 being real set
      st b1 <> 0 &
         0 <= delta(b1,b2,b3) &
         ((b1 * (b4 ^2)) + (b2 * b4)) + b3 = 0 &
         b4 <> ((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1)
   holds b4 = ((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1);

:: QUIN_1:th 16
theorem
for b1, b2, b3, b4 being real set
      st b1 <> 0 & 0 <= delta(b1,b2,b3)
   holds ((b1 * (b4 ^2)) + (b2 * b4)) + b3 = (b1 * (b4 - (((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1)))) * (b4 - (((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1)));

:: QUIN_1:th 17
theorem
for b1, b2, b3 being real set
      st b1 < 0 & 0 < delta(b1,b2,b3)
   holds ((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1) < ((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1);

:: QUIN_1:th 18
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 & 0 < delta(b1,b2,b3)
   holds    0 < ((b1 * (b4 ^2)) + (b2 * b4)) + b3
   iff
      ((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1) < b4 &
       b4 < ((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1);

:: QUIN_1:th 19
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 & 0 < delta(b1,b2,b3)
   holds    ((b1 * (b4 ^2)) + (b2 * b4)) + b3 < 0
   iff
      (((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1) <= b4 implies ((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1) < b4);

:: QUIN_1:th 22
theorem
for b1, b2, b3, b4 being complex set
      st b1 <> 0 &
         delta(b1,b2,b3) = 0 &
         ((b1 * (b4 ^2)) + (b2 * b4)) + b3 = 0
   holds b4 = - (b2 / (2 * b1));

:: QUIN_1:th 23
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 &
         0 < (((2 * b1) * b2) + b3) ^2 - delta(b1,b3,b4)
   holds 0 < ((b1 * (b2 ^2)) + (b3 * b2)) + b4;

:: QUIN_1:th 24
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & delta(b1,b2,b3) = 0
   holds    0 < ((b1 * (b4 ^2)) + (b2 * b4)) + b3
   iff
      b4 <> - (b2 / (2 * b1));

:: QUIN_1:th 25
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 &
         0 < (((2 * b1) * b2) + b3) ^2 - delta(b1,b3,b4)
   holds ((b1 * (b2 ^2)) + (b3 * b2)) + b4 < 0;

:: QUIN_1:th 26
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 & delta(b1,b2,b3) = 0
   holds    ((b1 * (b4 ^2)) + (b2 * b4)) + b3 < 0
   iff
      b4 <> - (b2 / (2 * b1));

:: QUIN_1:th 27
theorem
for b1, b2, b3 being real set
      st 0 < b1 & 0 < delta(b1,b2,b3)
   holds ((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1) < ((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1);

:: QUIN_1:th 28
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & 0 < delta(b1,b2,b3)
   holds    ((b1 * (b4 ^2)) + (b2 * b4)) + b3 < 0
   iff
      ((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1) < b4 &
       b4 < ((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1);

:: QUIN_1:th 29
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & 0 < delta(b1,b2,b3)
   holds    0 < ((b1 * (b4 ^2)) + (b2 * b4)) + b3
   iff
      (((- b2) - sqrt delta(b1,b2,b3)) / (2 * b1) <= b4 implies ((- b2) + sqrt delta(b1,b2,b3)) / (2 * b1) < b4);