### The Mizar article:

by
Jan Popiolek

Copyright (c) 1991 Association of Mizar Users

MML identifier: QUIN_1
[ MML identifier index ]

```environ

vocabulary ARYTM, FUNCT_3, SQUARE_1, ARYTM_1, ARYTM_3;
notation ORDINAL1, XCMPLX_0, XREAL_0, REAL_1, SQUARE_1;
constructors REAL_1, SQUARE_1, MEMBERED, XBOOLE_0;
clusters XREAL_0, SQUARE_1, MEMBERED, ZFMISC_1, XBOOLE_0;
requirements REAL, NUMERALS, SUBSET, ARITHM;
theorems XREAL_0, AXIOMS, REAL_1, REAL_2, SQUARE_1, XCMPLX_0, XCMPLX_1;

begin

reserve x, a, b, c for real number;

definition let a,b,c;
func delta(a,b,c) equals
:Def1:   b^2 - 4 * a * c;
coherence;
end;

definition let a,b,c;
cluster delta(a,b,c) -> real;
coherence
proof
delta(a,b,c) = b^2 - 4 * a * c by Def1;
hence thesis;
end;
end;

definition let a,b,c be Real;
redefine func delta(a,b,c) -> Real;
coherence by XREAL_0:def 1;
end;

theorem Th1:
a <> 0 implies
a * x^2 + b * x + c = a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a)
proof
assume
A1:   a <> 0;
then A2:   4 * a <> 0 by XCMPLX_1:6;
a * x^2 + b * x + c = a * x^2 + (b * x) * 1 + c
.= a * x^2 + (b * x) * (a * (1/a)) + c by A1,XCMPLX_1:107
.= a * x^2 + a * ((b * x) * (1/a)) + c by XCMPLX_1:4
.= a * (x^2 + ((b * x) * (1/a))) + c by XCMPLX_1:8
.= a * (x^2 + ((b * x)/a)) + c by XCMPLX_1:100
.= a * (x^2 + x * (b/a)) + c by XCMPLX_1:75
.= a * (x^2 + x * ((2 * b)/(2 * a))) + c by XCMPLX_1:92
.= a * (x^2 + x * (2 * (b/(2 * a)))) + c by XCMPLX_1:75
.= a * (x^2 + (2 * x) * (b/(2 * a))) + c by XCMPLX_1:4
.= a * (x^2 + 2 * x * (b/(2 * a))) + (b^2/(4 * a) + (c - b^2/(4 * a)))
by XCMPLX_1:27
.= (a * (x^2 + 2 * x * (b/(2 * a))) + b^2/(4 * a)) + (c - b^2/(4 * a))
by XCMPLX_1:1
.= (a * (x^2 + 2 * x * (b/(2 * a))) + a * (b^2/(4 * a) * (1/a))) + (c -
b^2/(4 * a)) by A1,XCMPLX_1:110
.= (a * (x^2 + 2 * x * (b/(2 * a))) + a * ((b^2 * 1)/((4 * a) * a))) +
(c - b^2/(4 * a)) by XCMPLX_1:77
.= (a * (x^2 + 2 * x * (b/(2 * a))) + a * (b^2/(4 * (a * a)))) + (c -
b^2/(4 * a)) by XCMPLX_1:4
.= (a * (x^2 + 2 * x * (b/(2 * a))) + a * (b^2/((2 * 2) * a^2))) + (c -
b^2/(4 * a)) by SQUARE_1:def 3
.= (a * (x^2 + 2 * x * (b/(2 * a))) + a * (b^2/(2^2 * a^2))) + (c -
b^2/(4 * a)) by SQUARE_1:def 3
.= (a * (x^2 + 2 * x * (b/(2 * a))) + a * (b^2/(2 * a)^2)) + (c -
b^2/(4 * a)) by SQUARE_1:68
.= (a * (x^2 + 2 * x * (b/(2 * a))) + a * (b/(2 * a))^2) + (c -
b^2/(4 * a)) by SQUARE_1:69
.= a * ((x^2 + 2 * x * (b/(2 * a))) + (b/(2 * a))^2) + (c -
b^2/(4 * a)) by XCMPLX_1:8
.= a * (x + b/(2 * a))^2 + (c - b^2/(4 * a)) by SQUARE_1:63
.= a * (x + b/(2 * a))^2 - (b^2/(4 * a) - c) by XCMPLX_1:38
.= a * (x + b/(2 * a))^2 - (b^2/(4 * a) - ((4 * a * c)/(4 * a)))
by A2,XCMPLX_1:90
.= a * (x + b/(2 * a))^2 - (b^2 - (4 * a * c))/(4 * a) by XCMPLX_1:121;
hence thesis by Def1;
end;

theorem
a > 0 & delta(a,b,c) <= 0 implies a * x^2 + b * x + c >= 0
proof
assume that
A1:   a > 0 and
A2:   delta(a,b,c) <= 0;
A3:   - delta(a,b,c) >= 0 by A2,REAL_1:26,REAL_2:109;
4 * a > 0 by A1,REAL_2:122;
then (- delta(a,b,c))/(4 * a) >= 0 by A3,REAL_2:125;
then - delta(a,b,c)/(4 * a) >= 0 by XCMPLX_1:188;
then a * (x + b/(2 * a))^2 + - delta(a,b,c)/(4 * a) >= a * (x + b/(2 * a)
)^2
+ 0
by REAL_1:55;
then A4:   a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) >= a * (x + b/(2 * a))
^2 by XCMPLX_0:def 8;
(x +b/(2 * a))^2 >= 0 by SQUARE_1:72;
then a * (x +b/(2 * a))^2 >= 0 by A1,REAL_2:121;
then a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) >= 0 by A4,AXIOMS:22;
hence thesis by A1,Th1;
end;

theorem
a > 0 & delta(a,b,c) < 0 implies a * x^2 + b * x + c > 0
proof
assume that
A1:   a > 0 and
A2:   delta(a,b,c) < 0;
A3:   - delta(a,b,c) > 0 by A2,REAL_1:26,REAL_2:110;
4 * a > 0 by A1,REAL_2:122;
then (- delta(a,b,c))/(4 * a) > 0 by A3,REAL_2:127;
then - delta(a,b,c)/(4 * a) > 0 by XCMPLX_1:188;
then a * (x + b/(2 * a))^2 + - delta(a,b,c)/(4 * a) > a * (x + b/(2 * a))
^2
by REAL_1:69;
then A4:   a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) > a * (x + b/(2 * a))^2
by XCMPLX_0:def 8;
(x +b/(2 * a))^2 >= 0 by SQUARE_1:72;
then a * (x +b/(2 * a))^2 >= 0 by A1,REAL_2:121;
then a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) > 0 by A4,AXIOMS:22;
hence thesis by A1,Th1;
end;

theorem
a < 0 & delta(a,b,c) <= 0 implies a * x^2 + b * x + c <= 0
proof
assume that
A1:   a < 0 and
A2:   delta(a,b,c) <= 0;
A3:   - delta(a,b,c) >= 0 by A2,REAL_1:26,REAL_2:109;
4 * a < 0 by A1,SQUARE_1:24;
then (- delta(a,b,c))/(4 * a) <= 0 by A3,REAL_2:126;
then - delta(a,b,c)/(4 * a) <= 0 by XCMPLX_1:188;
then a * (x + b/(2 * a))^2 + - delta(a,b,c)/(4 * a) <= a * (x + b/(2 * a)
)^2
+ 0
by REAL_1:55;
then A4:   a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) <= a * (x + b/(2 * a))
^2 by XCMPLX_0:def 8;
(x +b/(2 * a))^2 >= 0 by SQUARE_1:72;
then a * (x +b/(2 * a))^2 <= 0 by A1,REAL_2:123;
then a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) <= 0 by A4,AXIOMS:22;
hence thesis by A1,Th1;
end;

theorem
a < 0 & delta(a,b,c) < 0 implies a * x^2 + b * x + c < 0
proof
assume that
A1:   a < 0 and
A2:   delta(a,b,c) < 0;
A3:   - delta(a,b,c) > 0 by A2,REAL_1:66;
4 * a < 0 by A1,SQUARE_1:24;
then (- delta(a,b,c))/(4 * a) < 0 by A3,REAL_2:128;
then - delta(a,b,c)/(4 * a) < 0 by XCMPLX_1:188;
then a * (x + b/(2 * a))^2 + - delta(a,b,c)/(4 * a) < a * (x + b/(2 * a))
^2 + 0
by REAL_1:53;
then A4:   a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) < a * (x + b/(2 * a))^2
by XCMPLX_0:def 8;
(x +b/(2 * a))^2 >= 0 by SQUARE_1:72;
then a * (x +b/(2 * a))^2 <= 0 by A1,REAL_2:123;
then a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) < 0 by A4,AXIOMS:22;
hence thesis by A1,Th1;
end;

theorem Th6:
a > 0 & a * x^2 + b * x + c >= 0 implies (2 * a * x + b)^2
- delta(a,b,c) >= 0
proof
assume that
A1:   a > 0 and
A2:   a * x^2 + b * x + c >= 0;
A3:   4 * a > 0 by A1,REAL_2:122;
A4:   2 * a <> 0 by A1,REAL_2:122;
a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) >= 0 by A1,A2,Th1;
then (4 * a) * (a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a)) >= 0
by A3,REAL_2:121;
then ((4 * a) * (a * (x + b/(2 * a))^2)) - (4 * a) *
(delta(a,b,c)/(4 * a)) >= 0 by XCMPLX_1:40;
then ((4 * a) * a) * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a))
>= 0
by XCMPLX_1:4;
then (4 * (a * a)) * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a))
>= 0
by XCMPLX_1:4;
then ((2 * 2) * a^2) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) >= 0 by SQUARE_1:def 3;
then (2^2 * a^2) * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a))
>= 0
by SQUARE_1:def 3;
then (2 * a)^2 * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) >=
0
by SQUARE_1:68;
then ((2 * a) * (x + b/(2 * a)))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) >=
0
by SQUARE_1:68;
then ((2 * a) * x + (2 * a) * (b/(2 * a)))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) >= 0 by XCMPLX_1:8;
then A5:   (2 * a * x + b)^2 - (4 * a) * (delta(a,b,c)/(4 * a)) >= 0
by A4,XCMPLX_1:88;
4 * a <> 0 by A1,REAL_2:122;
hence thesis by A5,XCMPLX_1:88;
end;

theorem Th7:
a > 0 & a * x^2 + b * x + c > 0 implies (2 * a * x + b)^2 - delta(a,b,c) > 0
proof
assume that
A1:   a > 0 and
A2:   a * x^2 + b * x + c > 0;
A3:   4 * a > 0 by A1,REAL_2:122;
A4:   2 * a <> 0 by A1,REAL_2:122;
a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) > 0 by A1,A2,Th1;
then (4 * a) * (a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a)) > 0
by A3,REAL_2:122;
then ((4 * a) * (a * (x + b/(2 * a))^2)) - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by XCMPLX_1:40;
then ((4 * a) * a) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by XCMPLX_1:4;
then (4 * (a * a)) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by XCMPLX_1:4;
then ((2 * 2) * a^2) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by SQUARE_1:def 3;
then (2^2 * a^2) * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) >
0
by SQUARE_1:def 3;
then (2 * a)^2 * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) > 0
by SQUARE_1:68;
then ((2 * a) * (x + b/(2 * a)))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) > 0
by SQUARE_1:68;
then ((2 * a) * x + (2 * a) * (b/(2 * a)))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by XCMPLX_1:8;
then A5:   (2 * a * x + b)^2 - (4 * a) * (delta(a,b,c)/(4 * a)) > 0
by A4,XCMPLX_1:88;
4 * a <> 0 by A1,REAL_2:122;
hence thesis by A5,XCMPLX_1:88;
end;

theorem Th8:
a < 0 & a * x^2 + b * x + c <= 0 implies
(2 * a * x + b)^2 - delta(a,b,c) >= 0
proof
assume that
A1:   a < 0 and
A2:   a * x^2 + b * x + c <= 0;
A3:   4 * a < 0 by A1,SQUARE_1:24;
A4:   2 * a <> 0 by A1,SQUARE_1:24;
a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) <= 0 by A1,A2,Th1;
then (4 * a) * (a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a)) >= 0
by A3,REAL_2:121;
then ((4 * a) * (a * (x + b/(2 * a))^2)) - (4 * a) *
(delta(a,b,c)/(4 * a)) >= 0 by XCMPLX_1:40;
then ((4 * a) * a) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) >= 0 by XCMPLX_1:4;
then (4 * (a * a)) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) >= 0 by XCMPLX_1:4;
then ((2 * 2) * a^2) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) >= 0 by SQUARE_1:def 3;
then (2^2 * a^2) * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a))
>= 0
by SQUARE_1:def 3;
then (2 * a)^2 * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) >=
0
by SQUARE_1:68;
then ((2 * a) * (x + b/(2 * a)))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) >=
0
by SQUARE_1:68;
then ((2 * a) * x + (2 * a) * (b/(2 * a)))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) >= 0 by XCMPLX_1:8;
then A5:   (2 * a * x + b)^2 - (4 * a) * (delta(a,b,c)/(4 * a)) >= 0
by A4,XCMPLX_1:88;
4 * a <> 0 by A1,SQUARE_1:24;
hence thesis by A5,XCMPLX_1:88;
end;

theorem Th9:
a < 0 & a * x^2 + b * x + c < 0 implies (2 * a * x + b)^2 - delta(a,b,c) > 0
proof
assume that
A1:   a < 0 and
A2:   a * x^2 + b * x + c < 0;
A3:   4 * a < 0 by A1,SQUARE_1:24;
A4:   2 * a <> 0 by A1,SQUARE_1:24;
a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) < 0 by A1,A2,Th1;
then (4 * a) * (a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a)) > 0
by A3,REAL_2:122;
then ((4 * a) * (a * (x + b/(2 * a))^2)) - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by XCMPLX_1:40;
then ((4 * a) * a) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by XCMPLX_1:4;
then (4 * (a * a)) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by XCMPLX_1:4;
then ((2 * 2) * a^2) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by SQUARE_1:def 3;
then (2^2 * a^2) * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) >
0
by SQUARE_1:def 3;
then (2 * a)^2 * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) > 0
by SQUARE_1:68;
then ((2 * a) * (x + b/(2 * a)))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) > 0
by SQUARE_1:68;
then ((2 * a) * x + (2 * a) * (b/(2 * a)))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by XCMPLX_1:8;
then A5:   (2 * a * x + b)^2 - (4 * a) * (delta(a,b,c)/(4 * a)) > 0
by A4,XCMPLX_1:88;
4 * a <> 0 by A1,SQUARE_1:24;
hence thesis by A5,XCMPLX_1:88;
end;

theorem
( for x holds a * x^2 + b * x + c >= 0 ) & a > 0 implies delta(a,b,c) <= 0
proof
assume that
A1:   for x holds a * x^2 + b * x + c >= 0 and
A2:   a > 0;
A3:   2 * a <> 0 by A2,REAL_2:122;
now
a * (- b/(2 * a))^2 + b * (- b/(2 * a)) + c >= 0 by A1;
then (2 * a * (- b/(2 * a)) + b)^2 - delta(a,b,c) >= 0 by A2,Th6;
then (- (2 * a) * (b/(2 * a)) + b)^2 - delta(a,b,c) >= 0 by XCMPLX_1:175
;
then (- b + b)^2 - delta(a,b,c) >= 0 by A3,XCMPLX_1:88;
then 0 - delta(a,b,c) >= 0 by SQUARE_1:60,XCMPLX_0:def 6;
then - delta(a,b,c) >= 0 by XCMPLX_1:150;
hence thesis by REAL_1:26,50;
end;
hence thesis;
end;

theorem
( for x holds a * x^2 + b * x + c <= 0 ) & a < 0 implies delta(a,b,c) <= 0
proof
assume that
A1:   for x holds a * x^2 + b * x + c <= 0 and
A2:   a < 0;
A3:   2 * a <> 0 by A2,SQUARE_1:24;
now
a * (- b/(2 * a))^2 + b * (- b/(2 * a)) + c <= 0 by A1;
then (2 * a * (- b/(2 * a)) + b)^2 - delta(a,b,c) >= 0 by A2,Th8;
then (- (2 * a) * (b/(2 * a)) + b)^2 - delta(a,b,c) >= 0 by XCMPLX_1:175
;
then (- b + b)^2 - delta(a,b,c) >= 0 by A3,XCMPLX_1:88;
then 0 - delta(a,b,c) >= 0 by SQUARE_1:60,XCMPLX_0:def 6;
then - delta(a,b,c) >= 0 by XCMPLX_1:150;
hence thesis by REAL_1:26,50;
end;
hence thesis;
end;

theorem
( for x holds a * x^2 + b * x + c > 0 ) & a > 0 implies delta(a,b,c) < 0
proof
assume that
A1:   for x holds a * x^2 + b * x + c > 0 and
A2:   a > 0;
A3:   2 * a <> 0 by A2,REAL_2:122;
now
a * (- b/(2 * a))^2 + b * (- b/(2 * a)) + c > 0 by A1;
then (2 * a * (- b/(2 * a)) + b)^2 - delta(a,b,c) > 0 by A2,Th7;
then (- (2 * a) * (b/(2 * a)) + b)^2 - delta(a,b,c) > 0 by XCMPLX_1:175
;
then (- b + b)^2 - delta(a,b,c) > 0 by A3,XCMPLX_1:88;
then 0 - delta(a,b,c) > 0 by SQUARE_1:60,XCMPLX_0:def 6;
then - delta(a,b,c) > 0 by XCMPLX_1:150;
hence thesis by REAL_1:66;
end;
hence thesis;
end;

theorem
( for x holds a * x^2 + b * x + c < 0 ) & a < 0 implies delta(a,b,c) < 0
proof
assume that
A1:   for x holds a * x^2 + b * x + c < 0 and
A2:   a < 0;
A3:   2 * a <> 0 by A2,SQUARE_1:24;
now
a * (- b/(2 * a))^2 + b * (- b/(2 * a)) + c < 0 by A1;
then (2 * a * (- b/(2 * a)) + b)^2 - delta(a,b,c) > 0 by A2,Th9;
then (- (2 * a) * (b/(2 * a)) + b)^2 - delta(a,b,c) > 0 by XCMPLX_1:175
;
then (- b + b)^2 - delta(a,b,c) > 0 by A3,XCMPLX_1:88;
then 0 - delta(a,b,c) > 0 by SQUARE_1:60,XCMPLX_0:def 6;
then - delta(a,b,c) > 0 by XCMPLX_1:150;
hence thesis by REAL_1:26,50;
end;
hence thesis;
end;

theorem Th14:
a <> 0 & a * x^2 + b * x + c = 0 implies (2 * a * x + b)^2
- delta(a,b,c) = 0
proof
assume that
A1:   a <> 0 and
A2:   a * x^2 + b * x + c = 0;
A3:   2 * a <> 0 by A1,XCMPLX_1:6;
A4:   4 * a <> 0 by A1,XCMPLX_1:6;
a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) = 0 by A1,A2,Th1;
then (4 * a) * (a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a)) = 0;
then ((4 * a) * (a * (x + b/(2 * a))^2)) - (4 * a) *
(delta(a,b,c)/(4 * a)) = 0 by XCMPLX_1:40;
then ((4 * a) * a) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) = 0 by XCMPLX_1:4;
then (4 * (a * a)) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) = 0 by XCMPLX_1:4;
then ((2 * 2) * a^2) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) = 0 by SQUARE_1:def 3;
then (2^2 * a^2) * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) =
0
by SQUARE_1:def 3;
then (2 * a)^2 * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) = 0
by SQUARE_1:68;
then ((2 * a) * (x + b/(2 * a)))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) = 0
by SQUARE_1:68;
then ((2 * a) * x + (2 * a) * (b/(2 * a)))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) = 0 by XCMPLX_1:8;
then (2 * a * x + b)^2 - (4 * a) * (delta(a,b,c)/(4 * a)) = 0
by A3,XCMPLX_1:88;
hence thesis by A4,XCMPLX_1:88;
end;

Lm1:  a^2 = b^2 implies a = b or a = - b
proof
assume a^2 = b^2;
then a^2 - b^2 = 0 by XCMPLX_1:14;
then (a + b) * (a - b) = 0 by SQUARE_1:67;
then a + b = 0 or a - b = 0 by XCMPLX_1:6;
hence thesis by XCMPLX_0:def 6,XCMPLX_1:15;
end;

theorem
a <> 0 & delta(a,b,c) >= 0 & a * x^2 + b * x + c = 0 implies
x = (- b - sqrt delta(a,b,c))/(2 * a) or
x = (- b + sqrt delta(a,b,c))/(2 * a)
proof
assume that
A1:   a <> 0 and
A2:   delta(a,b,c) >= 0 and
A3:   a * x^2 + b * x + c = 0;
A4:   2 * a <> 0 by A1,XCMPLX_1:6;
(2 * a * x + b)^2 - delta(a,b,c) = 0 by A1,A3,Th14;
then (2 * a * x + b)^2 = delta(a,b,c) by XCMPLX_1:15;
then (2 * a * x + b)^2 = (sqrt delta(a,b,c))^2 by A2,SQUARE_1:def 4;
then 2 * a * x + b = sqrt delta(a,b,c) or 2 * a * x + b = - sqrt delta(a,
b,c)
by Lm1;
then 2 * a * x = - b + sqrt delta(a,b,c) or
2 * a * x = - b + - sqrt delta(a,b,c) by XCMPLX_1:137;
then x = (- b + sqrt delta(a,b,c))/(2 * a) or
2 * a * x = - b - sqrt delta(a,b,c) by A4,XCMPLX_0:def 8,XCMPLX_1:90;
hence thesis by A4,XCMPLX_1:90;
end;

theorem Th16:
a <> 0 & delta(a,b,c) >= 0 implies a * x^2 + b * x + c = a *
(x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a))
proof
assume that
A1:   a <> 0 and
A2:   delta(a,b,c) >= 0;
a * x^2 + b * x + c
= a * (x + b/(2 * a))^2 - 1 * (delta(a,b,c)/(4 * a)) by A1,Th1
.= a * (x + b/(2 * a))^2 - (a * (1/a)) * (delta(a,b,c)/(4 * a))
by A1,XCMPLX_1:107
.= a * (x + b/(2 * a))^2 - a * ((1/a) * (delta(a,b,c)/(4 * a))) by XCMPLX_1:4
.= a * ((x + b/(2 * a))^2 - (1/a) * (delta(a,b,c)/(4 * a))) by XCMPLX_1:40
.= a * ((x + b/(2 * a))^2 - (delta(a,b,c) * 1)/((4 * a) * a))
by XCMPLX_1:77
.= a * ((x + b/(2 * a))^2 - delta(a,b,c)/(4 * (a * a))) by XCMPLX_1:4
.= a * ((x + b/(2 * a))^2 - delta(a,b,c)/((2 * 2) * a^2)) by SQUARE_1:def 3
.= a * ((x + b/(2 * a))^2 - delta(a,b,c)/(2^2 * a^2)) by SQUARE_1:def 3
.= a * ((x + b/(2 * a))^2 - delta(a,b,c)/(2 * a)^2) by SQUARE_1:68
.= a * ((x + b/(2 * a))^2 - (sqrt delta(a,b,c))^2/(2 * a)^2) by A2,SQUARE_1:
def 4
.= a * ((x + b/(2 * a))^2 - (sqrt delta(a,b,c)/(2 * a))^2) by SQUARE_1:69
.= a * (((x + b/(2 * a)) - sqrt delta(a,b,c)/(2 * a)) *
((x + b/(2 * a)) + sqrt delta(a,b,c)/(2 * a))) by SQUARE_1:67
.= a * ((x + b/(2 * a)) - sqrt delta(a,b,c)/(2 * a)) *
((x + b/(2 * a)) + sqrt delta(a,b,c)/(2 * a)) by XCMPLX_1:4
.= a * ((x + b/(2 * a)) - sqrt delta(a,b,c)/(2 * a)) *
(x - (- b/(2 * a) - sqrt delta(a,b,c)/(2 * a))) by XCMPLX_1:157
.= a * (x + (b/(2 * a) - sqrt delta(a,b,c)/(2 * a))) *
(x - (- b/(2 * a) - sqrt delta(a,b,c)/(2 * a))) by XCMPLX_1:29
.= a * (x - (sqrt delta(a,b,c)/(2 * a) - b/(2 * a))) *
(x - (- b/(2 * a) - sqrt delta(a,b,c)/(2 * a))) by XCMPLX_1:38
.= a * (x - (- b/(2 * a) + sqrt delta(a,b,c)/(2 * a))) *
(x - (- b/(2 * a) - sqrt delta(a,b,c)/(2 * a))) by XCMPLX_0:def 8
.= a * (x - ((- b)/(2 * a) + sqrt delta(a,b,c)/(2 * a))) *
(x - (- b/(2 * a) - sqrt delta(a,b,c)/(2 * a))) by XCMPLX_1:188
.= a * (x - ((- b)/(2 * a) + sqrt delta(a,b,c)/(2 * a))) *
(x - ((- b)/(2 * a) - sqrt delta(a,b,c)/(2 * a))) by XCMPLX_1:188
.= a * (x - (- b + sqrt delta(a,b,c))/(2 * a)) *
(x - ((- b)/(2 * a) - sqrt delta(a,b,c)/(2 * a))) by XCMPLX_1:63
.= a * (x - (- b + sqrt delta(a,b,c))/(2 * a)) *
(x - (- b - sqrt delta(a,b,c))/(2 * a)) by XCMPLX_1:121;
hence thesis by XCMPLX_1:4;
end;

theorem Th17:
a < 0 & delta(a,b,c) > 0 implies
(- b + sqrt delta(a,b,c))/(2 * a) < (- b - sqrt delta(a,b,c))/(2 * a)
proof
assume that
A1:   a < 0 and
A2:   delta(a,b,c) > 0;
A3:   2 * a < 0 by A1,SQUARE_1:24;
sqrt delta(a,b,c) > 0 by A2,SQUARE_1:93;
then 2 * sqrt delta(a,b,c) > 0 by REAL_2:122;
then sqrt delta(a,b,c) + sqrt delta(a,b,c) > 0 by XCMPLX_1:11;
then sqrt delta(a,b,c) > - sqrt delta(a,b,c) by REAL_2:109;
then - b + sqrt delta(a,b,c) > - b + - sqrt delta(a,b,c) by REAL_1:53;
then - b + sqrt delta(a,b,c) > - b - sqrt delta(a,b,c) by XCMPLX_0:def 8;
hence thesis by A3,REAL_1:74;
end;

theorem
( a < 0 & delta(a,b,c) > 0 ) implies ( a * x^2 + b * x + c > 0 iff
(- b + sqrt delta(a,b,c))/(2 * a) < x &
x < (- b - sqrt delta(a,b,c))/(2 * a) )
proof
assume that
A1:   a < 0 and
A2:   delta(a,b,c) > 0;
thus a * x^2 + b * x + c > 0 implies
(- b + sqrt delta(a,b,c))/(2 * a) < x &
x < (- b - sqrt delta(a,b,c))/(2 * a)
proof
assume a * x^2 + b * x + c > 0;
then a * (x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a)) > 0 by A1,A2,Th16;
then a * ((x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a))) > 0 by XCMPLX_1:4;
then (x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a)) < 0/a by A1,REAL_2:178;
then x - (- b - sqrt delta(a,b,c))/(2 * a) > 0 &
x - (- b + sqrt delta(a,b,c))/(2 * a) < 0 or
x - (- b - sqrt delta(a,b,c))/(2 * a) < 0 &
x - (- b + sqrt delta(a,b,c))/(2 * a) > 0 by REAL_2:132;
then x > (- b - sqrt delta(a,b,c))/(2 * a) &
x < (- b + sqrt delta(a,b,c))/(2 * a) &
(- b + sqrt delta(a,b,c))/(2 * a) <
(- b - sqrt delta(a,b,c))/(2 * a) or
x < (- b - sqrt delta(a,b,c))/(2 * a) &
x > (- b + sqrt delta(a,b,c))/(2 * a)
by A1,A2,Th17,REAL_2:106,SQUARE_1:12;
hence thesis by AXIOMS:22;
end;
assume (- b + sqrt delta(a,b,c))/(2 * a) < x &
x < (- b - sqrt delta(a,b,c))/(2 * a);
then x - (- b + sqrt delta(a,b,c))/(2 * a) > 0 &
x - (- b - sqrt delta(a,b,c))/(2 * a) < 0
by REAL_2:105,SQUARE_1:11;
then (x - (- b + sqrt delta(a,b,c))/(2 * a)) *
(x - (- b - sqrt delta(a,b,c))/(2 * a)) < 0 by SQUARE_1:24;
then a * ((x - (- b + sqrt delta(a,b,c))/(2 * a)) *
(x - (- b - sqrt delta(a,b,c))/(2 * a))) > 0 by A1,REAL_2:122;
then a * (x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a)) > 0 by XCMPLX_1:4;
hence thesis by A1,A2,Th16;
end;

theorem
( a < 0 & delta(a,b,c) > 0 ) implies ( a * x^2 + b * x + c < 0 iff
x < (- b + sqrt delta(a,b,c))/(2 * a) or
x > (- b - sqrt delta(a,b,c))/(2 * a) )
proof
assume that
A1:   a < 0 and
A2:   delta(a,b,c) > 0;
thus a * x^2 + b * x + c < 0 implies
x < (- b + sqrt delta(a,b,c))/(2 * a)
or x > (- b - sqrt delta(a,b,c))/(2 * a)
proof
assume a * x^2 + b * x + c < 0;
then a * (x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a)) < 0 by A1,A2,Th16;
then a * ((x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a))) < 0 by XCMPLX_1:4;
then (x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a)) > 0/a
by A1,REAL_2:178;
then x - (- b - sqrt delta(a,b,c))/(2 * a) > 0 &
x - (- b + sqrt delta(a,b,c))/(2 * a) > 0 or
x - (- b - sqrt delta(a,b,c))/(2 * a) < 0 &
x - (- b + sqrt delta(a,b,c))/(2 * a) < 0 by SQUARE_1:26;
hence thesis by REAL_2:106,SQUARE_1:12;
end;
assume x < (- b + sqrt delta(a,b,c))/(2 * a) or
x > (- b - sqrt delta(a,b,c))/(2 * a);
then A3:  x - (- b + sqrt delta(a,b,c))/(2 * a) < 0 or
x - (- b - sqrt delta(a,b,c))/(2 * a) > 0 by REAL_2:105,SQUARE_1:11;
(- b + sqrt delta(a,b,c))/(2 * a) < (- b - sqrt delta(a,b,c))/(2 * a)
by A1,A2,Th17;
then x - (- b + sqrt delta(a,b,c))/(2 * a) >
x - (- b - sqrt delta(a,b,c))/(2 * a) by REAL_2:105;
then x - (- b + sqrt delta(a,b,c))/(2 * a) < 0 &
x - (- b - sqrt delta(a,b,c))/(2 * a) < 0 or
x - (- b - sqrt delta(a,b,c))/(2 * a) > 0 &
x - (- b + sqrt delta(a,b,c))/(2 * a) > 0 by A3,AXIOMS:22;
then (x - (- b + sqrt delta(a,b,c))/(2 * a)) *
(x - (- b - sqrt delta(a,b,c))/(2 * a)) > 0 by REAL_2:122;
then a * ((x - (- b + sqrt delta(a,b,c))/(2 * a)) *
(x - (- b - sqrt delta(a,b,c))/(2 * a))) < 0 by A1,SQUARE_1:24;
then a * (x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a)) < 0 by XCMPLX_1:4;
hence thesis by A1,A2,Th16;
end;

canceled 2;

theorem
a <> 0 & delta(a,b,c) = 0 & a * x^2 + b * x + c = 0 implies x = - b/(2 * a)
proof
assume that
A1:   a <> 0 and
A2:   delta(a,b,c) = 0 and
A3:   a * x^2 + b * x + c = 0;
A4:   2 * a <> 0 by A1,XCMPLX_1:6;
(2 * a * x + b)^2 - 0 = 0 by A1,A2,A3,Th14;
then 2 * a * x + b = 0 by SQUARE_1:73;
then 2 * a * x = - b by XCMPLX_0:def 6;
then x = (- b)/(2 * a) by A4,XCMPLX_1:90;
hence thesis by XCMPLX_1:188;
end;

theorem Th23:
a > 0 & (2 * a * x + b)^2 - delta(a,b,c) > 0 implies a * x^2 + b * x + c > 0
proof
assume that
A1:   a > 0 and
A2:   (2 * a * x + b)^2 - delta(a,b,c) > 0;
A3:   4 * a > 0 by A1,REAL_2:122;
A4:   2 * a <> 0 by A1,REAL_2:122;
4 * a <> 0 by A1,REAL_2:122;
then (2 * a * x + b)^2 - (4 * a) * (delta(a,b,c)/(4 * a)) > 0
by A2,XCMPLX_1:88;
then (2 * a * x + (2 * a) * (b/(2 * a)))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by A4,XCMPLX_1:88;
then ((2 * a) * (x + b/(2 * a)))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) > 0
by XCMPLX_1:8;
then (2 * a)^2 * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) > 0
by SQUARE_1:68;
then (2^2 * a^2) * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) >
0
by SQUARE_1:68;
then ((2 * 2) * a^2) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by SQUARE_1:def 3;
then (4 * (a * a)) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by SQUARE_1:def 3;
then ((4 * a) * a) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by XCMPLX_1:4;
then ((4 * a) * (a * (x + b/(2 * a))^2)) - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by XCMPLX_1:4;
then (4 * a) * (a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a)) > 0
by XCMPLX_1:40;
then a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) > 0/(4 * a)
by A3,REAL_2:178; hence thesis by A1,Th1;
end;

theorem
( a > 0 & delta(a,b,c) = 0 ) implies ( a * x^2 + b * x + c > 0 iff
x <> - b/(2 * a) )
proof
assume that
A1:   a > 0 and
A2:   delta(a,b,c) = 0;
A3:   2 * a <> 0 by A1,REAL_2:122;
thus a * x^2 + b * x + c > 0 implies x <> - b/(2 * a)
proof
assume a * x^2 + b * x + c > 0;
then (2 * a * x + b)^2 - 0 > 0 by A1,A2,Th7;
then (2 * a * x + b) * (2 * a * x + b) > 0 by SQUARE_1:def 3;
then 2 * a * x + b <> 0;
then 2 * a * x <> - b by XCMPLX_0:def 6;
then x <> (- b)/(2 * a) by A3,XCMPLX_1:88;
hence thesis by XCMPLX_1:188;
end;
assume x <> - b/(2 * a);
then x <> (- b)/(2 * a) by XCMPLX_1:188;
then 2 * a * x <> - b by A3,XCMPLX_1:90;
then 2 * a * x + b <> 0 by XCMPLX_0:def 6;
then (2 * a * x + b)^2 - delta(a,b,c) > 0 by A2,SQUARE_1:74;
hence thesis by A1,Th23;
end;

theorem Th25:
a < 0 & (2 * a * x + b)^2 - delta(a,b,c) > 0 implies a * x^2 + b * x + c < 0
proof
assume that
A1:   a < 0 and
A2:   (2 * a * x + b)^2 - delta(a,b,c) > 0;
A3:   4 * a < 0 by A1,SQUARE_1:24;
A4:   2 * a <> 0 by A1,SQUARE_1:24;
4 * a <> 0 by A1,SQUARE_1:24;
then (2 * a * x + b)^2 - (4 * a) * (delta(a,b,c)/(4 * a)) > 0
by A2,XCMPLX_1:88;
then (2 * a * x + (2 * a) * (b/(2 * a)))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by A4,XCMPLX_1:88;
then ((2 * a) * (x + b/(2 * a)))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) > 0
by XCMPLX_1:8;
then (2 * a)^2 * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) > 0
by SQUARE_1:68;
then (2^2 * a^2) * (x + b/(2 * a))^2 - (4 * a) * (delta(a,b,c)/(4 * a)) >
0
by SQUARE_1:68;
then ((2 * 2) * a^2) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by SQUARE_1:def 3;
then (4 * (a * a)) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by SQUARE_1:def 3;
then ((4 * a) * a) * (x + b/(2 * a))^2 - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by XCMPLX_1:4;
then ((4 * a) * (a * (x + b/(2 * a))^2)) - (4 * a) *
(delta(a,b,c)/(4 * a)) > 0 by XCMPLX_1:4;
then (4 * a) * (a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a)) > 0
by XCMPLX_1:40;
then a * (x + b/(2 * a))^2 - delta(a,b,c)/(4 * a) < 0/(4 * a)
by A3,REAL_2:178; hence thesis by A1,Th1;
end;

theorem
( a < 0 & delta(a,b,c) = 0 ) implies ( a * x^2 + b * x + c < 0 iff
x <> - b/(2 * a) )
proof
assume that
A1:   a < 0 and
A2:   delta(a,b,c) = 0;
A3:   2 * a <> 0 by A1,SQUARE_1:24;
thus a * x^2 + b * x + c < 0 implies x <> - b/(2 * a)
proof
assume
a * x^2 + b * x + c < 0;
then (2 * a * x + b)^2 - 0 > 0 by A1,A2,Th9;
then (2 * a * x + b) * (2 * a * x + b) > 0 by SQUARE_1:def 3;
then 2 * a * x + b <> 0;
then 2 * a * x <> - b by XCMPLX_0:def 6;
then x <> (- b)/(2 * a) by A3,XCMPLX_1:88;
hence thesis by XCMPLX_1:188;
end;
assume x <> - b/(2 * a);
then x <> (- b)/(2 * a) by XCMPLX_1:188;
then 2 * a * x <> - b by A3,XCMPLX_1:90;
then 2 * a * x + b <> 0 by XCMPLX_0:def 6;
then (2 * a * x + b)^2 - delta(a,b,c) > 0 by A2,SQUARE_1:74;
hence thesis by A1,Th25;
end;

theorem Th27:
a > 0 & delta(a,b,c) > 0 implies
(- b + sqrt delta(a,b,c))/(2 * a) > (- b - sqrt delta(a,b,c))/(2 * a)
proof
assume that
A1:   a > 0 and
A2:   delta(a,b,c) > 0;
A3:   2 * a > 0 by A1,REAL_2:122;
sqrt delta(a,b,c) > 0 by A2,SQUARE_1:93;
then 2 * sqrt delta(a,b,c) > 0 by REAL_2:122;
then sqrt delta(a,b,c) + sqrt delta(a,b,c) > 0 by XCMPLX_1:11;
then sqrt delta(a,b,c) > - sqrt delta(a,b,c) by REAL_2:109;
then - b + sqrt delta(a,b,c) > - b + - sqrt delta(a,b,c) by REAL_1:53;
then - b + sqrt delta(a,b,c) > - b - sqrt delta(a,b,c) by XCMPLX_0:def 8;
hence thesis by A3,REAL_1:73;
end;

theorem
( a > 0 & delta(a,b,c) > 0 ) implies ( a * x^2 + b * x + c < 0 iff
(- b - sqrt delta(a,b,c))/(2 * a) < x &
x < (- b + sqrt delta(a,b,c))/(2 * a) )
proof
assume that
A1:   a > 0 and
A2:   delta(a,b,c) > 0;
thus a * x^2 + b * x + c < 0 implies
(- b - sqrt delta(a,b,c))/(2 * a) < x &
x < (- b + sqrt delta(a,b,c))/(2 * a)
proof
assume a * x^2 + b * x + c < 0;
then a * (x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a)) < 0 by A1,A2,Th16;
then a * ((x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a))) < 0 by XCMPLX_1:4;
then (x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a)) < 0/a
by A1,REAL_2:178;
then x - (- b - sqrt delta(a,b,c))/(2 * a) > 0 &
x - (- b + sqrt delta(a,b,c))/(2 * a) < 0 or
x - (- b - sqrt delta(a,b,c))/(2 * a) < 0 &
x - (- b + sqrt delta(a,b,c))/(2 * a) > 0 by REAL_2:132;
then x > (- b - sqrt delta(a,b,c))/(2 * a) &
x < (- b + sqrt delta(a,b,c))/(2 * a) or
(- b + sqrt delta(a,b,c))/(2 * a) >
(- b - sqrt delta(a,b,c))/(2 * a) &
x < (- b - sqrt delta(a,b,c))/(2 * a) &
x > (- b + sqrt delta(a,b,c))/(2 * a)
by A1,A2,Th27,REAL_2:106,SQUARE_1:12;
hence thesis by AXIOMS:22;
end;
assume (- b - sqrt delta(a,b,c))/(2 * a) < x &
x < (- b + sqrt delta(a,b,c))/(2 * a);
then x - (- b - sqrt delta(a,b,c))/(2 * a) > 0 &
x - (- b + sqrt delta(a,b,c))/(2 * a) < 0
by REAL_2:105,SQUARE_1:11;
then (x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a)) < 0 by SQUARE_1:24;
then a * ((x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a))) < 0
by A1,SQUARE_1:24;
then a * (x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a)) < 0 by XCMPLX_1:4;
hence thesis by A1,A2,Th16;
end;

theorem
( a > 0 & delta(a,b,c) > 0 ) implies ( a * x^2 + b * x + c > 0 iff
x < (- b - sqrt delta(a,b,c))/(2 * a) or
x > (- b + sqrt delta(a,b,c))/(2 * a) )
proof
assume that
A1:   a > 0 and
A2:   delta(a,b,c) > 0;
thus a * x^2 + b * x + c > 0 implies
x < (- b - sqrt delta(a,b,c))/(2 * a) or
x > (- b + sqrt delta(a,b,c))/(2 * a)
proof
assume a * x^2 + b * x + c > 0;
then a * (x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a)) > 0 by A1,A2,Th16;
then a * ((x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a))) > 0 by XCMPLX_1:4;
then (x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a)) > 0/a
by A1,REAL_2:178;
then x - (- b - sqrt delta(a,b,c))/(2 * a) > 0 &
x - (- b + sqrt delta(a,b,c))/(2 * a) > 0 or
x - (- b - sqrt delta(a,b,c))/(2 * a) < 0 &
x - (- b + sqrt delta(a,b,c))/(2 * a) < 0 by SQUARE_1:26;
hence thesis by REAL_2:106,SQUARE_1:12;
end;
assume x < (- b - sqrt delta(a,b,c))/(2 * a) or
x > (- b + sqrt delta(a,b,c))/(2 * a);
then A3:   x - (- b - sqrt delta(a,b,c))/(2 * a) < 0 or
x - (- b + sqrt delta(a,b,c))/(2 * a) > 0
by REAL_2:105,SQUARE_1:11;
(- b + sqrt delta(a,b,c))/(2 * a) > (- b - sqrt delta(a,b,c))/(2 * a)
by A1,A2,Th27;
then x - (- b + sqrt delta(a,b,c))/(2 * a) <
x - (- b - sqrt delta(a,b,c))/(2 * a) by REAL_2:105;
then x - (- b - sqrt delta(a,b,c))/(2 * a) < 0 &
x - (- b + sqrt delta(a,b,c))/(2 * a) < 0 or
x - (- b - sqrt delta(a,b,c))/(2 * a) > 0 &
x - (- b + sqrt delta(a,b,c))/(2 * a) > 0 by A3,AXIOMS:22;
then (x - (- b + sqrt delta(a,b,c))/(2 * a)) *
(x - (- b - sqrt delta(a,b,c))/(2 * a)) > 0 by REAL_2:122;
then a * ((x - (- b + sqrt delta(a,b,c))/(2 * a)) *
(x - (- b - sqrt delta(a,b,c))/(2 * a))) > 0 by A1,REAL_2:122;
then a * (x - (- b - sqrt delta(a,b,c))/(2 * a)) *
(x - (- b + sqrt delta(a,b,c))/(2 * a)) > 0 by XCMPLX_1:4;
hence thesis by A1,A2,Th16;
end;

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