let a, b, c be Real; ( delta (a,b,c) < 0 implies for x being Real holds not Polynom (a,b,c,x) = 0 )
set e = a * c;
assume
delta (a,b,c) < 0
; for x being Real holds not Polynom (a,b,c,x) = 0
then
(b ^2) - ((4 * a) * c) < 0
by QUIN_1:def 1;
then A1:
((b ^2) - (4 * (a * c))) * (4 ") < 0
by XREAL_1:132;
given y being Real such that A2:
Polynom (a,b,c,y) = 0
; contradiction
set t = ((a ^2) * (y ^2)) + ((a * b) * y);
a * (((a * (y ^2)) + (b * y)) + c) = a * 0
by A2;
then
((((a ^2) * (y ^2)) + ((a * b) * y)) + ((b ^2) / 4)) - (((b ^2) * (4 ")) - ((4 * (a * c)) * (4 "))) = 0
;
then A3:
((a * y) + (b / 2)) ^2 = ((b ^2) - (4 * (a * c))) * (4 ")
;
then
(a * y) + (b / 2) > 0
by A1, XREAL_1:133;
hence
contradiction
by A3, A1, XREAL_1:133; verum