let x, a, b, c be Real; :: thesis: ( a > 0 & delta (a,b,c) < 0 implies ((a * (x ^2)) + (b * x)) + c > 0 )
assume that
A1: a > 0 and
A2: delta (a,b,c) < 0 ; :: thesis: ((a * (x ^2)) + (b * x)) + c > 0
( - (delta (a,b,c)) > - 0 & 4 * a > 0 ) by A1, A2, XREAL_1:26, XREAL_1:129;
then (- (delta (a,b,c))) / (4 * a) > 0 by XREAL_1:139;
then - ((delta (a,b,c)) / (4 * a)) > 0 by XCMPLX_1:187;
then A3: (a * ((x + (b / (2 * a))) ^2)) + (- ((delta (a,b,c)) / (4 * a))) > a * ((x + (b / (2 * a))) ^2) by XREAL_1:29;
(x + (b / (2 * a))) ^2 >= 0 by XREAL_1:63;
then a * ((x + (b / (2 * a))) ^2) >= 0 by A1, XREAL_1:127;
then (a * ((x + (b / (2 * a))) ^2)) - ((delta (a,b,c)) / (4 * a)) > 0 by A3, XXREAL_0:2;
hence ((a * (x ^2)) + (b * x)) + c > 0 by A1, Th1; :: thesis: verum