let a0, a1, a2, a3 be complex number ; :: thesis: ((((((1_root_of_quartic a0,a1,a2,a3) * (2_root_of_quartic a0,a1,a2,a3)) + ((1_root_of_quartic a0,a1,a2,a3) * (3_root_of_quartic a0,a1,a2,a3))) + ((1_root_of_quartic a0,a1,a2,a3) * (4_root_of_quartic a0,a1,a2,a3))) + ((2_root_of_quartic a0,a1,a2,a3) * (3_root_of_quartic a0,a1,a2,a3))) + ((2_root_of_quartic a0,a1,a2,a3) * (4_root_of_quartic a0,a1,a2,a3))) + ((3_root_of_quartic a0,a1,a2,a3) * (4_root_of_quartic a0,a1,a2,a3)) = a2

per cases ( ((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3) = 0 or ((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3) <> 0 ) ;

suppose A1: ((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3) = 0 ; :: thesis: ((((((1_root_of_quartic a0,a1,a2,a3) * (2_root_of_quartic a0,a1,a2,a3)) + ((1_root_of_quartic a0,a1,a2,a3) * (3_root_of_quartic a0,a1,a2,a3))) + ((1_root_of_quartic a0,a1,a2,a3) * (4_root_of_quartic a0,a1,a2,a3))) + ((2_root_of_quartic a0,a1,a2,a3) * (3_root_of_quartic a0,a1,a2,a3))) + ((2_root_of_quartic a0,a1,a2,a3) * (4_root_of_quartic a0,a1,a2,a3))) + ((3_root_of_quartic a0,a1,a2,a3) * (4_root_of_quartic a0,a1,a2,a3)) = a2

A2: ex p, r, s1 being complex number st
( p = ((8 * a2) - (3 * (a3 |^ 2))) / 32 & r = ((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024 & s1 = 2 -root ((p |^ 2) - r) & 1_root_of_quartic a0,a1,a2,a3 = (2 -root (- (2 * (p - s1)))) - (a3 / 4) ) by A1, Def5;
A3: ex p, r, s1 being complex number st
( p = ((8 * a2) - (3 * (a3 |^ 2))) / 32 & r = ((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024 & s1 = 2 -root ((p |^ 2) - r) & 2_root_of_quartic a0,a1,a2,a3 = (- (2 -root (- (2 * (p - s1))))) - (a3 / 4) ) by A1, Def6;
A4: ex p, r, s1 being complex number st
( p = ((8 * a2) - (3 * (a3 |^ 2))) / 32 & r = ((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024 & s1 = 2 -root ((p |^ 2) - r) & 3_root_of_quartic a0,a1,a2,a3 = (2 -root (- (2 * (p + s1)))) - (a3 / 4) ) by A1, Def7;
A5: ex p, r, s1 being complex number st
( p = ((8 * a2) - (3 * (a3 |^ 2))) / 32 & r = ((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024 & s1 = 2 -root ((p |^ 2) - r) & 4_root_of_quartic a0,a1,a2,a3 = (- (2 -root (- (2 * (p + s1))))) - (a3 / 4) ) by A1, Def8;
set p = ((8 * a2) - (3 * (a3 |^ 2))) / 32;
set r = ((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024;
set s1 = 2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024));
set t1 = 2 -root (- (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) - (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024))))));
set t2 = 2 -root (- (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) + (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024))))));
thus ((((((1_root_of_quartic a0,a1,a2,a3) * (2_root_of_quartic a0,a1,a2,a3)) + ((1_root_of_quartic a0,a1,a2,a3) * (3_root_of_quartic a0,a1,a2,a3))) + ((1_root_of_quartic a0,a1,a2,a3) * (4_root_of_quartic a0,a1,a2,a3))) + ((2_root_of_quartic a0,a1,a2,a3) * (3_root_of_quartic a0,a1,a2,a3))) + ((2_root_of_quartic a0,a1,a2,a3) * (4_root_of_quartic a0,a1,a2,a3))) + ((3_root_of_quartic a0,a1,a2,a3) * (4_root_of_quartic a0,a1,a2,a3)) = ((((((4 + 1) + 1) * a3) * a3) / 16) - ((2 -root (- (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) - (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024))))))) * (2 -root (- (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) - (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024))))))))) - ((2 -root (- (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) + (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024))))))) * (2 -root (- (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) + (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)))))))) by A2, A3, A4, A5
.= ((((3 * a3) * a3) / 8) - ((2 -root (- (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) - (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024))))))) |^ 2)) - ((2 -root (- (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) + (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024))))))) * (2 -root (- (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) + (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)))))))) by Th1
.= ((((3 * a3) * a3) / 8) - (- (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) - (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024))))))) - ((2 -root (- (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) + (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024))))))) * (2 -root (- (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) + (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)))))))) by Th8
.= ((((3 * a3) * a3) / 8) + (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) - (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)))))) - ((2 -root (- (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) + (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024))))))) |^ 2) by Th1
.= ((((3 * a3) * a3) / 8) + (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) - (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)))))) - (- (2 * ((((8 * a2) - (3 * (a3 |^ 2))) / 32) + (2 -root (((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)))))) by Th8
.= (((3 * a3) * a3) / 8) + (((8 * a2) - (3 * (a3 |^ 2))) / 8)
.= (((3 * a3) * a3) / 8) + (((8 * a2) - (3 * (a3 * a3))) / 8) by Th1
.= a2 ; :: thesis: verum

end;

suppose A6: ((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3) <> 0 ; :: thesis: ((((((1_root_of_quartic a0,a1,a2,a3) * (2_root_of_quartic a0,a1,a2,a3)) + ((1_root_of_quartic a0,a1,a2,a3) * (3_root_of_quartic a0,a1,a2,a3))) + ((1_root_of_quartic a0,a1,a2,a3) * (4_root_of_quartic a0,a1,a2,a3))) + ((2_root_of_quartic a0,a1,a2,a3) * (3_root_of_quartic a0,a1,a2,a3))) + ((2_root_of_quartic a0,a1,a2,a3) * (4_root_of_quartic a0,a1,a2,a3))) + ((3_root_of_quartic a0,a1,a2,a3) * (4_root_of_quartic a0,a1,a2,a3)) = a2

A7: ex p, q, r, s1, s2, s3 being complex number st
( p = ((8 * a2) - (3 * (a3 |^ 2))) / 32 & q = (((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64 & r = ((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024 & s1 = 2 -root (1_root_of_cubic (- (q |^ 2)),((p |^ 2) - r),(2 * p)) & s2 = 2 -root (2_root_of_cubic (- (q |^ 2)),((p |^ 2) - r),(2 * p)) & s3 = - (q / (s1 * s2)) & 1_root_of_quartic a0,a1,a2,a3 = ((s1 + s2) + s3) - (a3 / 4) ) by A6, Def5;
A8: ex p, q, r, s1, s2, s3 being complex number st
( p = ((8 * a2) - (3 * (a3 |^ 2))) / 32 & q = (((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64 & r = ((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024 & s1 = 2 -root (1_root_of_cubic (- (q |^ 2)),((p |^ 2) - r),(2 * p)) & s2 = 2 -root (2_root_of_cubic (- (q |^ 2)),((p |^ 2) - r),(2 * p)) & s3 = - (q / (s1 * s2)) & 2_root_of_quartic a0,a1,a2,a3 = (((- s1) - s2) + s3) - (a3 / 4) ) by A6, Def6;
A9: ex p, q, r, s1, s2, s3 being complex number st
( p = ((8 * a2) - (3 * (a3 |^ 2))) / 32 & q = (((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64 & r = ((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024 & s1 = 2 -root (1_root_of_cubic (- (q |^ 2)),((p |^ 2) - r),(2 * p)) & s2 = 2 -root (2_root_of_cubic (- (q |^ 2)),((p |^ 2) - r),(2 * p)) & s3 = - (q / (s1 * s2)) & 3_root_of_quartic a0,a1,a2,a3 = (((- s1) + s2) - s3) - (a3 / 4) ) by A6, Def7;
A10: ex p, q, r, s1, s2, s3 being complex number st
( p = ((8 * a2) - (3 * (a3 |^ 2))) / 32 & q = (((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64 & r = ((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024 & s1 = 2 -root (1_root_of_cubic (- (q |^ 2)),((p |^ 2) - r),(2 * p)) & s2 = 2 -root (2_root_of_cubic (- (q |^ 2)),((p |^ 2) - r),(2 * p)) & s3 = - (q / (s1 * s2)) & 4_root_of_quartic a0,a1,a2,a3 = ((s1 - s2) - s3) - (a3 / 4) ) by A6, Def8;
set p = ((8 * a2) - (3 * (a3 |^ 2))) / 32;
set q = (((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64;
set r = ((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024;
set s1 = 2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)));
set s2 = 2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)));
set s3 = - (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) / ((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))));
A11: ((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) * ((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) <> 0 by A6;
A12: (2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) = (2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) |^ 2 by Th1
.= 1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)) by Th8 ;
A13: (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) = (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) |^ 2 by Th1
.= 2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)) by Th8 ;
A14: (((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))) * ((2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))))) * (3_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))) = - (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)) by A12, A13, Th19;
then A15: ((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))) * ((2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))) <> 0 by A11, Th1;
A16: (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) / ((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))))) * (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) / ((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))))) = ((- ((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64)) / ((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))))) * (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) / ((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))))) by XCMPLX_1:188
.= ((- ((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64)) / ((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))))) * ((- ((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64)) / ((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))))) by XCMPLX_1:188
.= ((- ((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64)) * (- ((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64))) / (((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))) * ((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))))) by XCMPLX_1:77
.= (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) * ((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64)) / (((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))) * ((2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))))
.= ((3_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))) * (((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))) * ((2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))))) / (((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))) * ((2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))))) by A14, Th1
.= 3_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)) by A15, XCMPLX_1:90 ;
thus ((((((1_root_of_quartic a0,a1,a2,a3) * (2_root_of_quartic a0,a1,a2,a3)) + ((1_root_of_quartic a0,a1,a2,a3) * (3_root_of_quartic a0,a1,a2,a3))) + ((1_root_of_quartic a0,a1,a2,a3) * (4_root_of_quartic a0,a1,a2,a3))) + ((2_root_of_quartic a0,a1,a2,a3) * (3_root_of_quartic a0,a1,a2,a3))) + ((2_root_of_quartic a0,a1,a2,a3) * (4_root_of_quartic a0,a1,a2,a3))) + ((3_root_of_quartic a0,a1,a2,a3) * (4_root_of_quartic a0,a1,a2,a3)) = (- (2 * ((((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))) + ((2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))))) + ((- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) / ((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))))) * (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) / ((2 -root (1_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32)))) * (2 -root (2_root_of_cubic (- (((((8 * a1) - ((4 * a2) * a3)) + (a3 |^ 3)) / 64) |^ 2)),(((((8 * a2) - (3 * (a3 |^ 2))) / 32) |^ 2) - (((((256 * a0) - ((64 * a3) * a1)) + ((16 * (a3 |^ 2)) * a2)) - (3 * (a3 |^ 4))) / 1024)),(2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))))))))) + (((3 * a3) * a3) / 8) by A7, A8, A9, A10
.= (- (2 * (- (2 * (((8 * a2) - (3 * (a3 |^ 2))) / 32))))) + (((3 * a3) * a3) / 8) by Th18, A12, A13, A16
.= ((4 * ((8 * a2) - (3 * (a3 |^ 2)))) / 32) + (((3 * a3) * a3) / 8)
.= ((4 * ((8 * a2) - (3 * (a3 * a3)))) / 32) + (((3 * a3) * a3) / 8) by Th1
.= a2 ; :: thesis: verum

end;

end;