let x, a, b, c be Real; :: thesis: for n being Element of NAT st a <> 0 & b / a < 0 & c / a > 0 & n is even & n >= 1 & delta (a,b,c) >= 0 & Polynom (a,b,c,(x |^ n)) = 0 & not x = n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a)) & not x = - (n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a))) & not x = n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a)) holds
x = - (n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a)))
let n be Element of NAT ; :: thesis: ( a <> 0 & b / a < 0 & c / a > 0 & n is even & n >= 1 & delta (a,b,c) >= 0 & Polynom (a,b,c,(x |^ n)) = 0 & not x = n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a)) & not x = - (n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a))) & not x = n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a)) implies x = - (n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a))) )
assume that
A1: a <> 0 and
A2: ( b / a < 0 & c / a > 0 & n is even & n >= 1 ) and
A3: delta (a,b,c) >= 0 and
A4: Polynom (a,b,c,(x |^ n)) = 0 ; :: thesis: ( x = n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a))) or x = n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a))) )

now :: thesis: ( x = n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a))) or x = n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a))) )

per cases ( x |^ n = ((- b) + (sqrt (delta (a,b,c)))) / (2 * a) or x |^ n = ((- b) - (sqrt (delta (a,b,c)))) / (2 * a) ) by A1, A3, A4, POLYEQ_1:5;

suppose x |^ n = ((- b) + (sqrt (delta (a,b,c)))) / (2 * a) ; :: thesis: ( x = n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a))) or x = n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a))) )

then ( x = n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a))) ) by A2, A3, Th1, Th4;
hence ( x = n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a))) or x = n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a))) ) ; :: thesis: verum

end;

suppose x |^ n = ((- b) - (sqrt (delta (a,b,c)))) / (2 * a) ; :: thesis: ( x = n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a))) or x = n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a))) )

then ( x = n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a))) ) by A2, A3, Th1, Th4;
hence ( x = n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a))) or x = n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a))) ) ; :: thesis: verum

end;

end;

end;

hence ( x = n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) + (sqrt (delta (a,b,c)))) / (2 * a))) or x = n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a)) or x = - (n -root (((- b) - (sqrt (delta (a,b,c)))) / (2 * a))) ) ; :: thesis: verum