let a, b, c, x be Real; :: thesis: for n being Element of NAT st a <> 0 & not n is even & delta a,b,c >= 0 & Polynom a,b,c,(x |^ n) = 0 & 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 & not n is even & delta a,b,c >= 0 & Polynom a,b,c,(x |^ n) = 0 & 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: not n is even and
A3: ( delta a,b,c >= 0 & 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)) )
( x |^ n = ((- b) + (sqrt (delta a,b,c))) / (2 * a) or x |^ n = ((- b) - (sqrt (delta a,b,c))) / (2 * a) ) by A1, A3, POLYEQ_1:5;
hence ( 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, POWER:5; :: thesis: verum