set x = - 1;
let a1, a2, a3, a4, a5, b1, b2, b3, b4, b5 be Real; :: thesis: ( ( for x being Real holds Polynom (a1,a2,a3,a4,a5,x) = Polynom (b1,b2,b3,b4,b5,x) ) implies ( a5 = b5 & ((a1 - a2) + a3) - a4 = ((b1 - b2) + b3) - b4 & ((a1 + a2) + a3) + a4 = ((b1 + b2) + b3) + b4 ) )
A1: ( 0 |^ 3 = 0 & 0 |^ 4 = 0 ) by NEWTON:11;
assume A2: for x being Real holds Polynom (a1,a2,a3,a4,a5,x) = Polynom (b1,b2,b3,b4,b5,x) ; :: thesis: ( a5 = b5 & ((a1 - a2) + a3) - a4 = ((b1 - b2) + b3) - b4 & ((a1 + a2) + a3) + a4 = ((b1 + b2) + b3) + b4 )
then A3: Polynom (a1,a2,a3,a4,a5,(- 1)) = Polynom (b1,b2,b3,b4,b5,(- 1)) ;
A5: ( (- 1) |^ 3 = ((- 1) ^2) * (- 1) & ((- 1) |^ 3) * (- 1) = (- 1) |^ 4 ) by Th4;
( Polynom (a1,a2,a3,a4,a5,0) = Polynom (b1,b2,b3,b4,b5,0) & Polynom (a1,a2,a3,a4,a5,1) = Polynom (b1,b2,b3,b4,b5,1) ) by A2;
hence ( a5 = b5 & ((a1 - a2) + a3) - a4 = ((b1 - b2) + b3) - b4 & ((a1 + a2) + a3) + a4 = ((b1 + b2) + b3) + b4 ) by A1, A3, A5; :: thesis: verum