let f, g be Polynomial of F_Complex; :: thesis: ( f *' g is Hurwitz iff ( f is Hurwitz & g is Hurwitz ) )

A1: now

assume that
A2: f is Hurwitz and
A3: g is Hurwitz ; :: thesis: f *' g is Hurwitz

now

let z be Element of F_Complex; :: thesis: ( z is_a_root_of f *' g implies Re b₁ < 0 )
assume z is_a_root_of f *' g ; :: thesis: Re b₁ < 0
then A4: 0. F_Complex = eval ((f *' g),z) by POLYNOM5:def 6
.= (eval (f,z)) * (eval (g,z)) by POLYNOM4:27 ;

per cases ( eval (f,z) = 0. F_Complex or eval (g,z) = 0. F_Complex ) by A4, VECTSP_1:44;

suppose eval (f,z) = 0. F_Complex ; :: thesis: Re b₁ < 0

then z is_a_root_of f by POLYNOM5:def 6;
hence Re z < 0 by A2, Def8; :: thesis: verum

end;

suppose eval (g,z) = 0. F_Complex ; :: thesis: Re b₁ < 0

then z is_a_root_of g by POLYNOM5:def 6;
hence Re z < 0 by A3, Def8; :: thesis: verum

end;

end;

end;

hence f *' g is Hurwitz by Def8; :: thesis: verum

end;

now

assume A5: f *' g is Hurwitz ; :: thesis: ( f is Hurwitz & g is Hurwitz )

now

let z be Element of F_Complex; :: thesis: ( z is_a_root_of f implies Re z < 0 )
assume z is_a_root_of f ; :: thesis: Re z < 0
then z is_a_root_of f *' g by Lm13;
hence Re z < 0 by A5, Def8; :: thesis: verum

end;

hence f is Hurwitz by Def8; :: thesis: g is Hurwitz

now

let z be Element of F_Complex; :: thesis: ( z is_a_root_of g implies Re z < 0 )
assume z is_a_root_of g ; :: thesis: Re z < 0
then z is_a_root_of f *' g by Lm13;
hence Re z < 0 by A5, Def8; :: thesis: verum

end;

hence g is Hurwitz by Def8; :: thesis: verum

end;

hence ( f *' g is Hurwitz iff ( f is Hurwitz & g is Hurwitz ) ) by A1; :: thesis: verum