set p = 0_. L;
take 0_. L ; :: thesis: 0_. L is odd
set f = Polynomial-Function (L,(0_. L));

A1: now :: thesis: for x being Element of L holds (Polynomial-Function (L,(0_. L))) . x = 0. L

let x be Element of L; :: thesis: (Polynomial-Function (L,(0_. L))) . x = 0. L
thus (Polynomial-Function (L,(0_. L))) . x = eval ((0_. L),x) by POLYNOM5:def 13
.= 0. L by POLYNOM4:17 ; :: thesis: verum

end;

now :: thesis: for x being Element of L holds (Polynomial-Function (L,(0_. L))) . (- x) = - ((Polynomial-Function (L,(0_. L))) . x)

let x be Element of L; :: thesis: (Polynomial-Function (L,(0_. L))) . (- x) = - ((Polynomial-Function (L,(0_. L))) . x)
thus (Polynomial-Function (L,(0_. L))) . (- x) = 0. L by A1
.= - (0. L) by RLVECT_1:12
.= - ((Polynomial-Function (L,(0_. L))) . x) by A1 ; :: thesis: verum

end;

hence 0_. L is odd by Def4; :: thesis: verum