defpred S₁[ Nat] means for p being Polynomial of L st len p = L holds
odd_part p is odd ;

A12: now :: thesis: for k being Nat st ( for n being Nat st n < k holds
S₁[n] ) holds
S₁[k]

let k be Nat; :: thesis: ( ( for n being Nat st n < k holds
S₁[n] ) implies S₁[k] )
assume A13: for n being Nat st n < k holds
S₁[n] ; :: thesis: S₁[k]

now :: thesis: for p being Polynomial of L st len p = k holds
odd_part p is odd

let p be Polynomial of L; :: thesis: ( len p = k implies odd_part p is odd )
assume A14: len p = k ; :: thesis: odd_part p is odd

now :: thesis: ( ( k = 0 & odd_part p is odd ) or ( k <> 0 & odd_part p is odd ) )

per cases ( k = 0 or k <> 0 ) ;

case k = 0 ; :: thesis: odd_part p is odd

then p = 0_. L by A14, POLYNOM4:5;
hence odd_part p is odd by Th7; :: thesis: verum

end;

case A15: k <> 0 ; :: thesis: odd_part p is odd

set LMp = Leading-Monomial p;
set g = Polynomial-Function (L,(Leading-Monomial p));
(Leading-Monomial p) + (p - (Leading-Monomial p)) = ((Leading-Monomial p) + (- (Leading-Monomial p))) + p by POLYNOM3:26
.= ((Leading-Monomial p) - (Leading-Monomial p)) + p
.= (0_. L) + p by POLYNOM3:29
.= p by POLYNOM3:28 ;
then A16: odd_part p = (odd_part (Leading-Monomial p)) + (odd_part (p - (Leading-Monomial p))) by Th16;
consider t being Polynomial of L such that
A17: ( len t < len p & p = t + (Leading-Monomial p) & ( for n being Element of NAT st n < (len p) - 1 holds
t . n = p . n ) ) by A15, A14, POLYNOM4:16;
p - (Leading-Monomial p) = t + ((Leading-Monomial p) - (Leading-Monomial p)) by A17, POLYNOM3:26
.= t + (0_. L) by POLYNOM3:29
.= t by POLYNOM3:28 ;
then A18: odd_part (p - (Leading-Monomial p)) is odd by A17, A13, A14;

now :: thesis: ( ( deg p is odd & odd_part (Leading-Monomial p) is odd ) or ( deg p is even & odd_part (Leading-Monomial p) is odd ) )

per cases ( deg p is odd or deg p is even ) ;

case A19: deg p is odd ; :: thesis: odd_part (Leading-Monomial p) is odd

then A20: odd_part (Leading-Monomial p) = Leading-Monomial p by Th20;

now :: thesis: for x being Element of L holds - ((Polynomial-Function (L,(Leading-Monomial p))) . x) = (Polynomial-Function (L,(Leading-Monomial p))) . (- x)

let x be Element of L; :: thesis: - ((Polynomial-Function (L,(Leading-Monomial p))) . x) = (Polynomial-Function (L,(Leading-Monomial p))) . (- x)
(len p) + 1 > 0 + 1 by A14, A15, XREAL_1:8;
then len p >= 1 by NAT_1:13;
then A21: (len p) -' 1 = deg p by XREAL_1:233;
thus - ((Polynomial-Function (L,(Leading-Monomial p))) . x) = - (eval ((Leading-Monomial p),x)) by POLYNOM5:def 13
.= - ((p . ((len p) -' 1)) * ((power L) . (x,((len p) -' 1)))) by POLYNOM4:22
.= (p . ((len p) -' 1)) * (- ((power L) . (x,((len p) -' 1)))) by VECTSP_1:8
.= (p . ((len p) -' 1)) * ((power L) . ((- x),((len p) -' 1))) by A21, A19, Th4
.= eval ((Leading-Monomial p),(- x)) by POLYNOM4:22
.= (Polynomial-Function (L,(Leading-Monomial p))) . (- x) by POLYNOM5:def 13 ; :: thesis: verum

end;

hence odd_part (Leading-Monomial p) is odd by A20, Def4; :: thesis: verum

end;

case deg p is even ; :: thesis: odd_part (Leading-Monomial p) is odd

then odd_part (Leading-Monomial p) = 0_. L by Th19;
hence odd_part (Leading-Monomial p) is odd ; :: thesis: verum

end;

end;

end;

hence odd_part p is odd by A18, A16; :: thesis: verum

end;

end;

end;

hence odd_part p is odd ; :: thesis: verum

end;

hence S₁[k] ; :: thesis: verum

end;

A22: for k being Nat holds S₁[k] from NAT_1:sch 4(A12);
consider k being Nat such that
A23: len p = k ;
thus odd_part p is odd by A23, A22; :: thesis: verum