defpred S1[ Nat] means for p being Polynomial of L st len p = L holds
even_part p is even ;
A1: 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 A2: 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
even_part p is even
let p be Polynomial of L; :: thesis: ( len p = k implies even_part p is even )
assume A3: len p = k ; :: thesis:
now :: thesis: ( ( k = 0 & even_part p is even ) or ( k <> 0 & even_part p is even ) )
per cases ( k = 0 or k <> 0 ) ;
case A4: k <> 0 ; :: thesis:
set LMp = Leading-Monomial p;
set g = Polynomial-Function (L,());
() + (p - ()) = (() + ()) + p by POLYNOM3:26
.= () + p
.= (0_. L) + p by POLYNOM3:29
.= p by POLYNOM3:28 ;
then A5: even_part p = () + (even_part (p - ())) by Th15;
consider t being Polynomial of L such that
A6: ( len t < len p & p = t + () & ( for n being Element of NAT st n < (len p) - 1 holds
t . n = p . n ) ) by ;
p - () = t + () by
.= t + (0_. L) by POLYNOM3:29
.= t by POLYNOM3:28 ;
then A7: even_part (p - ()) is even by A6, A2, A3;
now :: thesis: ( ( deg p is even & even_part () is even ) or ( deg p is odd & even_part () is even ) )
per cases ( deg p is even or deg p is odd ) ;
case A8: deg p is even ; :: thesis:
now :: thesis: for x being Element of L holds () . x = () . (- x)
let x be Element of L; :: thesis: () . x = () . (- x)
(len p) + 1 > 0 + 1 by ;
then len p >= 1 by NAT_1:13;
then A9: (len p) -' 1 = deg p by XREAL_1:233;
thus () . x = eval ((),x) by POLYNOM5:def 13
.= (p . ((len p) -' 1)) * (() . (x,((len p) -' 1))) by POLYNOM4:22
.= (p . ((len p) -' 1)) * (() . ((- x),((len p) -' 1))) by A9, A8, Th3
.= eval ((),(- x)) by POLYNOM4:22
.= () . (- x) by POLYNOM5:def 13 ; :: thesis: verum
end;
then Polynomial-Function (L,()) is even ;
hence even_part () is even by ; :: thesis: verum
end;
end;
end;
hence even_part p is even by A7, A5; :: thesis: verum
end;
end;
end;
hence even_part p is even ; :: thesis: verum
end;
hence S1[k] ; :: thesis: verum
end;
A10: for k being Nat holds S1[k] from consider k being Nat such that
A11: len p = k ;
thus even_part p is even by ; :: thesis: verum