let L be non empty well-unital doubleLoopStr ; :: thesis: for p being Polynomial of L st deg p is odd holds
odd_part () = Leading-Monomial p

let p be Polynomial of L; :: thesis: ( deg p is odd implies odd_part () = Leading-Monomial p )
assume A1: deg p is odd ; :: thesis:
set LMp = Leading-Monomial p;
set o = odd_part ();
A2: dom () = NAT by FUNCT_2:def 1
.= dom () by FUNCT_2:def 1 ;
now :: thesis: for x being object st x in dom () holds
() . x = () . x
let x be object ; :: thesis: ( x in dom () implies () . x = () . x )
assume x in dom () ; :: thesis: () . x = () . x
then reconsider i = x as Element of NAT by FUNCT_2:def 1;
now :: thesis: ( ( len p = 0 & () . x = () . x ) or ( len p <> 0 & () . x = () . x ) )
per cases ( len p = 0 or len p <> 0 ) ;
case len p <> 0 ; :: thesis: () . x = () . x
then (len p) + 1 > 0 + 1 by XREAL_1:8;
then len p >= 1 by NAT_1:13;
then A3: (len p) -' 1 = deg p by XREAL_1:233;
now :: thesis: ( ( i is odd & () . i = () . i ) or ( i is even & () . i = () . i ) )
per cases ( i is odd or i is even ) ;
case i is odd ; :: thesis: () . i = () . i
hence (odd_part ()) . i = () . i by Def2; :: thesis: verum
end;
case A4: i is even ; :: thesis: () . i = () . i
hence () . i = 0. L by
.= () . i by ;
:: thesis: verum
end;
end;
end;
hence (odd_part ()) . x = () . x ; :: thesis: verum
end;
end;
end;
hence (odd_part ()) . x = () . x ; :: thesis: verum
end;
hence odd_part () = Leading-Monomial p by ; :: thesis: verum