theorem Th25:
  (p is even & q is even or p is odd & q is odd iff p*q is even)
proof
  reconsider pq=p*q as Element of Permutations(n) by MATRIX_9:39;
  now
    per cases;
    suppose
A1:   n<2;
      then pq is even by Lm3;
      hence thesis by A1,Lm3;
    end;
    suppose
      n>=2;
      then reconsider n2=n-2 as Nat by NAT_1:21;
      set K = the Fanoian non degenerated well-unital domRing-like
        commutative Ring;
      reconsider p9=p,q9=q,pq as Element of Permutations(n2+2);
      thus p is even&q is even or p is odd&q is odd implies p*q is even
      proof
        assume p is even & q is even or p is odd & q is odd;
        then sgn(p9,K)=1_K & sgn(q9,K)=1_K or sgn(p9,K)=-1_K & sgn(q9,K)=-1_K
        by Th23;
        then
A2:     sgn(p9,K)*sgn(q9,K)=1_K *1_K by VECTSP_1:10;
        sgn(pq,K)=1_K by A2,Th24;
        hence thesis by Th23;
      end;
      thus p*q is even implies ( p is even&q is even or p is odd&q is odd )
      proof
        assume p*q is even;
        then sgn(pq,K)=1_K by Th23;
        then
A3:     sgn(p9,K)*sgn(q9,K)=1_K by Th24;
        assume
A4:     not( p is even & q is even or p is odd & q is odd );
        now
          per cases by A4;
          suppose
A5:         p is even & q is odd;
            then
A6:         sgn(q9,K)=-1_K by Th23;
            sgn(p9,K)=1_K by A5,Th23;
            then sgn(p9,K)*sgn(q9,K)=-1_K by A6;
            hence thesis by A3,Th22;
          end;
          suppose
A7:         p is odd & q is even;
            then
A8:         sgn(q9,K)=1_K by Th23;
            sgn(p9,K)=-1_K by A7,Th23;
            then sgn(p9,K)*sgn(q9,K)=-1_K by A8;
            hence thesis by A3,Th22;
          end;
        end;
        hence thesis;
      end;
    end;
  end;
  hence thesis;
end;
