reserve n for Nat,
        i,j,i1,i2,i3,i4,i5,i6 for Element of n,
        p,q,r for n-element XFinSequence of NAT;
reserve i,j,n,n1,n2,m,k,l,u,e,p,t for Nat,
        a,b for non trivial Nat,
        x,y for Integer,
        r,q for Real;

theorem Th25:
  Py(a,2*n) = 2* Py(a,n)*Px(a,n)
proof
A1:Py(a,0)=0 by HILB10_1:3;
  per cases;
  suppose n=0;
    hence thesis by A1;
  end;
  suppose n>0;
    then
A2:   sgn(n)=1 & sgn(n+n) =1 by ABSVALUE:def 2;
    sgn(n+n)*Py(a,|.n+n.|) = Px(a,|.n.|)*sgn(n)*Py(a,|.n.|) +
    sgn(n)*Py(a,|.n.|)*Px(a,|.n.|) by HILB10_1:22;
    hence thesis by A2;
  end;
end;
