reserve i,j,n,n1,n2,m,k,u for Nat,
        r,r1,r2 for Real,
        x,y for Integer,
        a,b for non trivial Nat;

theorem Th24:
  x <= 0 implies sgn(x)*Py(a,|.x.|) = -Py(a,|.x.|)
proof
A1: Py(a,|.0.|)=0 by Th6;
  assume x<=0;
  then x=0 or x <0;
  then sgn(x) = -1 or (sgn(x)=0 & x=0 ) by ABSVALUE:def 2;
  hence thesis by A1;
end;
