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 Th32:
  sgn(x+y)*Py(a,|.x+y.|),sgn(x-y)*Py(a,|.x-y.|)
    are_congruent_mod Px(a,|.x.|)
proof
  x+y = 2*x+(-x+y);
  then
A1: sgn(x+y)*Py(a,|.x+y.|),-sgn(-x+y)*Py(a,|.-x+y.|)
  are_congruent_mod Px(a,|.x.|) by Th30;
A2: |. -x+y.|= |.-(-x+y).| by COMPLEX1:52;
  -x+y= (-1)*(x-y);
  then sgn(-x+y) = sgn(x-y)*sgn(-1) & sgn(-1)=-1 by ABSVALUE:18,def 2;
  hence thesis by A2,A1;
end;
