reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;

theorem Th8:
  (n mod 4 = 1 & m mod 4 = 1) or (n mod 4 = 3 & m mod 4 = 3)
  implies n * m mod 4 = 1
  proof
    assume (n mod 4 = 1 & m mod 4 = 1) or (n mod 4 = 3 & m mod 4 = 3);
    then per cases;
    suppose
A1:   n mod 4 = 1 & m mod 4 = 1;
      then n * m mod 4 = (1*m) mod 4 by RADIX_2:3;
      hence thesis by A1;
    end;
    suppose n mod 4 = 3 & m mod 4 = 3;
      then n * m mod 4 = (3*m) mod 4 &  (3*m) mod 4 = 3*3 mod 4 & 3*3 = 4*2+1
      by RADIX_2:3;
      hence thesis by NUMBER02:16;
    end;
  end;
