theorem
  a = i mod p & b = j mod p implies a-b = (i-j) mod p
  proof
    assume A1: a = i mod p & b = j mod p; then
    -b = (p-j) mod p by Th16;
    then a-b = (i+(p-j)) mod p by A1,Th15
    .= (i-j+1*p) mod p;
    hence thesis by NAT_D:61;
  end;
