reserve a,b,c,k,m,n for Nat;
reserve p for Prime;

theorem Th33:
  c <> 0 & c < b implies not (a*b+c) / b is integer
  proof
    assume that
A1: c <> 0 and
A2: c < b;
A3: (a*b+c) mod b = c mod b by NAT_D:21;
    c mod b = c by A2,NAT_D:24;
    then not b divides a*b+c by A1,A2,A3,INT_1:62;
    hence thesis by A2,WSIERP_1:17;
end;
