theorem Th15:
  T is IPC_theory iff CnIPC(T) = T
proof
  hereby
    assume
A1: T is IPC_theory;
A2: CnIPC(T) c= T
    by A1,Def15;
    T c= CnIPC(T) by Th12;
    hence CnIPC(T) = T by A2,XBOOLE_0:def 10;
  end;
  thus thesis;
end;
