reserve i,j,k for Element of NAT;
reserve a,b for Element of REAL;
reserve r,s,t for Element of RAT+;
reserve i,j,k for Element of omega;

theorem Th7:
  NAT c< INT
proof
  0 in {0} by TARSKI:def 1;
  then [0,1] in [:{0},NAT:] by ZFMISC_1:87;
  then
A1: [0,1] in NAT \/ [:{0},NAT:] by XBOOLE_0:def 3;
A2: not [0,1] in NAT by ARYTM_3:32;
  [0,1] <> [0,0] by XTUPLE_0:1;
  then not [0,1] in {[0,0]} by TARSKI:def 1;
  then [0,1] in INT by A1,XBOOLE_0:def 5;
  hence thesis by A2,Lm5;
end;
