reserve A,B,C for Ordinal;
reserve a,b,c,d for natural Ordinal;
reserve l,m,n for natural Ordinal;
reserve i,j,k for Element of omega;
reserve x,y,z for Element of RAT+;

theorem Th33:
  [i,j] in RAT+ iff i,j are_coprime & j <> {} & j <> 1
proof
A1: not [i,j] in omega by Th32;
  hereby
    assume [i,j] in RAT+;
    then
A2: [i,j] in RATplus \ the set of all [k,1] by A1,XBOOLE_0:def 3;
    hence i,j are_coprime & j <> {} by Lm5;
    not [i,j] in the set of all [k,1] by A2,XBOOLE_0:def 5;
    hence j <> 1;
  end;
  assume i,j are_coprime & j <> {};
  then
A3: [i,j] in RATplus;
  assume j <> 1;
  then not ex k st [i,j] = [k,1] by XTUPLE_0:1;
  then not [i,j] in the set of all [k,1];
  then [i,j] in RATplus \ the set of all [k,1] by A3,XBOOLE_0:def 5;
  hence thesis by XBOOLE_0:def 3;
end;
