theorem Th29:
  x in omega or ex i,j st x = [i,j] & i,j are_coprime & j <> {} & j <> 1
proof
  assume not x in omega;
  then
A1: x in RATplus \ the set of all [k,1] by XBOOLE_0:def 3;
  then
A2: not x in the set of all [k,1] by XBOOLE_0:def 5;
  x in RATplus by A1;
  then consider a,b being Element of omega such that
A3: x = [a,b] & a,b are_coprime & b <> {};
  [a,1] in the set of all [k,1];
  hence thesis by A2,A3;
end;
