reserve r,s,t,x9,y9,z9,p,q for Element of RAT+;
reserve x,y,z for Element of REAL+;

theorem
  for A being Subset of REAL+ st 0 in A & for x,y st x in A & y = one
  holds x + y in A holds omega c= A
proof
  let A be Subset of REAL+;
  defpred P[set] means $1 in A;
  assume that
A1: P[0] and
A2: for x,y st x in A & y = one holds x + y in A;
  let e be object;
  assume e in omega;
  then reconsider a = e as natural Ordinal;
A3: for a being natural Ordinal st P[a] holds P[succ a]
  proof
    reconsider rone = one as Element of REAL+ by Th1;
    let a be natural Ordinal;
    assume
A4: a in A;
    reconsider i = a as Element of omega by ORDINAL1:def 12;
A5: a in omega by ORDINAL1:def 12;
    then a in RAT+ by Lm5;
    then reconsider x = a as Element of REAL+ by Th1;
    consider x9, y9 being Element of RAT+ such that
A6: x = x9 & rone = y9 and
A7: x + rone = x9 + y9 by A5,Lm5,Lm47;
    x9 + y9 = i +^ 1 by A6,Lm45
      .= succ i by ORDINAL2:31;
    hence thesis by A2,A4,A7;
  end;
  P[a] from ORDINAL2:sch 17(A1,A3);
  hence thesis;
end;
