reserve x for set;
reserve a, b, c for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p, q for Rational;
reserve s1, s2 for Real_Sequence;

theorem Th85:
  a >= 1 & b >= 0 implies a #R b >= 1
proof
  assume that
A1: a >= 1 and
A2: b >= 0;
  consider s being Rational_Sequence such that
A3: s is convergent and
A4: b = lim s and
A5: for n holds s.n>=b by Th68;
A6: now
    let n;
A7: a #Q s .n = a #Q (s.n) by Def5;
    s.n>=b by A5;
    hence a #Q s .n >= 1 by A1,A2,A7,Th60;
  end;
  a #Q s is convergent by A1,A3,Th69;
  then a #R b = lim (a #Q s) by A1,A3,A4,Def6;
  hence thesis by A1,A3,A6,Th1,Th69;
end;
