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 Th83:
  a>1 & c>b implies a #R c > a #R b
proof
  assume that
A1: a>1 and
A2: c>b;
  consider p such that
A3: b<p and
A4: p<c by A2,RAT_1:7;
  consider q such that
A5: b<q and
A6: q<p by A3,RAT_1:7;
  consider s2 being Rational_Sequence such that
A7: s2 is convergent and
A8: b = lim s2 and
A9: for n holds s2.n<=b by Th67;
A10: a #Q q < a #Q p by A1,A6,Th64;
  now
    let n;
    s2.n<=b by A9;
    then s2.n<=q by A5,XXREAL_0:2;
    then a #Q (s2.n) <= a #Q q by A1,Th63;
    hence a #Q s2 .n <= a #Q q by Def5;
  end;
  then
A11: lim (a #Q s2) <= a #Q q by A1,A7,Th2,Th69;
  a #Q s2 is convergent by A1,A7,Th69;
  then a #R b <= a #Q q by A1,A7,A8,A11,Def6;
  then
A12: a #R b < a #Q p by A10,XXREAL_0:2;
  consider s1 being Rational_Sequence such that
A13: s1 is convergent and
A14: c = lim s1 and
A15: for n holds s1.n>=c by Th68;
  now
    let n;
    s1.n>=c by A15;
    then s1.n>=p by A4,XXREAL_0:2;
    then a #Q (s1.n) >= a #Q p by A1,Th63;
    hence a #Q s1 .n >= a #Q p by Def5;
  end;
  then
A16: lim (a #Q s1) >= a #Q p by A1,A13,Th1,Th69;
  a #Q s1 is convergent by A1,A13,Th69;
  then a #R c >= a #Q p by A1,A13,A14,A16,Def6;
  hence thesis by A12,XXREAL_0:2;
end;
