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 Th82:
  a>=1 & c>=b implies a #R c >= a #R b
proof
  assume that
A1: a>=1 and
A2: c>=b;
  consider s1 being Rational_Sequence such that
A3: s1 is convergent and
A4: c = lim s1 and
A5: for n holds s1.n>=c by Th68;
A6: a #Q s1 is convergent by A1,A3,Th69;
  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 s2 is convergent by A1,A7,Th69;
  now
    let n;
    s1.n>=c by A5;
    then
A11: s1.n>=b by A2,XXREAL_0:2;
    s2.n<=b by A9;
    then s1.n >= s2.n by A11,XXREAL_0:2;
    then a #Q (s1.n) >= a #Q (s2.n) by A1,Th63;
    then a #Q (s1.n) >= a #Q s2 .n by Def5;
    hence a #Q s1 .n >= a #Q s2 .n by Def5;
  end;
  then lim (a #Q s1) >= lim (a #Q s2) by A6,A10,SEQ_2:18;
  then a #R c >= lim (a #Q s2) by A1,A3,A4,A6,Def6;
  hence thesis by A1,A7,A8,A10,Def6;
end;
