reserve D for non empty set,
  i,j,k,l for Nat,
  n for Nat,
  x for set,
  a,b,c,r,r1,r2 for Real,
  p,q for FinSequence of REAL,
  MR,MR1 for Matrix of REAL;

theorem Th3:
  r > 0 implies log(number_e,r) <= r - 1 & (r = 1 iff log(number_e,
  r) = r - 1) & (r <> 1 iff log(number_e,r) < r - 1)
proof
  assume
A1: r>0;
  then r in {g where g is Real : 0<g};
  then r in right_open_halfline(0) by XXREAL_1:230;
  then log(number_e,r) = ln.r by TAYLOR_1:def 2;
  hence thesis by A1,Th2;
end;
