
theorem
for f be PartFunc of REAL,REAL, a be Real st
 f is_left_convergent_in a & f is non-increasing holds
  for x be Real st x in dom f & x < a holds f.x >= lim_left(f,a)
proof
    let f be PartFunc of REAL,REAL, a be Real;
    assume that
A1:  f is_left_convergent_in a and
A2:  f is non-increasing;
    let x be Real;
    assume that
A3:  x in dom f and
A4:  x < a;

    hereby assume
A5:  f.x < lim_left(f,a); then
A6:  lim_left(f,a) - f.x > 0 by XREAL_1:50;
     set g1 = lim_left(f,a) - f.x;
     consider r be Real such that
A7:   r < a and
A8:   for r1 be Real st r < r1 & r1 < a & r1 in dom f
        holds |.f.r1-lim_left(f,a).| < g1 by A6,A1,LIMFUNC2:41;

     consider R be Real such that
A9:   max(x,r) < R & R < a & R in dom f by A1,A4,A7,XXREAL_0:29,LIMFUNC2:def 1;
A10: x <= max(x,r) & r <= max(x,r) by XXREAL_0:25; then
A11: r < R by A9,XXREAL_0:2;

     x < R by A9,A10,XXREAL_0:2; then
A12: f.x >= f.R by A2,A3,A9,RFUNCT_2:def 4; then
     lim_left(f,a) > f.R by A5,XXREAL_0:2; then
     |.f.R - lim_left(f,a).| = -(f.R - lim_left(f,a))
       by ABSVALUE:def 1,XREAL_1:49; then
     lim_left(f,a) - f.R < lim_left(f,a) - f.x by A8,A9,A11;
     hence contradiction by A12,XREAL_1:10;
    end;
end;
