
theorem Th6:
for f be PartFunc of REAL,REAL, a be Real st
 f is_left_convergent_in a & f is non-decreasing 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-decreasing;
    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:  f.x - lim_left(f,a) > 0 by XREAL_1:50;
     set g1 = f.x - lim_left(f,a);
     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
     r < R by A9,XXREAL_0:2; then
A11: |.f.R - lim_left(f,a).| < g1 by A8,A9;

     x < R by A9,A10,XXREAL_0:2; then
A12: f.x <= f.R by A2,A3,A9,RFUNCT_2:def 3; then
     lim_left(f,a) < f.R by A5,XXREAL_0:2; then
     f.R - lim_left(f,a) > 0 by XREAL_1:50; then
     f.R - lim_left(f,a) < g1 by A11,ABSVALUE:def 1;
     hence contradiction by A12,XREAL_1:9;
    end;
end;
