
theorem Th9:
for f be PartFunc of REAL,REAL, a be Real st
 f is_right_convergent_in a & f is non-increasing holds
  for x be Real st x in dom f & a < x holds f.x <= lim_right(f,a)
proof
    let f be PartFunc of REAL,REAL, a be Real;
    assume that
A1:  f is_right_convergent_in a and
A2:  f is non-increasing;
    let x be Real;
    assume that
A3:  x in dom f and
A4:  a < x;

    hereby assume
A5:  f.x > lim_right(f,a); then
A6:  f.x - lim_right(f,a) > 0 by XREAL_1:50;
     set g1 = f.x - lim_right(f,a);
     consider r be Real such that
A7:   a < r and
A8:   for r1 be Real st r1 < r & a < r1 & r1 in dom f
        holds |.f.r1-lim_right(f,a).| < g1 by A6,A1,LIMFUNC2:42;

     consider R be Real such that
A9:   min(x,r) > R & R > a & R in dom f by A1,A4,A7,XXREAL_0:21,LIMFUNC2:def 4;
A10: x >= min(x,r) & r >= min(x,r) by XXREAL_0:17; then
     r > R by A9,XXREAL_0:2; then
A11: |.f.R - lim_right(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 4; then
     lim_right(f,a) < f.R by A5,XXREAL_0:2; then
     f.R - lim_right(f,a) > 0 by XREAL_1:50; then
     |.f.R - lim_right(f,a) .| = f.R - lim_right(f,a) by ABSVALUE:def 1;
     hence contradiction by A11,A12,XREAL_1:9;
    end;
end;
