
theorem Th14:
for f be PartFunc of REAL,REAL, x0 be Real st
 f is_left_divergent_to+infty_in x0 holds
  ex r be Real st 0<r & f|(].x0-r,x0.[) is bounded_below
proof
    let f be PartFunc of REAL,REAL, x0 be Real;
    assume
A1:  f is_left_divergent_to+infty_in x0;
    consider r be Real such that
A2:  r < x0 and
A3:  for r1 be Real st r < r1 & r1 < x0 & r1 in dom f holds 1 < f.r1
       by A1,LIMFUNC2:8;

    set R = x0-r;

    for r1 be object st r1 in dom(f|(].x0-R,x0.[))
     holds 1 < (f|(].x0-R,x0.[)).r1
    proof
     let r1 be object;
     assume A4: r1 in dom(f|(].x0-R,x0.[)); then
     reconsider r1 as Real;
     r1 in dom f /\ ].x0-R,x0.[ by A4,RELAT_1:61; then
A5:  r1 in dom f & r1 in ].x0-R,x0.[ by XBOOLE_0:def 4; then
     x0-R < r1 & r1 < x0 by XXREAL_1:4; then
     1 < f.r1 by A3,A5;
     hence thesis by A5,FUNCT_1:49;
    end; then
    f|(].x0-R,x0.[) is bounded_below by SEQ_2:def 2;
    hence ex r be Real st 0<r & f|(].x0-r,x0.[) is bounded_below
      by A2,XREAL_1:50;
end;
