
theorem Th13:
for f be PartFunc of REAL,REAL, x0 be Real st
 f is_right_divergent_to-infty_in x0 holds
  ex r be Real st 0<r & f|(].x0,x0+r.[) is bounded_above
proof
    let f be PartFunc of REAL,REAL, x0 be Real;
    assume
A1:  f is_right_divergent_to-infty_in x0;
    consider r be Real such that
A2:  x0 < r and
A3:  for r1 be Real st r1 < r & x0 < r1 & r1 in dom f holds f.r1 < 1
       by A1,LIMFUNC2:12;

    set R = r-x0;

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