
theorem Th10:
for f be PartFunc of REAL,REAL st
 f is divergent_in+infty_to-infty holds
  ex r be Real st f|right_open_halfline r is bounded_above
proof
    let f be PartFunc of REAL,REAL;
    assume f is divergent_in+infty_to-infty; then
    consider r be Real such that
A1:  for r1 be Real st r < r1 & r1 in dom f holds f.r1 < 1 by LIMFUNC1:47;

    for r1 be object st r1 in dom(f|right_open_halfline r)
     holds (f|right_open_halfline r).r1 < 1
    proof
     let r1 be object;
     assume A2: r1 in dom(f|right_open_halfline r); then
     reconsider r1 as Real;
     r1 in dom f /\ right_open_halfline r by A2,RELAT_1:61; then
A3:  r1 in dom f & r1 in right_open_halfline r by XBOOLE_0:def 4; then
     r < r1 by XXREAL_1:4; then
     f.r1 < 1 by A1,A3;
     hence thesis by A3,FUNCT_1:49;
    end; then
    f|right_open_halfline r is bounded_above by SEQ_2:def 1;
    hence ex r be Real st f|right_open_halfline r is bounded_above;
end;
