
theorem Th7:
for f be PartFunc of REAL,REAL st
 f is divergent_in-infty_to+infty holds
  ex r be Real st f|left_open_halfline r is bounded_below
proof
    let f be PartFunc of REAL,REAL;
    assume
A1:  f is divergent_in-infty_to+infty;
    consider r be Real such that
A2:  for r1 be Real st r1 < r & r1 in dom f holds 1 < f.r1
       by A1,LIMFUNC1:48;

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