
theorem Th1:
for f be PartFunc of REAL,REAL st f is divergent_in-infty_to+infty holds
   not f is convergent_in-infty & not f is divergent_in-infty_to-infty
proof
    let f be PartFunc of REAL,REAL;
    assume
A1:  f is divergent_in-infty_to+infty;

    hereby assume f is convergent_in-infty; then
     consider g be Real such that
A2:   for g1 be Real st 0 < g1 ex r be Real st for r1 be Real st r1 < r &
       r1 in dom f holds |. f.r1 - g .| < g1 by LIMFUNC1:45;
     consider R1 be Real such that
A3:   for r1 be Real st r1 < R1 & r1 in dom f holds |. f.r1 - g .| < 1 by A2;

     consider R2 be Real such that
A4:   for r1 be Real st r1 < R2 & r1 in dom f holds g+1 < f.r1
     by A1,LIMFUNC1:48;

     set R = min(R1,R2);
A5:  R <= R1 & R <= R2 by XXREAL_0:17;
     consider r be Real such that
A6:  r < R & r in dom f by A1,LIMFUNC1:48;
A7: r < R1 by A5,A6,XXREAL_0:2;
     r < R2 by A5,A6,XXREAL_0:2; then
A8: 1 < f.r - g by A4,A6,XREAL_1:20;
     f.r - g <= |. f.r - g .| by ABSVALUE:4; then
     1 < |. f.r - g .| by A8,XXREAL_0:2;
     hence contradiction by A3,A6,A7;
    end;

    hereby assume f is divergent_in-infty_to-infty; then
     consider R1 be Real such that
A9:  for r1 be Real st r1 < R1 & r1 in dom f holds f.r1 < 0 by LIMFUNC1:49;
     consider R2 be Real such that
A10:  for r1 be Real st r1 < R2 & r1 in dom f holds 0 < f.r1 by A1,LIMFUNC1:48;
     set R = min(R1,R2);
A11:  R <= R1 & R <= R2 by XXREAL_0:17;
     consider r be Real such that
A12:  r < R & r in dom f by A1,LIMFUNC1:48;
A13: r < R1 & r < R2 by A11,A12,XXREAL_0:2; then
     f.r > 0 by A10,A12;
     hence contradiction by A9,A12,A13;
    end;
end;
