reserve r,r1,r2,g,g1,g2,x0,t for Real;
reserve n,k for Nat;
reserve seq for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem
  (f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in
  x0) implies abs(f) is_left_divergent_to+infty_in x0
proof
  assume
A1: f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0;
  now
    per cases by A1;
    suppose
A2:   f is_left_divergent_to+infty_in x0;
A3:   now
        let seq;
        assume that
A4:     seq is convergent and
A5:     lim seq=x0 and
A6:     rng seq c=dom abs(f)/\left_open_halfline(x0);
A7:     rng seq c=dom f/\left_open_halfline(x0) by A6,VALUED_1:def 11;
        then f/*seq is divergent_to+infty by A2,A4,A5;
        then
A8:     abs(f/*seq) is divergent_to+infty by LIMFUNC1:25;
        dom f/\left_open_halfline(x0)c=dom f by XBOOLE_1:17;
        then rng seq c=dom f by A7,XBOOLE_1:1;
        hence (abs f)/*seq is divergent_to+infty by A8,RFUNCT_2:10;
      end;
      now
        let r;
        assume r<x0;
        then consider g such that
A9:     r<g and
A10:    g<x0 and
A11:    g in dom f by A2;
        take g;
        thus r<g & g<x0 & g in dom abs(f) by A9,A10,A11,VALUED_1:def 11;
      end;
      hence thesis by A3;
    end;
    suppose
A12:  f is_left_divergent_to-infty_in x0;
A13:  now
        let seq;
        assume that
A14:    seq is convergent and
A15:    lim seq=x0 and
A16:    rng seq c=dom abs(f)/\left_open_halfline(x0);
A17:    rng seq c=dom f/\left_open_halfline(x0) by A16,VALUED_1:def 11;
        then f/*seq is divergent_to-infty by A12,A14,A15;
        then
A18:    abs(f/*seq) is divergent_to+infty by LIMFUNC1:25;
        dom f/\left_open_halfline(x0)c=dom f by XBOOLE_1:17;
        then rng seq c=dom f by A17,XBOOLE_1:1;
        hence (abs f)/*seq is divergent_to+infty by A18,RFUNCT_2:10;
      end;
      now
        let r;
        assume r<x0;
        then consider g such that
A19:    r<g and
A20:    g<x0 and
A21:    g in dom f by A12;
        take g;
        thus r<g & g<x0 & g in dom abs(f) by A19,A20,A21,VALUED_1:def 11;
      end;
      hence thesis by A13;
    end;
  end;
  hence thesis;
end;
