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

theorem
  (f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0)
  implies abs(f) is_divergent_to+infty_in x0
proof
  assume
A1: f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0;
  now
    per cases by A1;
    suppose
A2:   f is_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)\{x0};
A7:     rng seq c=dom f\{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;
        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 r1,r2;
        assume that
A9:     r1<x0 and
A10:    x0<r2;
        consider g1,g2 such that
A11:    r1<g1 and
A12:    g1<x0 and
A13:    g1 in dom f and
A14:    g2<r2 and
A15:    x0<g2 and
A16:    g2 in dom f by A2,A9,A10;
        take g1;
        take g2;
        thus r1<g1 & g1<x0 & g1 in dom abs(f) & g2<r2 & x0<g2 & g2 in dom abs(
        f) by A11,A12,A13,A14,A15,A16,VALUED_1:def 11;
      end;
      hence thesis by A3;
    end;
    suppose
A17:  f is_divergent_to-infty_in x0;
A18:  now
        let seq;
        assume that
A19:    seq is convergent and
A20:    lim seq=x0 and
A21:    rng seq c=dom abs(f)\{x0};
A22:    rng seq c=dom f\{x0} by A21,VALUED_1:def 11;
        then f/*seq is divergent_to-infty by A17,A19,A20;
        then
A23:    abs(f/*seq) is divergent_to+infty by LIMFUNC1:25;
        rng seq c=dom f by A22,XBOOLE_1:1;
        hence (abs f)/*seq is divergent_to+infty by A23,RFUNCT_2:10;
      end;
      now
        let r1,r2;
        assume that
A24:    r1<x0 and
A25:    x0<r2;
        consider g1,g2 such that
A26:    r1<g1 and
A27:    g1<x0 and
A28:    g1 in dom f and
A29:    g2<r2 and
A30:    x0<g2 and
A31:    g2 in dom f by A17,A24,A25;
        take g1;
        take g2;
        thus r1<g1 & g1<x0 & g1 in dom abs(f) & g2<r2 & x0<g2 & g2 in dom abs(
        f) by A26,A27,A28,A29,A30,A31,VALUED_1:def 11;
      end;
      hence thesis by A18;
    end;
  end;
  hence thesis;
end;
