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_right_convergent_in x0 implies abs(f) is_right_convergent_in x0 &
  lim_right(abs(f),x0)=|.lim_right(f,x0).|
proof
  assume
A1: f is_right_convergent_in x0;
A2: now
    let seq;
    assume that
A3: seq is convergent and
A4: lim seq=x0 and
A5: rng seq c=dom(abs(f))/\right_open_halfline(x0);
A6: rng seq c=dom f/\right_open_halfline(x0) by A5,VALUED_1:def 11;
    dom f/\right_open_halfline(x0)c=dom f by XBOOLE_1:17;
    then rng seq c=dom f by A6,XBOOLE_1:1;
    then
A7: abs(f/*seq)=(abs f)/*seq by RFUNCT_2:10;
A8: f/*seq is convergent by A1,A3,A4,A6;
    hence (abs f)/*seq is convergent by A7;
    lim(f/*seq)=lim_right(f,x0) by A1,A3,A4,A6,Def8;
    hence lim((abs f)/*seq)=|.lim_right(f,x0).| by A8,A7,SEQ_4:14;
  end;
  now
    let r;
    assume x0<r;
    then consider g such that
A9: g<r and
A10: x0<g and
A11: g in dom f by A1;
    take g;
    thus g<r & x0<g & g in dom abs(f) by A9,A10,A11,VALUED_1:def 11;
  end;
  hence abs(f) is_right_convergent_in x0 by A2;
  hence thesis by A2,Def8;
end;
