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 Th33:
  f1 is_convergent_in x0 & f2 is_convergent_in x0 & (for r1,r2 st
r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom(f1+f2) & g2<r2 & x0<g2 & g2
in dom(f1+f2)) implies f1+f2 is_convergent_in x0 & lim(f1+f2,x0)=lim(f1,x0)+lim
  (f2,x0)
proof
  assume that
A1: f1 is_convergent_in x0 and
A2: f2 is_convergent_in x0 and
A3: for r1,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom(f1+
  f2) & g2<r2 & x0<g2 & g2 in dom(f1+f2);
A4: now
    let seq;
    assume that
A5: seq is convergent and
A6: lim seq=x0 and
A7: rng seq c=dom(f1+f2)\{x0};
A8: dom(f1+f2)=dom f1/\dom f2 by A7,Lm4;
A9: rng seq c=dom f1\ {x0} by A7,Lm4;
A10: rng seq c=dom f2\{x0} by A7,Lm4;
    then
A11: lim(f2/*seq)=lim(f2,x0) by A2,A5,A6,Def4;
A12: f2/*seq is convergent by A2,A5,A6,A10;
    rng seq c=dom(f1+f2) by A7,Lm4;
    then
A13: f1/*seq+f2/*seq=(f1+f2)/*seq by A8,RFUNCT_2:8;
A14: f1/*seq is convergent by A1,A5,A6,A9;
    hence (f1+f2)/*seq is convergent by A12,A13,SEQ_2:5;
    lim(f1/*seq)=lim(f1,x0) by A1,A5,A6,A9,Def4;
    hence lim((f1+f2)/*seq)=lim(f1,x0)+lim(f2,x0) by A14,A12,A11,A13,SEQ_2:6;
  end;
  hence f1+f2 is_convergent_in x0 by A3;
  hence thesis by A4,Def4;
end;
