reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem Th82:
  f1 is convergent_in+infty & f2 is convergent_in+infty & (for r
  ex g st r<g & g in dom(f1+f2)) implies f1+f2 is convergent_in+infty &
  lim_in+infty(f1+f2) = lim_in+infty f1 + lim_in+infty f2
proof
  assume that
A1: f1 is convergent_in+infty and
A2: f2 is convergent_in+infty and
A3: for r ex g st r<g & g in dom(f1+f2);
A4: now
    let seq;
    assume that
A5: seq is divergent_to+infty and
A6: rng seq c=dom(f1+f2);
A7: dom(f1+f2)=dom f1/\dom f2 by A6,Lm2;
A8: rng seq c=dom f2 by A6,Lm2;
    then
A9: lim(f2/*seq)=lim_in+infty f2 by A2,A5,Def12;
A10: f2/*seq is convergent by A2,A5,A8;
A11: rng seq c=dom f1 by A6,Lm2;
    then
A12: lim(f1/*seq)=lim_in+infty f1 by A1,A5,Def12;
A13: f1/*seq is convergent by A1,A5,A11;
    then f1/*seq+f2/*seq is convergent by A10;
    hence (f1+f2)/*seq is convergent by A6,A7,RFUNCT_2:8;
    thus lim((f1+f2)/*seq)=lim(f1/*seq+f2/*seq) by A6,A7,RFUNCT_2:8
      .=lim_in+infty f1+lim_in+infty f2 by A13,A12,A10,A9,SEQ_2:6;
  end;
  hence f1+f2 is convergent_in+infty by A3;
  hence thesis by A4,Def12;
end;
