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
  f1 is divergent_in-infty_to-infty & f2 is divergent_in-infty_to-infty
  & (for r ex g st g<r & g in dom f1 /\ dom f2) implies f1+f2 is
  divergent_in-infty_to-infty & f1(#)f2 is divergent_in-infty_to+infty
proof
  assume that
A1: f1 is divergent_in-infty_to-infty and
A2: f2 is divergent_in-infty_to-infty and
A3: for r ex g st g<r & g in dom f1/\dom 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;
    rng seq c=dom f2 by A6,Lm2;
    then
A8: f2/*seq is divergent_to-infty by A2,A5;
    rng seq c=dom f1 by A6,Lm2;
    then f1/*seq is divergent_to-infty by A1,A5;
    then f1/*seq+f2/*seq is divergent_to-infty by A8,Th11;
    hence (f1+f2)/*seq is divergent_to-infty by A6,A7,RFUNCT_2:8;
  end;
A9: now
    let seq;
    assume that
A10: seq is divergent_to-infty and
A11: rng seq c=dom(f1(#) f2);
A12: dom(f1(#)f2)=dom f1/\ dom f2 by A11,Lm3;
    rng seq c=dom f2 by A11,Lm3;
    then
A13: f2/*seq is divergent_to-infty by A2,A10;
    rng seq c=dom f1 by A11,Lm3;
    then f1/*seq is divergent_to-infty by A1,A10;
    then (f1/*seq)(#)(f2/*seq) is divergent_to+infty by A13,Th24;
    hence (f1(#)f2)/*seq is divergent_to+infty by A11,A12,RFUNCT_2:8;
  end;
  now
    let r;
    consider g such that
A14: g<r & g in dom f1/\dom f2 by A3;
    take g;
    thus g<r & g in dom(f1+f2) by A14,VALUED_1:def 1;
  end;
  hence f1+f2 is divergent_in-infty_to-infty by A4;
  now
    let r;
    consider g such that
A15: g<r & g in dom f1/\dom f2 by A3;
    take g;
    thus g<r & g in dom(f1(#)f2) by A15,VALUED_1:def 4;
  end;
  hence thesis by A9;
end;
