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
  f1 is_divergent_to+infty_in x0 & f2 is_divergent_to+infty_in x0 & (for
r1,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom f1 /\ dom f2 & g2<
r2 & x0<g2 & g2 in dom f1 /\ dom f2) implies f1+f2 is_divergent_to+infty_in x0
  & f1(#)f2 is_divergent_to+infty_in x0
proof
  assume that
A1: f1 is_divergent_to+infty_in x0 and
A2: f2 is_divergent_to+infty_in x0 and
A3: for r1,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom f1
  /\dom f2 & g2<r2 & x0<g2 & g2 in dom f1/\dom f2;
A4: now
    let s be Real_Sequence;
    assume that
A5: s is convergent and
A6: lim s=x0 and
A7: rng s c=dom(f1+f2)\{x0};
    rng s c=dom f2\{x0} by A7,Lm4;
    then
A8: f2/*s is divergent_to+infty by A2,A5,A6;
    rng s c=dom f1\{ x0} by A7,Lm4;
    then f1/*s is divergent_to+infty by A1,A5,A6;
    then
A9: f1/*s+f2/*s is divergent_to+infty by A8,LIMFUNC1:8;
A10: dom(f1+f2)=dom f1/\dom f2 by A7,Lm4;
    rng s c=dom(f1+f2) by A7,Lm4;
    hence (f1+f2)/*s is divergent_to+infty by A10,A9,RFUNCT_2:8;
  end;
A11: now
    let s be Real_Sequence;
    assume that
A12: s is convergent and
A13: lim s=x0 and
A14: rng s c=dom(f1(#)f2)\{x0};
    rng s c=dom f2\{x0} by A14,Lm2;
    then
A15: f2/*s is divergent_to+infty by A2,A12,A13;
    rng s c=dom f1\{x0} by A14,Lm2;
    then f1/*s is divergent_to+infty by A1,A12,A13;
    then
A16: (f1/*s)(#)(f2/*s) is divergent_to+infty by A15,LIMFUNC1:10;
A17: dom(f1(#) f2)=dom f1/\dom f2 by A14,Lm2;
    rng s c=dom(f1(#)f2) by A14,Lm2;
    hence (f1(#)f2)/*s is divergent_to+infty by A17,A16,RFUNCT_2:8;
  end;
  now
    let r1,r2;
    assume that
A18: r1<x0 and
A19: x0<r2;
    consider g1,g2 such that
A20: r1<g1 and
A21: g1<x0 and
A22: g1 in dom f1/\dom f2 and
A23: g2<r2 and
A24: x0<g2 and
A25: g2 in dom f1/\dom f2 by A3,A18,A19;
    take g1;
    take g2;
    thus r1<g1 & g1<x0 & g1 in dom(f1+f2) & g2<r2 & x0<g2 & g2 in dom(f1+f2)
    by A20,A21,A22,A23,A24,A25,VALUED_1:def 1;
  end;
  hence f1+f2 is_divergent_to+infty_in x0 by A4;
  now
    let r1,r2;
    assume that
A26: r1<x0 and
A27: x0<r2;
    consider g1,g2 such that
A28: r1<g1 and
A29: g1<x0 and
A30: g1 in dom f1/\dom f2 and
A31: g2<r2 and
A32: x0<g2 and
A33: g2 in dom f1/\dom f2 by A3,A26,A27;
    take g1;
    take g2;
    thus r1<g1 & g1<x0 & g1 in dom(f1(#)f2) & g2<r2 & x0<g2 & g2 in dom(f1(#)
    f2) by A28,A29,A30,A31,A32,A33,VALUED_1:def 4;
  end;
  hence thesis by A11;
end;
