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 Th33:
  f1 is_left_divergent_to+infty_in x0 & (for r st r<x0 ex g st r<g
& g<x0 & g in dom f) & (ex r st 0<r & dom f /\ ].x0-r,x0.[ c= dom f1 /\ ].x0-r,
  x0.[ & for g st g in dom f /\ ].x0-r,x0.[ holds f1.g<=f.g) implies f
  is_left_divergent_to+infty_in x0
proof
  assume that
A1: f1 is_left_divergent_to+infty_in x0 and
A2: for r st r<x0 ex g st r<g & g<x0 & g in dom f;
  given r such that
A3: 0<r and
A4: dom f/\].x0-r,x0.[c=dom f1/\].x0-r,x0.[ and
A5: for g st g in dom f/\].x0-r,x0.[ holds f1.g<=f.g;
  thus for r st r<x0 ex g st r<g & g<x0 & g in dom f by A2;
  let seq such that
A6: seq is convergent and
A7: lim seq=x0 and
A8: rng seq c=dom f/\left_open_halfline(x0);
  x0-r<x0 by A3,Lm1;
  then consider k such that
A9: for n st k<=n holds x0-r<seq.n by A6,A7,Th1;
A10: rng(seq^\k)c= rng seq by VALUED_0:21;
  dom f/\left_open_halfline(x0)c=left_open_halfline(x0) by XBOOLE_1:17;
  then rng seq c=left_open_halfline(x0) by A8,XBOOLE_1:1;
  then
A11: rng(seq^\k)c=left_open_halfline(x0) by A10,XBOOLE_1:1;
  now
    let x be object;
    assume
A12: x in rng(seq^\k);
    then consider n being Element of NAT such that
A13: (seq^\k).n=x by FUNCT_2:113;
    (seq^\k).n in left_open_halfline(x0) by A11,A12,A13;
    then (seq^\k).n in {g: g<x0} by XXREAL_1:229;
    then
A14: ex r1 st r1=(seq^\k).n & r1<x0;
    x0-r<seq.(n+k) by A9,NAT_1:12;
    then x0-r<(seq^\k).n by NAT_1:def 3;
    then x in {g1: x0-r<g1 & g1<x0} by A13,A14;
    hence x in ].x0-r,x0.[ by RCOMP_1:def 2;
  end;
  then
A15: rng(seq^\k)c=].x0-r,x0.[ by TARSKI:def 3;
A16: dom f/\left_open_halfline(x0)c=dom f by XBOOLE_1:17;
  then
A17: rng seq c=dom f by A8,XBOOLE_1:1;
  then rng(seq^\k)c=dom f by A10,XBOOLE_1:1;
  then
A18: rng(seq^\k)c=dom f/\].x0-r,x0.[ by A15,XBOOLE_1:19;
  then
A19: rng(seq^\k)c=dom f1/\].x0-r,x0.[ by A4,XBOOLE_1:1;
A20: dom f1/\].x0-r,x0.[c=dom f1 by XBOOLE_1:17;
  then rng(seq^\k)c=dom f1 by A19,XBOOLE_1:1;
  then
A21: rng(seq^\k)c=dom f1/\left_open_halfline(x0) by A11,XBOOLE_1:19;
A22: now
    let n;
A23: n in NAT by ORDINAL1:def 12;
    (seq^\k).n in rng(seq^\k) by VALUED_0:28;
    then f1.((seq^\k).n)<=f.((seq^\k).n) by A5,A18;
    then (f1/*(seq^\k)).n<=f.((seq^\k).n)
          by A19,A20,FUNCT_2:108,XBOOLE_1:1,A23;
    hence (f1/*(seq^\k)).n<=(f/*(seq^\k)).n
by A17,A10,FUNCT_2:108,XBOOLE_1:1,A23;
  end;
  lim(seq^\k)=x0 by A6,A7,SEQ_4:20;
  then f1/*(seq^\k) is divergent_to+infty by A1,A6,A21;
  then f/*(seq^\k) is divergent_to+infty by A22,LIMFUNC1:42;
  then (f/*seq)^\k is divergent_to+infty by A8,A16,VALUED_0:27,XBOOLE_1:1;
  hence thesis by LIMFUNC1:7;
end;
