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 & (ex r st 0<r & ].x0-r,x0.[ \/ ].x0,x0
+r.[ c= dom f /\ dom f1 & for g st g in ].x0-r,x0.[ \/ ].x0,x0+r.[ holds f.g<=
  f1.g) implies f is_divergent_to-infty_in x0
proof
  assume
A1: f1 is_divergent_to-infty_in x0;
  given r such that
A2: 0<r and
A3: ].x0-r,x0.[ \/ ].x0,x0+r.[c=dom f/\dom f1 and
A4: for g st g in ].x0-r,x0.[ \/ ].x0,x0+r.[ holds f.g<=f1.g;
A5: ].x0-r,x0.[\/].x0,x0+r.[=dom f/\(].x0-r,x0.[\/].x0,x0+r.[) by A3,
XBOOLE_1:18,28;
A6: ].x0-r,x0.[\/].x0,x0+r.[=dom f1/\(].x0-r,x0.[\/].x0,x0+r.[) by A3,
XBOOLE_1:18,28;
  for r1,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom f & g2<
  r2 & x0<g2 & g2 in dom f by A2,A3,Th5,XBOOLE_1:18;
  hence thesis by A1,A2,A4,A5,A6,Th25;
end;
