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
  f1 is_right_divergent_to-infty_in x0 & (ex r st 0<r & ].x0,x0+r.[ c=
  dom f /\ dom f1 & for g st g in ].x0,x0+r.[ holds f.g<= f1.g) implies f
  is_right_divergent_to-infty_in x0
proof
  assume
A1: f1 is_right_divergent_to-infty_in x0;
  given r such that
A2: 0<r and
A3: ].x0,x0+r.[c=dom f/\dom f1 and
A4: for g st g in ].x0,x0+r.[ holds f.g<=f1.g;
A5: dom f/\dom f1 c=dom f by XBOOLE_1:17;
  then
A6: ].x0,x0+r.[=dom f/\].x0,x0+r.[ by A3,XBOOLE_1:1,28;
  dom f/\dom f1 c=dom f1 by XBOOLE_1:17;
  then
A7: ].x0,x0+r.[=dom f1/\].x0,x0+r.[ by A3,XBOOLE_1:1,28;
  for r st x0<r ex g st g<r & x0<g & g in dom f by A2,A3,A5,Th4,XBOOLE_1:1;
  hence thesis by A1,A2,A4,A6,A7,Th36;
end;
