reserve n,m,k for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,p,x0,x1,x2 for Real;
reserve s1,s2,q1 for Real_Sequence;
reserve Y for Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem
  x0 in dom f2 & f1 is_continuous_in x0 & f1.x0<>0 & f2 is_continuous_in
  x0 implies f2/f1 is_continuous_in x0
proof
  assume
A1: x0 in dom f2;
  assume that
A2: f1 is_continuous_in x0 and
A3: f1.x0<>0 and
A4: f2 is_continuous_in x0;
  not f1.x0 in {0} by A3,TARSKI:def 1;
  then
A5: not x0 in f1"{0} by FUNCT_1:def 7;
  x0 in dom f1 by A3,FUNCT_1:def 2;
  then x0 in dom f1 \ f1"{0} by A5,XBOOLE_0:def 5;
  then x0 in dom(f1^) by RFUNCT_1:def 2;
  then
A6: x0 in dom(f1^) /\ dom f2 by A1,XBOOLE_0:def 4;
  f1^ is_continuous_in x0 by A2,A3,Th10;
  then f2(#)(f1^) is_continuous_in x0 by A4,A6,Th7;
  hence thesis by RFUNCT_1:31;
end;
