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
  X c= dom f1 /\ dom f2 & f1|X is continuous & f1"{0} = {} & f2|X is
  continuous implies (f2/f1)|X is continuous
proof
  assume
A1: X c= dom f1 /\ dom f2;
  assume that
A2: f1|X is continuous and
A3: f1"{0} = {} and
A4: f2|X is continuous;
A5: dom(f1^) = dom f1 \ {} by A3,RFUNCT_1:def 2
    .= dom f1;
  (f1^)|X is continuous by A2,A3,Th22;
  then (f2(#)(f1^))|X is continuous by A1,A4,A5,Th18;
  hence thesis by RFUNCT_1:31;
end;
