reserve T,T1,T2,S for non empty TopSpace;
reserve p,q for Point of TOP-REAL 2;

theorem Th27:
  for X being non empty TopSpace, f1,f2 being Function of X,R^1 st
  f1 is continuous & f2 is continuous & (for q being Point of X holds f2.q<>0)
holds ex g being Function of X,R^1 st (for p being Point of X,r1,r2 being Real
 st f1.p=r1 & f2.p=r2 holds g.p=r1/r2) & g is continuous
proof
  let X be non empty TopSpace, f1,f2 be Function of X,R^1;
  assume that
A1: f1 is continuous and
A2: f2 is continuous & for q being Point of X holds f2.q<>0;
  consider g1 being Function of X,R^1 such that
A3: for p being Point of X,r2 being Real st f2.p=r2 holds g1.p=1/ r2 and
A4: g1 is continuous by A2,Th26;
  consider g2 being Function of X,R^1 such that
A5: for p being Point of X,r1,r2 being Real st f1.p=r1 & g1.p=r2
  holds g2.p=r1*r2 and
A6: g2 is continuous by A1,A4,Th25;
  for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
  holds g2.p=r1/r2
  proof
    let p be Point of X,r1,r2 be Real;
    assume that
A7: f1.p=r1 and
A8: f2.p=r2;
    g1.p=1/r2 by A3,A8;
    then g2.p=r1*(1/r2) by A5,A7
      .=r1/r2;
    hence thesis;
  end;
  hence thesis by A6;
end;
