let X be non empty TopSpace; :: thesis: for 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
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 / r2 ) & g is continuous )
let f1, f2 be Function of X,R^1 ; :: thesis: ( f1 is continuous & f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) implies ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 / r2 ) & g is continuous ) )
assume A1: ( f1 is continuous & f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) ) ; :: thesis: ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 / r2 ) & g is continuous )
then consider g1 being Function of X,R^1 such that
A2: ( ( for p being Point of X
for r2 being real number st f2 . p = r2 holds
g1 . p = 1 / r2 ) & g1 is continuous ) by Th36;
consider g2 being Function of X,R^1 such that
A3: ( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & g1 . p = r2 holds
g2 . p = r1 * r2 ) & g2 is continuous ) by A1, A2, Th35;
for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g2 . p = r1 / r2

proof

let p be Point of X; :: thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g2 . p = r1 / r2
let r1, r2 be real number ; :: thesis: ( f1 . p = r1 & f2 . p = r2 implies g2 . p = r1 / r2 )
assume A4: ( f1 . p = r1 & f2 . p = r2 ) ; :: thesis: g2 . p = r1 / r2
then g1 . p = 1 / r2 by A2;
then g2 . p = r1 * (1 / r2) by A3, A4
.= r1 / r2 ;
hence g2 . p = r1 / r2 ; :: thesis: verum

end;

hence ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 / r2 ) & g is continuous ) by A3; :: thesis: verum