let X be non empty TopSpace; 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 st f1 . p = r1 & f2 . p = r2 holds
g . p = sqrt (1 + ((r1 / r2) ^2)) ) & g is continuous )
let f1, f2 be Function of X,R^1; ( 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 st f1 . p = r1 & f2 . p = r2 holds
g . p = sqrt (1 + ((r1 / r2) ^2)) ) & g is continuous ) )
assume
( f1 is continuous & f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) )
; ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds
g . p = sqrt (1 + ((r1 / r2) ^2)) ) & g is continuous )
then consider g2 being Function of X,R^1 such that
A1:
for p being Point of X
for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds
g2 . p = 1 + ((r1 / r2) ^2)
and
A2:
g2 is continuous
by Th7;
for q being Point of X ex r being Real st
( g2 . q = r & r >= 0 )
then consider g3 being Function of X,R^1 such that
A3:
for p being Point of X
for r1 being Real st g2 . p = r1 holds
g3 . p = sqrt r1
and
A4:
g3 is continuous
by A2, Th5;
for p being Point of X
for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds
g3 . p = sqrt (1 + ((r1 / r2) ^2))
hence
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds
g . p = sqrt (1 + ((r1 / r2) ^2)) ) & g is continuous )
by A4; verum