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 st f1 . p = r1 & f2 . p = r2 holds

g . p = (r1 / r2) ^2 ) & 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 st f1 . p = r1 & f2 . p = r2 holds

g . p = (r1 / r2) ^2 ) & g is continuous ) )

assume ( 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 st f1 . p = r1 & f2 . p = r2 holds

g . p = (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 = r1 / r2 and

A2: g2 is continuous by JGRAPH_2:27;

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 = r1 * r1 and

A4: g3 is continuous by A2, JGRAPH_2:22;

for p being Point of X

for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds

g3 . p = (r1 / r2) ^2

( ( for p being Point of X

for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds

g . p = (r1 / r2) ^2 ) & g is continuous ) by A4; :: thesis: verum

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 = (r1 / r2) ^2 ) & 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 st f1 . p = r1 & f2 . p = r2 holds

g . p = (r1 / r2) ^2 ) & g is continuous ) )

assume ( 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 st f1 . p = r1 & f2 . p = r2 holds

g . p = (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 = r1 / r2 and

A2: g2 is continuous by JGRAPH_2:27;

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 = r1 * r1 and

A4: g3 is continuous by A2, JGRAPH_2:22;

for p being Point of X

for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds

g3 . p = (r1 / r2) ^2

proof

hence
ex g being Function of X,R^1 st
let p be Point of X; :: thesis: for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds

g3 . p = (r1 / r2) ^2

let r1, r2 be Real; :: thesis: ( f1 . p = r1 & f2 . p = r2 implies g3 . p = (r1 / r2) ^2 )

assume ( f1 . p = r1 & f2 . p = r2 ) ; :: thesis: g3 . p = (r1 / r2) ^2

then g2 . p = r1 / r2 by A1;

hence g3 . p = (r1 / r2) ^2 by A3; :: thesis: verum

end;g3 . p = (r1 / r2) ^2

let r1, r2 be Real; :: thesis: ( f1 . p = r1 & f2 . p = r2 implies g3 . p = (r1 / r2) ^2 )

assume ( f1 . p = r1 & f2 . p = r2 ) ; :: thesis: g3 . p = (r1 / r2) ^2

then g2 . p = r1 / r2 by A1;

hence g3 . p = (r1 / r2) ^2 by A3; :: thesis: verum

( ( for p being Point of X

for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds

g . p = (r1 / r2) ^2 ) & g is continuous ) by A4; :: thesis: verum