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 / (sqrt (1 + ((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 / (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous ) )

assume that

A1: f1 is continuous and

A2: ( 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 / (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous )

consider g2 being Function of X,R^1 such that

A3: for p being Point of X

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

g2 . p = sqrt (1 + ((r1 / r2) ^2)) and

A4: g2 is continuous by A1, A2, Th8;

for q being Point of X holds g2 . q <> 0

A5: for p being Point of X

for r1, r0 being Real st f1 . p = r1 & g2 . p = r0 holds

g3 . p = r1 / r0 and

A6: g3 is continuous by A1, A4, JGRAPH_2:27;

for p being Point of X

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

g3 . p = r1 / (sqrt (1 + ((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 / (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous ) by A6; :: 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 / (sqrt (1 + ((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 / (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous ) )

assume that

A1: f1 is continuous and

A2: ( 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 / (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous )

consider g2 being Function of X,R^1 such that

A3: for p being Point of X

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

g2 . p = sqrt (1 + ((r1 / r2) ^2)) and

A4: g2 is continuous by A1, A2, Th8;

for q being Point of X holds g2 . q <> 0

proof

then consider g3 being Function of X,R^1 such that
let q be Point of X; :: thesis: g2 . q <> 0

reconsider r1 = f1 . q, r2 = f2 . q as Real ;

sqrt (1 + ((r1 / r2) ^2)) > 0 by Lm1, SQUARE_1:25;

hence g2 . q <> 0 by A3; :: thesis: verum

end;reconsider r1 = f1 . q, r2 = f2 . q as Real ;

sqrt (1 + ((r1 / r2) ^2)) > 0 by Lm1, SQUARE_1:25;

hence g2 . q <> 0 by A3; :: thesis: verum

A5: for p being Point of X

for r1, r0 being Real st f1 . p = r1 & g2 . p = r0 holds

g3 . p = r1 / r0 and

A6: g3 is continuous by A1, A4, JGRAPH_2:27;

for p being Point of X

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

g3 . p = r1 / (sqrt (1 + ((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 / (sqrt (1 + ((r1 / r2) ^2)))

let r1, r2 be Real; :: thesis: ( f1 . p = r1 & f2 . p = r2 implies g3 . p = r1 / (sqrt (1 + ((r1 / r2) ^2))) )

assume that

A7: f1 . p = r1 and

A8: f2 . p = r2 ; :: thesis: g3 . p = r1 / (sqrt (1 + ((r1 / r2) ^2)))

g2 . p = sqrt (1 + ((r1 / r2) ^2)) by A3, A7, A8;

hence g3 . p = r1 / (sqrt (1 + ((r1 / r2) ^2))) by A5, A7; :: thesis: verum

end;g3 . p = r1 / (sqrt (1 + ((r1 / r2) ^2)))

let r1, r2 be Real; :: thesis: ( f1 . p = r1 & f2 . p = r2 implies g3 . p = r1 / (sqrt (1 + ((r1 / r2) ^2))) )

assume that

A7: f1 . p = r1 and

A8: f2 . p = r2 ; :: thesis: g3 . p = r1 / (sqrt (1 + ((r1 / r2) ^2)))

g2 . p = sqrt (1 + ((r1 / r2) ^2)) by A3, A7, A8;

hence g3 . p = r1 / (sqrt (1 + ((r1 / r2) ^2))) by A5, A7; :: thesis: verum

( ( for p being Point of X

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

g . p = r1 / (sqrt (1 + ((r1 / r2) ^2))) ) & g is continuous ) by A6; :: thesis: verum