let X be non empty TopSpace; :: thesis: for f1, f2 being Function of X,R^1
for a, b being Real st f1 is continuous & f2 is continuous & b <> 0 & ( 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) - a) / b ) & g is continuous )
let f1, f2 be Function of X,R^1; :: thesis: for a, b being Real st f1 is continuous & f2 is continuous & b <> 0 & ( 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) - a) / b ) & g is continuous )
let a, b be Real; :: thesis: ( f1 is continuous & f2 is continuous & b <> 0 & ( 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) - a) / b ) & g is continuous ) )
assume that
A1: ( f1 is continuous & f2 is continuous ) and
A2: b <> 0 and
A3: 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) - a) / b ) & g is continuous )
consider g3 being Function of X,R^1 such that
A4: for p being Point of X
for r1, r0 being Real st f1 . p = r1 & f2 . p = r0 holds
g3 . p = r1 / r0 and
A5: g3 is continuous by A1, A3, JGRAPH_2:27;
consider g1 being Function of X,R^1 such that
A6: for p being Point of X holds
( g1 . p = b & g1 is continuous ) by JGRAPH_2:20;
consider g2 being Function of X,R^1 such that
A7: for p being Point of X holds
( g2 . p = a & g2 is continuous ) by JGRAPH_2:20;
consider g4 being Function of X,R^1 such that
A8: for p being Point of X
for r1, r0 being Real st g3 . p = r1 & g2 . p = r0 holds
g4 . p = r1 - r0 and
A9: g4 is continuous by A7, A5, JGRAPH_2:21;
for q being Point of X holds g1 . q <> 0 by A2, A6;
then consider g5 being Function of X,R^1 such that
A10: for p being Point of X
for r1, r0 being Real st g4 . p = r1 & g1 . p = r0 holds
g5 . p = r1 / r0 and
A11: g5 is continuous by A6, A9, JGRAPH_2:27;
for p being Point of X
for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds
g5 . p = ((r1 / r2) - a) / b

proof

let p be Point of X; :: thesis: for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds
g5 . p = ((r1 / r2) - a) / b
let r1, r2 be Real; :: thesis: ( f1 . p = r1 & f2 . p = r2 implies g5 . p = ((r1 / r2) - a) / b )
set r8 = r1 / r2;
A12: g1 . p = b by A6;
assume ( f1 . p = r1 & f2 . p = r2 ) ; :: thesis: g5 . p = ((r1 / r2) - a) / b
then A13: g3 . p = r1 / r2 by A4;
g2 . p = a by A7;
then g4 . p = (r1 / r2) - a by A8, A13;
hence g5 . p = ((r1 / r2) - a) / b by A10, A12; :: thesis: verum

end;

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 = ((r1 / r2) - a) / b ) & g is continuous ) by A11; :: thesis: verum