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 = r2 * (((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 = r2 * (((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 = r2 * (((r1 / r2) - a) / b) ) & g is continuous ) )
assume A1: ( f1 is continuous & f2 is continuous & b <> 0 & ( 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 = r2 * (((r1 / r2) - a) / b) ) & g is continuous )
consider g1 being Function of X,R^1 such that
A2: for p being Point of X holds
( g1 . p = b & g1 is continuous ) by JGRAPH_2:30;
consider g2 being Function of X,R^1 such that
A3: for p being Point of X holds
( g2 . p = a & g2 is continuous ) by JGRAPH_2:30;
consider g3 being Function of X,R^1 such that
A4: ( ( for p being Point of X
for r1, r0 being real number st f1 . p = r1 & f2 . p = r0 holds
g3 . p = r1 / r0 ) & g3 is continuous ) by A1, JGRAPH_2:37;
consider g4 being Function of X,R^1 such that
A5: ( ( for p being Point of X
for r1, r0 being real number st g3 . p = r1 & g2 . p = r0 holds
g4 . p = r1 - r0 ) & g4 is continuous ) by A3, A4, JGRAPH_2:31;
for q being Point of X holds g1 . q <> 0 by A1, A2;
then consider g5 being Function of X,R^1 such that
A6: ( ( for p being Point of X
for r1, r0 being real number st g4 . p = r1 & g1 . p = r0 holds
g5 . p = r1 / r0 ) & g5 is continuous ) by A2, A5, JGRAPH_2:37;
consider g6 being Function of X,R^1 such that
A7: ( ( for p being Point of X
for r1, r0 being real number st f2 . p = r1 & g5 . p = r0 holds
g6 . p = r1 * r0 ) & g6 is continuous ) by A1, A6, JGRAPH_2:35;
for p being Point of X
for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds
g6 . p = r2 * (((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
g6 . p = r2 * (((r1 / r2) - a) / b)
let r1, r2 be Real; :: thesis: ( f1 . p = r1 & f2 . p = r2 implies g6 . p = r2 * (((r1 / r2) - a) / b) )
assume A8: ( f1 . p = r1 & f2 . p = r2 ) ; :: thesis: g6 . p = r2 * (((r1 / r2) - a) / b)
A9: g1 . p = b by A2;
A10: g2 . p = a by A3;
set r8 = r1 / r2;
g3 . p = r1 / r2 by A4, A8;
then g4 . p = (r1 / r2) - a by A5, A10;
then g5 . p = ((r1 / r2) - a) / b by A6, A9;
hence g6 . p = r2 * (((r1 / r2) - a) / b) by A7, A8; :: 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 = r2 * (((r1 / r2) - a) / b) ) & g is continuous ) by A7; :: thesis: verum