let X be non empty TopSpace; :: thesis: for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous 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 ) & g is continuous )
let f1, f2 be Function of X,R^1; :: thesis: ( f1 is continuous & f2 is continuous 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 ) & g is continuous ) )
assume A1: ( f1 is continuous & f2 is continuous ) ; :: 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 ) & g is continuous )
then consider g1 being Function of X,R^1 such that
A2: for p being Point of X
for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds
g1 . p = r1 + r2 and
A3: g1 is continuous by Th19;
consider g3 being Function of X,R^1 such that
A4: for p being Point of X
for r1 being Real st g1 . p = r1 holds
g3 . p = r1 * r1 and
A5: g3 is continuous by A3, Th22;
consider g2 being Function of X,R^1 such that
A6: for p being Point of X
for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds
g2 . p = r1 - r2 and
A7: g2 is continuous by A1, Th21;
consider g4 being Function of X,R^1 such that
A8: for p being Point of X
for r1 being Real st g2 . p = r1 holds
g4 . p = r1 * r1 and
A9: g4 is continuous by A7, Th22;
consider g5 being Function of X,R^1 such that
A10: for p being Point of X
for r1, r2 being Real st g3 . p = r1 & g4 . p = r2 holds
g5 . p = r1 - r2 and
A11: g5 is continuous by A5, A9, Th21;
consider g6 being Function of X,R^1 such that
A12: for p being Point of X
for r1 being Real st g5 . p = r1 holds
g6 . p = (1 / 4) * r1 and
A13: g6 is continuous by A11, Th23;
for p being Point of X
for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds
g6 . p = r1 * r2

proof

let p be Point of X; :: thesis: for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds
g6 . p = r1 * r2
let r1, r2 be Real; :: thesis: ( f1 . p = r1 & f2 . p = r2 implies g6 . p = r1 * r2 )
assume A14: ( f1 . p = r1 & f2 . p = r2 ) ; :: thesis: g6 . p = r1 * r2
then g2 . p = r1 - r2 by A6;
then A15: g4 . p = (r1 - r2) ^2 by A8;
g1 . p = r1 + r2 by A2, A14;
then g3 . p = (r1 + r2) ^2 by A4;
then g5 . p = ((r1 + r2) ^2) - ((r1 - r2) ^2) by A10, A15;
then g6 . p = (1 / 4) * ((((r1 ^2) + ((2 * r1) * r2)) + (r2 ^2)) - ((r1 - r2) ^2)) by A12
.= r1 * r2 ;
hence g6 . p = r1 * r2 ; :: 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 ) & g is continuous ) by A13; :: thesis: verum