let X be non empty TopSpace; :: thesis: for f1 being Function of X,R^1 st f1 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = abs r1 ) & g is continuous )
let f1 be Function of X,R^1; :: thesis: ( f1 is continuous implies ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = abs r1 ) & g is continuous ) )
assume f1 is continuous ; :: thesis: ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = abs r1 ) & g is continuous )
then consider g1 being Function of X,R^1 such that
A1: for p being Point of X
for r1 being real number st f1 . p = r1 holds
g1 . p = r1 ^2 and
A2: g1 is continuous by Th11;
for q being Point of X ex r being real number st
( g1 . q = r & r >= 0 )

proof

let q be Point of X; :: thesis: ex r being real number st
( g1 . q = r & r >= 0 )
reconsider r11 = f1 . q as Real by TOPMETR:17;
g1 . q = r11 ^2 by A1;
hence ex r being real number st
( g1 . q = r & r >= 0 ) by XREAL_1:63; :: thesis: verum

end;

then consider g2 being Function of X,R^1 such that
A3: for p being Point of X
for r1 being real number st g1 . p = r1 holds
g2 . p = sqrt r1 and
A4: g2 is continuous by A2, JGRAPH_3:5;
for p being Point of X
for r1 being real number st f1 . p = r1 holds
g2 . p = abs r1

proof

let p be Point of X; :: thesis: for r1 being real number st f1 . p = r1 holds
g2 . p = abs r1
let r1 be real number ; :: thesis: ( f1 . p = r1 implies g2 . p = abs r1 )
assume f1 . p = r1 ; :: thesis: g2 . p = abs r1
then g1 . p = r1 ^2 by A1;
then g2 . p = sqrt (r1 ^2) by A3
.= abs r1 by COMPLEX1:72 ;
hence g2 . p = abs r1 ; :: thesis: verum

end;

hence ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = abs r1 ) & g is continuous ) by A4; :: thesis: verum