let X be non empty TopSpace; :: thesis: for f1 being Function of X,R^1
for a being Real st f1 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being Real st f1 . p = r1 holds
g . p = a - r1 ) & g is continuous )
let f1 be Function of X,R^1; :: thesis: for a being Real st f1 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being Real st f1 . p = r1 holds
g . p = a - r1 ) & g is continuous )
let a be Real; :: thesis: ( f1 is continuous implies ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being Real st f1 . p = r1 holds
g . p = a - 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 st f1 . p = r1 holds
g . p = a - 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 st f1 . p = r1 holds
g1 . p = r1 - a and
A2: g1 is continuous by Th7;
consider g2 being Function of X,R^1 such that
A3: for p being Point of X
for r1 being Real st g1 . p = r1 holds
g2 . p = - r1 and
A4: g2 is continuous by A2, JGRAPH_4:8;
for p being Point of X
for r1 being Real st f1 . p = r1 holds
g2 . p = a - r1

proof

let p be Point of X; :: thesis: for r1 being Real st f1 . p = r1 holds
g2 . p = a - r1
let r1 be Real; :: thesis: ( f1 . p = r1 implies g2 . p = a - r1 )
assume f1 . p = r1 ; :: thesis: g2 . p = a - r1
then g1 . p = r1 - a by A1;
then g2 . p = - (r1 - a) by A3
.= a - r1 ;
hence g2 . p = a - r1 ; :: thesis: verum

end;

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