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 = r1 - a ) & 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 = r1 - a ) & 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 = r1 - a ) & 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 = r1 - a ) & 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 JGRAPH_2:24;
for p being Point of X
for r1 being Real st f1 . p = r1 holds
g1 . p = r1 - a
proof
let p be Point of X; :: thesis: for r1 being Real st f1 . p = r1 holds
g1 . p = r1 - a

let r1 be Real; :: thesis: ( f1 . p = r1 implies g1 . p = r1 - a )
assume f1 . p = r1 ; :: thesis: g1 . p = r1 - a
then g1 . p = r1 + (- a) by A1;
hence g1 . p = r1 - a ; :: 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 = r1 - a ) & g is continuous ) by A2; :: thesis: verum