let X be non empty TopSpace; :: thesis:
let f1, f2 be Function of X,R^1; :: thesis:
assume that
A1: f1 is continuous and
A2: f2 is continuous ; :: thesis:
defpred S₁[ set , set ] means for r1, r2 being Real st f1 . $1 = r1 & f2 . $1 = r2 holds
$2 = r1 - r2;
A3: for x being Element of X ex y being Element of REAL st S₁[x,y]

let x be Element of X; :: thesis:
reconsider r1 = f1 . x as Element of REAL by TOPMETR:17;
reconsider r2 = f2 . x as Element of REAL by TOPMETR:17;
set r3 = r1 - r2;
for r1, r2 being Real st f1 . x = r1 & f2 . x = r2 holds
r1 - r2 = r1 - r2 ;
hence ex y being Element of REAL st
for r1, r2 being Real st f1 . x = r1 & f2 . x = r2 holds
y = r1 - r2 ; :: thesis:

end;

ex f being Function of the carrier of X,REAL st
for x being Element of X holds S₁[x,f . x] from FUNCT_2:sch 3(A3);
then consider f being Function of the carrier of X,REAL such that
A4: for x being Element of X
for r1, r2 being Real st f1 . x = r1 & f2 . x = r2 holds
f . x = r1 - r2 ;
reconsider g0 = f as Function of X,R^1 by TOPMETR:17;
for p being Point of X
for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )

let p be Point of X; :: thesis:
let V be Subset of R^1; :: thesis:
reconsider r = g0 . p as Real ;
reconsider r1 = f1 . p as Real ;
reconsider r2 = f2 . p as Real ;
assume ( g0 . p in V & V is open ) ; :: thesis:
then consider r0 being Real such that
A5: r0 > 0 and
A6: ].(r - r0),(r + r0).[ c= V by FRECHET:8;
reconsider G1 = ].(r1 - (r0 / 2)),(r1 + (r0 / 2)).[ as Subset of R^1 by TOPMETR:17;
A7: r1 < r1 + (r0 / 2) by A5, XREAL_1:29, XREAL_1:215;
then r1 - (r0 / 2) < r1 by XREAL_1:19;
then A8: f1 . p in G1 by A7, XXREAL_1:4;
reconsider G2 = ].(r2 - (r0 / 2)),(r2 + (r0 / 2)).[ as Subset of R^1 by TOPMETR:17;
A9: r2 < r2 + (r0 / 2) by A5, XREAL_1:29, XREAL_1:215;
then r2 - (r0 / 2) < r2 by XREAL_1:19;
then A10: f2 . p in G2 by A9, XXREAL_1:4;
G2 is open by JORDAN6:35;
then consider W2 being Subset of X such that
A11: ( p in W2 & W2 is open ) and
A12: f2 .: W2 c= G2 by A2, A10, Th10;
G1 is open by JORDAN6:35;
then consider W1 being Subset of X such that
A13: ( p in W1 & W1 is open ) and
A14: f1 .: W1 c= G1 by A1, A8, Th10;
set W = W1 /\ W2;
A15: g0 .: (W1 /\ W2) c= ].(r - r0),(r + r0).[

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in g0 .: (W1 /\ W2) ; :: thesis:
then consider z being object such that
A16: z in dom g0 and
A17: z in W1 /\ W2 and
A18: g0 . z = x by FUNCT_1:def 6;
reconsider pz = z as Point of X by A16;
reconsider aa2 = f2 . pz as Real ;
reconsider aa1 = f1 . pz as Real ;
A19: pz in the carrier of X ;
then A20: pz in dom f1 by FUNCT_2:def 1;
A21: pz in dom f2 by A19, FUNCT_2:def 1;
z in W2 by A17, XBOOLE_0:def 4;
then A22: f2 . pz in f2 .: W2 by A21, FUNCT_1:def 6;
then A23: r2 - (r0 / 2) < aa2 by A12, XXREAL_1:4;
A24: aa2 < r2 + (r0 / 2) by A12, A22, XXREAL_1:4;
z in W1 by A17, XBOOLE_0:def 4;
then A25: f1 . pz in f1 .: W1 by A20, FUNCT_1:def 6;
then r1 - (r0 / 2) < aa1 by A14, XXREAL_1:4;
then (r1 - (r0 / 2)) - (r2 + (r0 / 2)) < aa1 - aa2 by A24, XREAL_1:14;
then (r1 - r2) - ((r0 / 2) + (r0 / 2)) < aa1 - aa2 ;
then A26: r - r0 < aa1 - aa2 by A4;
A27: x = aa1 - aa2 by A4, A18;
then reconsider rx = x as Real ;
aa1 < r1 + (r0 / 2) by A14, A25, XXREAL_1:4;
then aa1 - aa2 < (r1 + (r0 / 2)) - (r2 - (r0 / 2)) by A23, XREAL_1:14;
then aa1 - aa2 < (r1 - r2) + ((r0 / 2) + (r0 / 2)) ;
then rx < r + r0 by A4, A27;
hence x in ].(r - r0),(r + r0).[ by A27, A26, XXREAL_1:4; :: thesis:

end;

( W1 /\ W2 is open & p in W1 /\ W2 ) by A13, A11, XBOOLE_0:def 4;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A6, A15, XBOOLE_1:1; :: thesis:

end;

then A28: g0 is continuous by Th10;
for p being Point of X
for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds
g0 . p = r1 - r2 by A4;
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 A28; :: thesis: