let m be Nat; :: thesis:
let f be Function of (TOP-REAL m),R^1 ; :: thesis:
A1: TopStruct(# the U1 of (TOP-REAL m),the topology of (TOP-REAL m) #) = TopSpaceMetr (Euclid m) by EUCLID:def 8;
then reconsider f1 = f as Function of (TopSpaceMetr (Euclid m)),R^1 ;
A2: m in NAT by ORDINAL1:def 13;

hereby :: thesis:

assume A3: f is continuous ; :: thesis:
let p be Point of (TOP-REAL m); :: thesis:
let r be real positive number ; :: thesis:
reconsider p1 = p as Point of (Euclid m) by EUCLID:71;
reconsider q1 = f . p as Point of RealSpace ;
f1 is continuous by A1, A3, YELLOW12:36;
then consider s being real positive number such that
A4: f .: (Ball p1,s) c= Ball q1,r by A1, Th17;
take s = s; :: thesis:
( q1 in REAL & r in REAL ) by XREAL_0:def 1;
then ( Ball p1,s = Ball p,s & Ball q1,r = ].((f . p) - r),((f . p) + r).[ ) by A2, TOPREAL9:13, FRECHET:7;
hence f .: (Ball p,s) c= ].((f . p) - r),((f . p) + r).[ by A4; :: thesis:

end;

assume A5: for p being Point of (TOP-REAL m)
for r being real positive number ex s being real positive number st f .: (Ball p,s) c= ].((f . p) - r),((f . p) + r).[ ; :: thesis:
for p being Point of (Euclid m)
for q being Point of RealSpace
for r being real positive number st q = f1 . p holds
ex s being real positive number st f1 .: (Ball p,s) c= Ball q,r

proof

let p be Point of (Euclid m); :: thesis:
let q be Point of RealSpace ; :: thesis:
let r be real positive number ; :: thesis:
assume A6: q = f1 . p ; :: thesis:
reconsider p1 = p as Point of (TOP-REAL m) by EUCLID:71;
consider s being real positive number such that
A7: f .: (Ball p1,s) c= ].((f . p) - r),((f . p) + r).[ by A5;
take s ; :: thesis:
( q in REAL & r in REAL ) by XREAL_0:def 1;
then ( Ball p1,s = Ball p,s & ].((f . p) - r),((f . p) + r).[ = Ball q,r ) by A2, A6, TOPREAL9:13, FRECHET:7;
hence f1 .: (Ball p,s) c= Ball q,r by A7; :: thesis:

end;

then f1 is continuous by Th17;
hence f is continuous by A1, YELLOW12:36; :: thesis: