let m be Nat; :: thesis: for f being Function of R^1,(TOP-REAL m) holds
( f is open iff for p being Point of R^1
for r being real positive number ex s being real positive number st Ball ((f . p),s) c= f .: ].(p - r),(p + r).[ )
let f be Function of R^1,(TOP-REAL m); :: thesis: ( f is open iff for p being Point of R^1
for r being real positive number ex s being real positive number st Ball ((f . p),s) c= f .: ].(p - r),(p + r).[ )
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 R^1,(TopSpaceMetr (Euclid m)) ;
A2: m in NAT by ORDINAL1:def 12;
thus ( f is open implies for p being Point of R^1
for r being real positive number ex s being real positive number st Ball ((f . p),s) c= f .: ].(p - r),(p + r).[ ) :: thesis: ( ( for p being Point of R^1
for r being real positive number ex s being real positive number st Ball ((f . p),s) c= f .: ].(p - r),(p + r).[ ) implies f is open )

proof

assume A3: f is open ; :: thesis: for p being Point of R^1
for r being real positive number ex s being real positive number st Ball ((f . p),s) c= f .: ].(p - r),(p + r).[
let p be Point of R^1; :: thesis: for r being real positive number ex s being real positive number st Ball ((f . p),s) c= f .: ].(p - r),(p + r).[
let r be real positive number ; :: thesis: ex s being real positive number st Ball ((f . p),s) c= f .: ].(p - r),(p + r).[
reconsider p1 = p as Point of RealSpace ;
reconsider q1 = f . p as Point of (Euclid m) by EUCLID:67;
f1 is open by A1, A3, Th1;
then consider s being real positive number such that
A4: Ball (q1,s) c= f1 .: (Ball (p1,r)) by Th6;
r in REAL by XREAL_0:def 1;
then ( ].(p - r),(p + r).[ = Ball (p1,r) & Ball ((f . p),s) = Ball (q1,s) ) by A2, FRECHET:7, TOPREAL9:13;
hence ex s being real positive number st Ball ((f . p),s) c= f .: ].(p - r),(p + r).[ by A4; :: thesis: verum

end;

assume A5: for p being Point of R^1
for r being real positive number ex s being real positive number st Ball ((f . p),s) c= f .: ].(p - r),(p + r).[ ; :: thesis: f is open
for p being Point of RealSpace
for q being Point of (Euclid m)
for r being real positive number st q = f1 . p holds
ex s being real positive number st Ball (q,s) c= f1 .: (Ball (p,r))

proof

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

end;

then f1 is open by Th6;
hence f is open by A1, Th1; :: thesis: verum