let M1, M2 be non empty MetrSpace; :: thesis: for f being Function of (),() holds
( f is open iff for p being Point of M1
for q being Point of M2
for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r)) )

let f be Function of (),(); :: thesis: ( f is open iff for p being Point of M1
for q being Point of M2
for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r)) )

thus ( f is open implies for p being Point of M1
for q being Point of M2
for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r)) ) :: thesis: ( ( for p being Point of M1
for q being Point of M2
for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r)) ) implies f is open )
proof
assume A1: f is open ; :: thesis: for p being Point of M1
for q being Point of M2
for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r))

let p be Point of M1; :: thesis: for q being Point of M2
for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r))

let q be Point of M2; :: thesis: for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r))

let r be positive Real; :: thesis: ( q = f . p implies ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r)) )
assume A2: q = f . p ; :: thesis: ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r))
reconsider V = Ball (p,r) as Subset of () ;
A3: p in V by GOBOARD6:1;
V is open by TOPMETR:14;
hence ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r)) by A1, A2, A3, Th4; :: thesis: verum
end;
assume A4: for p being Point of M1
for q being Point of M2
for r being positive Real st q = f . p holds
ex s being positive Real st Ball (q,s) c= f .: (Ball (p,r)) ; :: thesis: f is open
for p being Point of ()
for V being open Subset of ()
for q being Point of M2 st q = f . p & p in V holds
ex r being positive Real st Ball (q,r) c= f .: V
proof
let p be Point of (); :: thesis: for V being open Subset of ()
for q being Point of M2 st q = f . p & p in V holds
ex r being positive Real st Ball (q,r) c= f .: V

let V be open Subset of (); :: thesis: for q being Point of M2 st q = f . p & p in V holds
ex r being positive Real st Ball (q,r) c= f .: V

let q be Point of M2; :: thesis: ( q = f . p & p in V implies ex r being positive Real st Ball (q,r) c= f .: V )
assume A5: q = f . p ; :: thesis: ( not p in V or ex r being positive Real st Ball (q,r) c= f .: V )
reconsider p1 = p as Point of M1 ;
assume p in V ; :: thesis: ex r being positive Real st Ball (q,r) c= f .: V
then consider r being Real such that
A6: r > 0 and
A7: Ball (p1,r) c= V by TOPMETR:15;
A8: f .: (Ball (p1,r)) c= f .: V by ;
ex s being positive Real st Ball (q,s) c= f .: (Ball (p1,r)) by A4, A5, A6;
hence ex r being positive Real st Ball (q,r) c= f .: V by ; :: thesis: verum
end;
hence f is open by Th4; :: thesis: verum