let r be real number ; :: thesis: for M being MetrStruct
for p being Element of holds Sphere p,r c= cl_Ball p,r
let M be MetrStruct ; :: thesis: for p being Element of holds Sphere p,r c= cl_Ball p,r
let p be Element of ; :: thesis: Sphere p,r c= cl_Ball p,r

per cases ( not M is empty or M is empty ) ;

suppose not M is empty ; :: thesis: Sphere p,r c= cl_Ball p,r

then consider M' being non empty MetrStruct , p' being Element of such that
A1: ( M' = M & p' = p ) and
Sphere p,r = { q where q is Element of : dist p',q = r } by Def17;

now

let x be Element of ; :: thesis: ( x in Sphere p',r implies x in cl_Ball p',r )
assume x in Sphere p',r ; :: thesis: x in cl_Ball p',r
then dist p',x = r by Th14;
then x in { q where q is Element of : dist p',q <= r } ;
hence x in cl_Ball p',r by Lm6; :: thesis: verum

end;

hence Sphere p,r c= cl_Ball p,r by A1, SUBSET_1:7; :: thesis: verum

end;

suppose A2: M is empty ; :: thesis: Sphere p,r c= cl_Ball p,r

then Sphere p,r is empty by Def17;
hence Sphere p,r c= cl_Ball p,r by A2, Def16; :: thesis: verum

end;

end;