let X be RealUnitarySpace; :: thesis: for x being Point of X

for r being Real holds Sphere (x,r) c= cl_Ball (x,r)

let x be Point of X; :: thesis: for r being Real holds Sphere (x,r) c= cl_Ball (x,r)

let r be Real; :: thesis: Sphere (x,r) c= cl_Ball (x,r)

for r being Real holds Sphere (x,r) c= cl_Ball (x,r)

let x be Point of X; :: thesis: for r being Real holds Sphere (x,r) c= cl_Ball (x,r)

let r be Real; :: thesis: Sphere (x,r) c= cl_Ball (x,r)

now :: thesis: for y being Point of X st y in Sphere (x,r) holds

y in cl_Ball (x,r)

hence
Sphere (x,r) c= cl_Ball (x,r)
by SUBSET_1:2; :: thesis: verumy in cl_Ball (x,r)

let y be Point of X; :: thesis: ( y in Sphere (x,r) implies y in cl_Ball (x,r) )

assume y in Sphere (x,r) ; :: thesis: y in cl_Ball (x,r)

then ||.(x - y).|| = r by Th51;

hence y in cl_Ball (x,r) ; :: thesis: verum

end;assume y in Sphere (x,r) ; :: thesis: y in cl_Ball (x,r)

then ||.(x - y).|| = r by Th51;

hence y in cl_Ball (x,r) ; :: thesis: verum