let r be real number ; :: thesis: for M being MetrStruct
for p being Element of M holds (Sphere (p,r)) \/ (Ball (p,r)) = cl_Ball (p,r)
let M be MetrStruct ; :: thesis: for p being Element of M holds (Sphere (p,r)) \/ (Ball (p,r)) = cl_Ball (p,r)
let p be Element of M; :: thesis: (Sphere (p,r)) \/ (Ball (p,r)) = cl_Ball (p,r)
( Sphere (p,r) c= cl_Ball (p,r) & Ball (p,r) c= cl_Ball (p,r) ) by Th15, Th16;
hence (Sphere (p,r)) \/ (Ball (p,r)) c= cl_Ball (p,r) by XBOOLE_1:8; :: according to XBOOLE_0:def 10 :: thesis: cl_Ball (p,r) c= (Sphere (p,r)) \/ (Ball (p,r))

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

suppose not M is empty ; :: thesis: cl_Ball (p,r) c= (Sphere (p,r)) \/ (Ball (p,r))

then consider M9 being non empty MetrStruct , p9 being Element of M9 such that
A1: ( M9 = M & p9 = p ) and
Sphere (p,r) = { q where q is Element of M9 : dist (p9,q) = r } by Def17;

now

let x be Element of M9; :: thesis: ( x in cl_Ball (p9,r) implies x in (Sphere (p9,r)) \/ (Ball (p9,r)) )
assume x in cl_Ball (p9,r) ; :: thesis: x in (Sphere (p9,r)) \/ (Ball (p9,r))
then A2: dist (p9,x) <= r by Th13;

now

per cases ( dist (p9,x) < r or dist (p9,x) = r ) by A2, XXREAL_0:1;

case dist (p9,x) < r ; :: thesis: x in Ball (p9,r)

hence x in Ball (p9,r) by Th12; :: thesis: verum

end;

case dist (p9,x) = r ; :: thesis: x in Sphere (p9,r)

hence x in Sphere (p9,r) by Th14; :: thesis: verum

end;

end;

end;

hence x in (Sphere (p9,r)) \/ (Ball (p9,r)) by XBOOLE_0:def 3; :: thesis: verum

end;

hence cl_Ball (p,r) c= (Sphere (p,r)) \/ (Ball (p,r)) by A1, SUBSET_1:2; :: thesis: verum

end;

suppose A3: M is empty ; :: thesis: cl_Ball (p,r) c= (Sphere (p,r)) \/ (Ball (p,r))

then ( Ball (p,r) is empty & cl_Ball (p,r) is empty ) by Def15, Def16;
hence cl_Ball (p,r) c= (Sphere (p,r)) \/ (Ball (p,r)) by A3, Def17; :: thesis: verum

end;

end;