let r be Real; :: 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 Th14, Th15;
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 A1: not M is empty ; :: thesis: cl_Ball (p,r) c= (Sphere (p,r)) \/ (Ball (p,r))

now :: thesis: for x being Element of M st x in cl_Ball (p,r) holds
x in (Sphere (p,r)) \/ (Ball (p,r))

let x be Element of M; :: thesis: ( x in cl_Ball (p,r) implies x in (Sphere (p,r)) \/ (Ball (p,r)) )
assume x in cl_Ball (p,r) ; :: thesis: x in (Sphere (p,r)) \/ (Ball (p,r))
then A2: dist (p,x) <= r by Th12;

now :: thesis: ( ( dist (p,x) < r & x in Ball (p,r) ) or ( dist (p,x) = r & x in Sphere (p,r) ) )

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

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

hence x in Ball (p,r) by A1, Th11; :: thesis: verum

end;

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

hence x in Sphere (p,r) by A1, Th13; :: thesis: verum

end;

end;

end;

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

end;

hence cl_Ball (p,r) c= (Sphere (p,r)) \/ (Ball (p,r)) by 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 ) ;
hence cl_Ball (p,r) c= (Sphere (p,r)) \/ (Ball (p,r)) by A3, Def16; :: thesis: verum

end;

end;