let x be real number ; :: thesis: for a, r being real positive number holds (Ball |[x,(r * a)]|,(r * a)) \/ {|[x,0 ]|} c= (+ x,r) " [.0 ,a.[
let a, r be real positive number ; :: thesis: (Ball |[x,(r * a)]|,(r * a)) \/ {|[x,0 ]|} c= (+ x,r) " [.0 ,a.[
let u be set ; :: according to TARSKI:def 3 :: thesis: ( not u in (Ball |[x,(r * a)]|,(r * a)) \/ {|[x,0 ]|} or u in (+ x,r) " [.0 ,a.[ )
assume A1: u in (Ball |[x,(r * a)]|,(r * a)) \/ {|[x,0 ]|} ; :: thesis: u in (+ x,r) " [.0 ,a.[
then A2: ( u in Ball |[x,(r * a)]|,(r * a) or u in {|[x,0 ]|} ) by XBOOLE_0:def 3;
reconsider p = u as Point of (TOP-REAL 2) by A1;
A3: |[x,0 ]| in y>=0-plane by Th22;
Ball |[x,(r * a)]|,(r * a) c= y>=0-plane by Th24;
then reconsider q = p as Point of Niemytzki-plane by A3, A2, Lm1, TARSKI:def 1;
A5: (+ x,r) . q <= 1 by BORSUK_1:83, XXREAL_1:1;
A6: (+ x,r) . q >= 0 by BORSUK_1:83, XXREAL_1:1;

per cases ( a > 1 or ( a <= 1 & u in Ball |[x,(r * a)]|,(r * a) ) or u = |[x,0 ]| ) by A2, TARSKI:def 1;

suppose a > 1 ; :: thesis: u in (+ x,r) " [.0 ,a.[

then (+ x,r) . q < a by A5, XXREAL_0:2;
then (+ x,r) . q in [.0 ,a.[ by A6, XXREAL_1:3;
hence u in (+ x,r) " [.0 ,a.[ by FUNCT_2:46; :: thesis: verum

end;

suppose A7: ( a <= 1 & u in Ball |[x,(r * a)]|,(r * a) ) ; :: thesis: u in (+ x,r) " [.0 ,a.[

then |.(p - |[x,(r * a)]|).| < r * a by TOPREAL9:7;
then (+ x,r) . p < a by A7, Th67;
then (+ x,r) . q in [.0 ,a.[ by A6, XXREAL_1:3;
hence u in (+ x,r) " [.0 ,a.[ by FUNCT_2:46; :: thesis: verum

end;

suppose u = |[x,0 ]| ; :: thesis: u in (+ x,r) " [.0 ,a.[

then (+ x,r) . u = 0 by Def5;
then (+ x,r) . q in [.0 ,a.[ by XXREAL_1:3;
hence u in (+ x,r) " [.0 ,a.[ by FUNCT_2:46; :: thesis: verum

end;

end;