let x be real number ; :: thesis: for a, r being real positive number holds Ball |[x,(r * a)]|,(r * a) c= (+ x,r) " ].0 ,a.[
let a, r be real positive number ; :: thesis: Ball |[x,(r * a)]|,(r * a) c= (+ x,r) " ].0 ,a.[
let u be set ; :: according to TARSKI:def 3 :: thesis: ( not u in Ball |[x,(r * a)]|,(r * a) or u in (+ x,r) " ].0 ,a.[ )
assume A1: u in Ball |[x,(r * a)]|,(r * a) ; :: thesis: u in (+ x,r) " ].0 ,a.[
then reconsider p = u as Point of (TOP-REAL 2) ;
Ball |[x,(r * a)]|,(r * a) c= y>=0-plane by Th24;
then reconsider q = p as Point of Niemytzki-plane by A1, Def3;
q = |[(p `1 ),(p `2 )]| by EUCLID:57;
then A2: p `2 >= 0 by Lm1, Th22;

A3: now

assume (+ x,r) . p = 0 ; :: thesis: contradiction
then p = |[x,0 ]| by A2, Th64;
then A4: p in y=0-line by Th19;
Ball |[x,(r * a)]|,(r * a) misses y=0-line by Th25;
hence contradiction by A4, A1, XBOOLE_0:3; :: thesis: verum

end;

A5: (+ x,r) . q in the carrier of I[01] by FUNCT_2:7;
then A6: (+ x,r) . q <= 1 by BORSUK_1:83, XXREAL_1:1;

per cases ( a > 1 or a <= 1 ) ;

suppose A7: a > 1 ; :: thesis: u in (+ x,r) " ].0 ,a.[

A8: (+ x,r) . q > 0 by A5, A3, BORSUK_1:83, XXREAL_1:1;
(+ x,r) . q < a by A7, A6, XXREAL_0:2;
then (+ x,r) . q in ].0 ,a.[ by A8, XXREAL_1:4;
hence u in (+ x,r) " ].0 ,a.[ by FUNCT_2:46; :: thesis: verum

end;

suppose A9: a <= 1 ; :: thesis: u in (+ x,r) " ].0 ,a.[

|.(p - |[x,(r * a)]|).| < r * a by A1, TOPREAL9:7;
then A10: (+ x,r) . p < a by A9, Th67;
(+ x,r) . q > 0 by A5, A3, BORSUK_1:83, XXREAL_1:1;
then (+ x,r) . q in ].0 ,a.[ by A10, XXREAL_1:4;
hence u in (+ x,r) " ].0 ,a.[ by FUNCT_2:46; :: thesis: verum

end;

end;