let x be Point of RealSpace; :: thesis: for x9, r being Real st x9 = x holds
Ball (x,r) = ].(x9 - r),(x9 + r).[
let x9, r be Real; :: thesis: ( x9 = x implies Ball (x,r) = ].(x9 - r),(x9 + r).[ )
assume A1: x9 = x ; :: thesis: Ball (x,r) = ].(x9 - r),(x9 + r).[
thus Ball (x,r) c= ].(x9 - r),(x9 + r).[ :: according to XBOOLE_0:def 10 :: thesis: ].(x9 - r),(x9 + r).[ c= Ball (x,r)

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in Ball (x,r) or y in ].(x9 - r),(x9 + r).[ )
assume A2: y in Ball (x,r) ; :: thesis: y in ].(x9 - r),(x9 + r).[
then reconsider y1 = y as Element of RealSpace ;
reconsider x2 = x, y2 = y1 as Element of REAL by METRIC_1:def 14;
A3: dist (x,y1) = real_dist . (x2,y2) by METRIC_1:def 1, METRIC_1:def 14
.= abs (x2 - y2) by METRIC_1:def 13
.= abs (- (y2 - x2))
.= abs (y2 - x2) by COMPLEX1:138 ;
dist (x,y1) < r by A2, METRIC_1:12;
hence y in ].(x9 - r),(x9 + r).[ by A1, A3, RCOMP_1:8; :: thesis: verum

end;

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in ].(x9 - r),(x9 + r).[ or y in Ball (x,r) )
assume A4: y in ].(x9 - r),(x9 + r).[ ; :: thesis: y in Ball (x,r)
then reconsider y2 = y as Real ;
reconsider x1 = x9, y1 = y2 as Element of RealSpace by METRIC_1:def 14;
abs (y2 - x9) = abs (- (y2 - x9)) by COMPLEX1:138
.= abs (x9 - y2)
.= real_dist . (x9,y2) by METRIC_1:def 13 ;
then A5: real_dist . (x9,y2) < r by A4, RCOMP_1:8;
dist (x1,y1) = real_dist . (x9,y2) by METRIC_1:def 1, METRIC_1:def 14;
hence y in Ball (x,r) by A1, A5, METRIC_1:12; :: thesis: verum