let x be Point of RealSpace ; :: thesis: for x', r being Real st x' = x & r > 0 holds
Ball x,r = ].(x' - r),(x' + r).[
let x', r be Real; :: thesis: ( x' = x & r > 0 implies Ball x,r = ].(x' - r),(x' + r).[ )
assume A1: ( x' = x & r > 0 ) ; :: thesis: Ball x,r = ].(x' - r),(x' + r).[
thus Ball x,r c= ].(x' - r),(x' + r).[ :: according to XBOOLE_0:def 10 :: thesis: ].(x' - r),(x' + 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 ].(x' - r),(x' + r).[ )
assume A2: y in Ball x,r ; :: thesis: y in ].(x' - r),(x' + r).[
then reconsider y1 = y as Element of RealSpace ;
A3: dist x,y1 < r by A2, METRIC_1:12;
reconsider x2 = x, y2 = y1 as Element of REAL by METRIC_1:def 14;
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 ;
hence y in ].(x' - r),(x' + r).[ by A1, A3, RCOMP_1:8; :: thesis: verum

end;

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