let n be Element of NAT ; :: thesis: for p being Point of
for x, p' being Point of
for r being real number st p = p' & |.(x - p').| < r holds
x in Ball p,r
let p be Point of ; :: thesis: for x, p' being Point of
for r being real number st p = p' & |.(x - p').| < r holds
x in Ball p,r
let x, p' be Point of ; :: thesis: for r being real number st p = p' & |.(x - p').| < r holds
x in Ball p,r
let r be real number ; :: thesis: ( p = p' & |.(x - p').| < r implies x in Ball p,r )
reconsider x' = x as Point of by TOPREAL3:13;
assume ( p = p' & |.(x - p').| < r ) ; :: thesis: x in Ball p,r
then dist x',p < r by SPPOL_1:62;
hence x in Ball p,r by METRIC_1:12; :: thesis: verum