let n be Element of NAT ; :: thesis: for p being Point of (Euclid n)
for x, p' being Point of (TOP-REAL n)
for r being real number st p = p' & x in Ball p,r holds
|.(x - p').| < r
let p be Point of (Euclid n); :: thesis: for x, p' being Point of (TOP-REAL n)
for r being real number st p = p' & x in Ball p,r holds
|.(x - p').| < r
let x, p' be Point of (TOP-REAL n); :: thesis: for r being real number st p = p' & x in Ball p,r holds
|.(x - p').| < r
let r be real number ; :: thesis: ( p = p' & x in Ball p,r implies |.(x - p').| < r )
reconsider x' = x as Point of (Euclid n) by TOPREAL3:13;
assume A1: ( p = p' & x in Ball p,r ) ; :: thesis: |.(x - p').| < r
then dist x',p < r by METRIC_1:12;
hence |.(x - p').| < r by A1, SPPOL_1:62; :: thesis: verum