let p be Point of (TOP-REAL 2); :: thesis: for e being Point of (Euclid 2)
for D being non empty Subset of (TOP-REAL 2)
for r being real number st D = Ball e,r & p = e holds
W-bound D = (p `1 ) - r
let e be Point of (Euclid 2); :: thesis: for D being non empty Subset of (TOP-REAL 2)
for r being real number st D = Ball e,r & p = e holds
W-bound D = (p `1 ) - r
let D be non empty Subset of (TOP-REAL 2); :: thesis: for r being real number st D = Ball e,r & p = e holds
W-bound D = (p `1 ) - r
let r be real number ; :: thesis: ( D = Ball e,r & p = e implies W-bound D = (p `1 ) - r )
assume that
A1: D = Ball e,r and
A2: p = e ; :: thesis: W-bound D = (p `1 ) - r
A3: r > 0 by A1, TBSP_1:17;
then A4: (p `1 ) - r < (p `1 ) - 0 by XREAL_1:17;
(p `1 ) + 0 < (p `1 ) + r by A3, XREAL_1:8;
then (p `1 ) - r < (p `1 ) + r by A4, XXREAL_0:2;
then A5: inf ].((p `1 ) - r),((p `1 ) + r).[ = (p `1 ) - r by Th22;
proj1 .: D = ].((p `1 ) - r),((p `1 ) + r).[ by A1, A2, Th51;
hence W-bound D = (p `1 ) - r by A5, SPRECT_1:48; :: thesis: verum