let a be Real; :: thesis: for X being non empty compact Subset of (TOP-REAL 2)
for p being Point of (Euclid 2) st p = 0. (TOP-REAL 2) & a > 0 holds
X c= Ball (p,((((|.(E-bound X).| + |.(N-bound X).|) + |.(W-bound X).|) + |.(S-bound X).|) + a))
let X be non empty compact Subset of (TOP-REAL 2); :: thesis: for p being Point of (Euclid 2) st p = 0. (TOP-REAL 2) & a > 0 holds
X c= Ball (p,((((|.(E-bound X).| + |.(N-bound X).|) + |.(W-bound X).|) + |.(S-bound X).|) + a))
let p be Point of (Euclid 2); :: thesis: ( p = 0. (TOP-REAL 2) & a > 0 implies X c= Ball (p,((((|.(E-bound X).| + |.(N-bound X).|) + |.(W-bound X).|) + |.(S-bound X).|) + a)) )
assume that
A1: p = 0. (TOP-REAL 2) and
A2: a > 0 ; :: thesis: X c= Ball (p,((((|.(E-bound X).| + |.(N-bound X).|) + |.(W-bound X).|) + |.(S-bound X).|) + a))
set A = X;
set n = N-bound X;
set s = S-bound X;
set e = E-bound X;
set w = W-bound X;
set r = (((|.(E-bound X).| + |.(N-bound X).|) + |.(W-bound X).|) + |.(S-bound X).|) + a;
A3: (((|.(E-bound X).| + |.(N-bound X).|) + |.(W-bound X).|) + |.(S-bound X).|) + 0 < (((|.(E-bound X).| + |.(N-bound X).|) + |.(W-bound X).|) + |.(S-bound X).|) + a by A2, XREAL_1:8;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in Ball (p,((((|.(E-bound X).| + |.(N-bound X).|) + |.(W-bound X).|) + |.(S-bound X).|) + a)) )
assume A4: x in X ; :: thesis: x in Ball (p,((((|.(E-bound X).| + |.(N-bound X).|) + |.(W-bound X).|) + |.(S-bound X).|) + a))
then reconsider b = x as Point of (Euclid 2) by TOPREAL3:8;
reconsider P = p, B = b as Point of (TOP-REAL 2) by TOPREAL3:8;
A5: P `1 = 0 by A1, Th22;
A6: B `1 <= E-bound X by A4, PSCOMP_1:24;
A7: B `2 <= N-bound X by A4, PSCOMP_1:24;
A8: S-bound X <= B `2 by A4, PSCOMP_1:24;
A9: P `2 = 0 by A1, Th22;
A10: dist (p,b) = (Pitag_dist 2) . (p,b) by METRIC_1:def 1
.= sqrt ((((P `1) - (B `1)) ^2) + (((P `2) - (B `2)) ^2)) by TOPREAL3:7
.= sqrt (((B `1) ^2) + ((B `2) ^2)) by A5, A9 ;
A11: 0 <= (B `2) ^2 by XREAL_1:63;
0 <= (B `1) ^2 by XREAL_1:63;
then sqrt (((B `1) ^2) + ((B `2) ^2)) <= (sqrt ((B `1) ^2)) + (sqrt ((B `2) ^2)) by A11, SQUARE_1:59;
then sqrt (((B `1) ^2) + ((B `2) ^2)) <= |.(B `1).| + (sqrt ((B `2) ^2)) by COMPLEX1:72;
then A12: sqrt (((B `1) ^2) + ((B `2) ^2)) <= |.(B `1).| + |.(B `2).| by COMPLEX1:72;
A13: 0 <= |.(N-bound X).| by COMPLEX1:46;
A14: 0 <= |.(E-bound X).| by COMPLEX1:46;
A15: 0 <= |.(W-bound X).| by COMPLEX1:46;
A16: 0 <= |.(S-bound X).| by COMPLEX1:46;
A17: W-bound X <= B `1 by A4, PSCOMP_1:24;
hence x in Ball (p,((((|.(E-bound X).| + |.(N-bound X).|) + |.(W-bound X).|) + |.(S-bound X).|) + a)) by METRIC_1:11; :: thesis: verum