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,(((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (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,(((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a)
let p be Point of (Euclid 2); :: thesis: ( p = 0. (TOP-REAL 2) & a > 0 implies X c= Ball p,(((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a) )
assume that
A1: p = 0. (TOP-REAL 2) and
A2: a > 0 ; :: thesis: X c= Ball p,(((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (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 = ((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a;
A3: ((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + 0 < ((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a by A2, XREAL_1:10;
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in Ball p,(((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a) )
assume A4: x in X ; :: thesis: x in Ball p,(((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a)
then reconsider b = x as Point of (Euclid 2) by TOPREAL3:13;
reconsider P = p, B = b as Point of (TOP-REAL 2) by TOPREAL3:13;
A5: P `1 = 0 by A1, Th32;
A6: B `1 <= E-bound X by A4, PSCOMP_1:71;
A7: B `2 <= N-bound X by A4, PSCOMP_1:71;
A8: S-bound X <= B `2 by A4, PSCOMP_1:71;
A9: P `2 = 0 by A1, Th32;
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 TOPREAL3:12
.= sqrt (((B `1 ) ^2 ) + ((B `2 ) ^2 )) by A5, A9 ;
A11: 0 <= (B `2 ) ^2 by XREAL_1:65;
0 <= (B `1 ) ^2 by XREAL_1:65;
then sqrt (((B `1 ) ^2 ) + ((B `2 ) ^2 )) <= (sqrt ((B `1 ) ^2 )) + (sqrt ((B `2 ) ^2 )) by A11, Th6;
then sqrt (((B `1 ) ^2 ) + ((B `2 ) ^2 )) <= (abs (B `1 )) + (sqrt ((B `2 ) ^2 )) by COMPLEX1:158;
then A12: sqrt (((B `1 ) ^2 ) + ((B `2 ) ^2 )) <= (abs (B `1 )) + (abs (B `2 )) by COMPLEX1:158;
A13: 0 <= abs (N-bound X) by COMPLEX1:132;
A14: 0 <= abs (E-bound X) by COMPLEX1:132;
A15: 0 <= abs (W-bound X) by COMPLEX1:132;
A16: 0 <= abs (S-bound X) by COMPLEX1:132;
A17: W-bound X <= B `1 by A4, PSCOMP_1:71;

now

per cases ( ( B `1 >= 0 & B `2 >= 0 ) or ( B `1 < 0 & B `2 >= 0 ) or ( B `1 >= 0 & B `2 < 0 ) or ( B `1 < 0 & B `2 < 0 ) ) ;

case A18: ( B `1 >= 0 & B `2 >= 0 ) ; :: thesis: dist p,b < ((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a

((abs (E-bound X)) + (abs (N-bound X))) + 0 <= ((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X)) by A15, XREAL_1:9;
then (abs (E-bound X)) + (abs (N-bound X)) <= (((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X)) by A16, XREAL_1:9;
then A19: (abs (E-bound X)) + (abs (N-bound X)) < ((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a by A3, XXREAL_0:2;
A20: abs (B `2 ) <= abs (N-bound X) by A7, A18, Th7;
abs (B `1 ) <= abs (E-bound X) by A6, A18, Th7;
then (abs (B `1 )) + (abs (B `2 )) <= (abs (E-bound X)) + (abs (N-bound X)) by A20, XREAL_1:9;
then dist p,b <= (abs (E-bound X)) + (abs (N-bound X)) by A10, A12, XXREAL_0:2;
hence dist p,b < ((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a by A19, XXREAL_0:2; :: thesis: verum

end;

case A21: ( B `1 < 0 & B `2 >= 0 ) ; :: thesis: dist p,b < ((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a

0 + ((abs (N-bound X)) + (abs (W-bound X))) <= (abs (E-bound X)) + ((abs (N-bound X)) + (abs (W-bound X))) by A14, XREAL_1:9;
then (abs (W-bound X)) + (abs (N-bound X)) <= (((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X)) by A16, XREAL_1:9;
then A22: (abs (W-bound X)) + (abs (N-bound X)) < ((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a by A3, XXREAL_0:2;
A23: abs (B `2 ) <= abs (N-bound X) by A7, A21, Th7;
abs (B `1 ) <= abs (W-bound X) by A17, A21, Th8;
then (abs (B `1 )) + (abs (B `2 )) <= (abs (W-bound X)) + (abs (N-bound X)) by A23, XREAL_1:9;
then dist p,b <= (abs (W-bound X)) + (abs (N-bound X)) by A10, A12, XXREAL_0:2;
hence dist p,b < ((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a by A22, XXREAL_0:2; :: thesis: verum

end;

case A24: ( B `1 >= 0 & B `2 < 0 ) ; :: thesis: dist p,b < ((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a

A25: (((abs (E-bound X)) + (abs (N-bound X))) + (abs (S-bound X))) + 0 <= (((abs (E-bound X)) + (abs (N-bound X))) + (abs (S-bound X))) + (abs (W-bound X)) by A15, XREAL_1:9;
((abs (E-bound X)) + (abs (S-bound X))) + 0 <= ((abs (E-bound X)) + (abs (S-bound X))) + (abs (N-bound X)) by A13, XREAL_1:9;
then (abs (E-bound X)) + (abs (S-bound X)) <= (((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X)) by A25, XXREAL_0:2;
then A26: (abs (E-bound X)) + (abs (S-bound X)) < ((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a by A3, XXREAL_0:2;
A27: abs (B `2 ) <= abs (S-bound X) by A8, A24, Th8;
abs (B `1 ) <= abs (E-bound X) by A6, A24, Th7;
then (abs (B `1 )) + (abs (B `2 )) <= (abs (E-bound X)) + (abs (S-bound X)) by A27, XREAL_1:9;
then dist p,b <= (abs (E-bound X)) + (abs (S-bound X)) by A10, A12, XXREAL_0:2;
hence dist p,b < ((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a by A26, XXREAL_0:2; :: thesis: verum

end;

case A28: ( B `1 < 0 & B `2 < 0 ) ; :: thesis: dist p,b < ((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a

then A29: abs (B `2 ) <= abs (S-bound X) by A8, Th8;
abs (B `1 ) <= abs (W-bound X) by A17, A28, Th8;
then (abs (B `1 )) + (abs (B `2 )) <= (abs (W-bound X)) + (abs (S-bound X)) by A29, XREAL_1:9;
then A30: dist p,b <= (abs (W-bound X)) + (abs (S-bound X)) by A10, A12, XXREAL_0:2;
0 + ((abs (W-bound X)) + (abs (S-bound X))) <= ((abs (E-bound X)) + (abs (N-bound X))) + ((abs (W-bound X)) + (abs (S-bound X))) by A14, A13, XREAL_1:9;
then (abs (W-bound X)) + (abs (S-bound X)) < ((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a by A2, XREAL_1:10;
hence dist p,b < ((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a by A30, XXREAL_0:2; :: thesis: verum

end;

end;

end;

hence x in Ball p,(((((abs (E-bound X)) + (abs (N-bound X))) + (abs (W-bound X))) + (abs (S-bound X))) + a) by METRIC_1:12; :: thesis: verum