let p be Point of (TOP-REAL 2); :: thesis: for e being Point of (Euclid 2)
for P being Subset of (TOP-REAL 2)
for r being Real st P = Ball (e,r) & p = e holds
proj1 .: P = ].((p `1) - r),((p `1) + r).[
let e be Point of (Euclid 2); :: thesis: for P being Subset of (TOP-REAL 2)
for r being Real st P = Ball (e,r) & p = e holds
proj1 .: P = ].((p `1) - r),((p `1) + r).[
let P be Subset of (TOP-REAL 2); :: thesis: for r being Real st P = Ball (e,r) & p = e holds
proj1 .: P = ].((p `1) - r),((p `1) + r).[
let r be Real; :: thesis: ( P = Ball (e,r) & p = e implies proj1 .: P = ].((p `1) - r),((p `1) + r).[ )
assume that
A1: P = Ball (e,r) and
A2: p = e ; :: thesis: proj1 .: P = ].((p `1) - r),((p `1) + r).[

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: ].((p `1) - r),((p `1) + r).[ c= proj1 .: P

let a be object ; :: thesis: ( a in proj1 .: P implies a in ].((p `1) - r),((p `1) + r).[ )
assume a in proj1 .: P ; :: thesis: a in ].((p `1) - r),((p `1) + r).[
then consider x being object such that
A3: x in the carrier of (TOP-REAL 2) and
A4: x in P and
A5: a = proj1 . x by FUNCT_2:64;
reconsider b = a as Real by A5;
reconsider x = x as Point of (TOP-REAL 2) by A3;
A6: a = x `1 by A5, PSCOMP_1:def 5;
then A7: b < (p `1) + r by A1, A2, A4, Th37;
(p `1) - r < b by A1, A2, A4, A6, Th37;
hence a in ].((p `1) - r),((p `1) + r).[ by A7, XXREAL_1:4; :: thesis: verum

end;

let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in ].((p `1) - r),((p `1) + r).[ or a in proj1 .: P )
assume A8: a in ].((p `1) - r),((p `1) + r).[ ; :: thesis: a in proj1 .: P
then reconsider b = a as Real ;
reconsider f = |[b,(p `2)]| as Point of (Euclid 2) by TOPREAL3:8;
A9: dist (f,e) = (Pitag_dist 2) . (f,e) by METRIC_1:def 1
.= sqrt ((((|[b,(p `2)]| `1) - (p `1)) ^2) + (((|[b,(p `2)]| `2) - (p `2)) ^2)) by A2, TOPREAL3:7
.= sqrt (((b - (p `1)) ^2) + (((|[b,(p `2)]| `2) - (p `2)) ^2))
.= sqrt (((b - (p `1)) ^2) + (((p `2) - (p `2)) ^2))
.= sqrt (((b - (p `1)) ^2) + 0) ;
b < (p `1) + r by A8, XXREAL_1:4;
then A10: b - (p `1) < ((p `1) + r) - (p `1) by XREAL_1:9;

now :: thesis: ( ( 0 <= b - (p `1) & dist (f,e) < r ) or ( 0 > b - (p `1) & dist (f,e) < r ) )

per cases ( 0 <= b - (p `1) or 0 > b - (p `1) ) ;

case 0 <= b - (p `1) ; :: thesis: dist (f,e) < r

hence dist (f,e) < r by A10, A9, SQUARE_1:22; :: thesis: verum

end;

case A11: 0 > b - (p `1) ; :: thesis: dist (f,e) < r

(p `1) - r < b by A8, XXREAL_1:4;
then ((p `1) - r) + r < b + r by XREAL_1:6;
then A12: (p `1) - b < (r + b) - b by XREAL_1:9;
sqrt ((b - (p `1)) ^2) = sqrt ((- (b - (p `1))) ^2)
.= - (b - (p `1)) by A11, SQUARE_1:22 ;
hence dist (f,e) < r by A9, A12; :: thesis: verum

end;

end;

end;

then A13: |[b,(p `2)]| in P by A1, METRIC_1:11;
a = |[b,(p `2)]| `1
.= proj1 . |[b,(p `2)]| by PSCOMP_1:def 5 ;
hence a in proj1 .: P by A13, FUNCT_2:35; :: thesis: verum