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 number 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 number 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 number st P = Ball e,r & p = e holds
proj1 .: P = ].((p `1 ) - r),((p `1 ) + r).[
let r be real number ; :: 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 set ; :: 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 set such that
A3: x in the carrier of (TOP-REAL 2) and
A4: x in P and
A5: a = proj1 . x by FUNCT_2:115;
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 28;
then A7: b < (p `1 ) + r by A1, A2, A4, Th47;
(p `1 ) - r < b by A1, A2, A4, A6, Th47;
hence a in ].((p `1 ) - r),((p `1 ) + r).[ by A7, XXREAL_1:4; :: thesis: verum

end;

let a be set ; :: 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 TOPREAL3:13;
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, TOPREAL3:12
.= sqrt (((b - (p `1 )) ^2 ) + (((|[b,(p `2 )]| `2 ) - (p `2 )) ^2 )) by EUCLID:56
.= sqrt (((b - (p `1 )) ^2 ) + (((p `2 ) - (p `2 )) ^2 )) by EUCLID:56
.= 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_1:11;

now

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_1:89; :: 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_1:8;
then A12: (p `1 ) - b < (r + b) - b by XREAL_1:11;
sqrt ((b - (p `1 )) ^2 ) = sqrt ((- (b - (p `1 ))) ^2 )
.= - (b - (p `1 )) by A11, SQUARE_1:89 ;
hence dist f,e < r by A9, A12; :: thesis: verum

end;

end;

end;

then A13: |[b,(p `2 )]| in P by A1, METRIC_1:12;
a = |[b,(p `2 )]| `1 by EUCLID:56
.= proj1 . |[b,(p `2 )]| by PSCOMP_1:def 28 ;
hence a in proj1 .: P by A13, FUNCT_2:43; :: thesis: verum