let C be compact Subset of (TOP-REAL 2); :: thesis: ( BDD C <> {} implies W-bound C <= W-bound (BDD C) )
set WC = W-bound C;
set WB = W-bound (BDD C);
set hal = ((W-bound (BDD C)) + (W-bound C)) / 2;
assume that
A1: BDD C <> {} and
A2: W-bound C > W-bound (BDD C) ; :: thesis: contradiction
A3: ( ((W-bound (BDD C)) + (W-bound C)) / 2 > W-bound (BDD C) & ((W-bound (BDD C)) + (W-bound C)) / 2 < W-bound C ) by A2, XREAL_1:228;

now

per cases ( for q1 being Point of (TOP-REAL 2) st q1 in BDD C holds
q1 `1 >= ((W-bound (BDD C)) + (W-bound C)) / 2 or ex q1 being Point of (TOP-REAL 2) st
( q1 in BDD C & q1 `1 < ((W-bound (BDD C)) + (W-bound C)) / 2 ) ) ;

suppose for q1 being Point of (TOP-REAL 2) st q1 in BDD C holds
q1 `1 >= ((W-bound (BDD C)) + (W-bound C)) / 2 ; :: thesis: contradiction

hence contradiction by A1, A2, Lm9, XREAL_1:228; :: thesis: verum

end;

suppose ex q1 being Point of (TOP-REAL 2) st
( q1 in BDD C & q1 `1 < ((W-bound (BDD C)) + (W-bound C)) / 2 ) ; :: thesis: contradiction

then consider q1 being Point of (TOP-REAL 2) such that
A4: ( q1 in BDD C & q1 `1 < ((W-bound (BDD C)) + (W-bound C)) / 2 ) ;
A5: q1 `1 < W-bound C by A3, A4, XXREAL_0:2;
set Q = |[(((W-bound C) + (q1 `1 )) / 2),(q1 `2 )]|;
set WH = west_halfline |[(((W-bound C) + (q1 `1 )) / 2),(q1 `2 )]|;
A6: |[(((W-bound C) + (q1 `1 )) / 2),(q1 `2 )]| `1 = ((W-bound C) + (q1 `1 )) / 2 by EUCLID:56;
then ( q1 `2 = |[(((W-bound C) + (q1 `1 )) / 2),(q1 `2 )]| `2 & q1 `1 < |[(((W-bound C) + (q1 `1 )) / 2),(q1 `2 )]| `1 ) by A5, EUCLID:56, XREAL_1:228;
then A7: q1 in west_halfline |[(((W-bound C) + (q1 `1 )) / 2),(q1 `2 )]| by TOPREAL1:def 15;
west_halfline |[(((W-bound C) + (q1 `1 )) / 2),(q1 `2 )]| misses C

proof

assume west_halfline |[(((W-bound C) + (q1 `1 )) / 2),(q1 `2 )]| meets C ; :: thesis: contradiction
then consider y being set such that
A8: y in (west_halfline |[(((W-bound C) + (q1 `1 )) / 2),(q1 `2 )]|) /\ C by XBOOLE_0:4;
A9: ( y in west_halfline |[(((W-bound C) + (q1 `1 )) / 2),(q1 `2 )]| & y in C ) by A8, XBOOLE_0:def 4;
reconsider y = y as Point of (TOP-REAL 2) by A8;
A10: ( y `1 <= |[(((W-bound C) + (q1 `1 )) / 2),(q1 `2 )]| `1 & y `2 = |[(((W-bound C) + (q1 `1 )) / 2),(q1 `2 )]| `2 ) by A9, TOPREAL1:def 15;
|[(((W-bound C) + (q1 `1 )) / 2),(q1 `2 )]| `1 < W-bound C by A5, A6, XREAL_1:228;
then y `1 < W-bound C by A10, XXREAL_0:2;
hence contradiction by A9, PSCOMP_1:71; :: thesis: verum

end;

then west_halfline |[(((W-bound C) + (q1 `1 )) / 2),(q1 `2 )]| c= UBD C by JORDAN2C:134;
then q1 in (BDD C) /\ (UBD C) by A4, A7, XBOOLE_0:def 4;
then BDD C meets UBD C by XBOOLE_0:def 7;
hence contradiction by JORDAN2C:28; :: thesis: verum

end;

end;

end;

hence contradiction ; :: thesis: verum