let C be compact Subset of (TOP-REAL 2); :: thesis: ( BDD C <> {} implies E-bound C >= E-bound (BDD C) )
set WC = E-bound (BDD C);
set WB = E-bound C;
set hal = ((E-bound C) + (E-bound (BDD C))) / 2;
assume that
A1: BDD C <> {} and
A2: E-bound (BDD C) > E-bound C ; :: thesis: contradiction
A3: ((E-bound C) + (E-bound (BDD C))) / 2 > E-bound C by A2, XREAL_1:226;

now :: thesis: contradiction

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

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

hence contradiction by A1, A2, Lm10, XREAL_1:226; :: thesis: verum

end;

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

then consider q1 being Point of (TOP-REAL 2) such that
A4: q1 in BDD C and
A5: q1 `1 > ((E-bound C) + (E-bound (BDD C))) / 2 ;
set Q = |[(((E-bound C) + (q1 `1)) / 2),(q1 `2)]|;
set WH = east_halfline |[(((E-bound C) + (q1 `1)) / 2),(q1 `2)]|;
A6: |[(((E-bound C) + (q1 `1)) / 2),(q1 `2)]| `1 = ((E-bound C) + (q1 `1)) / 2 by EUCLID:52;
A7: q1 `1 > E-bound C by A3, A5, XXREAL_0:2;
east_halfline |[(((E-bound C) + (q1 `1)) / 2),(q1 `2)]| misses C

proof

A8: |[(((E-bound C) + (q1 `1)) / 2),(q1 `2)]| `1 > E-bound C by A7, A6, XREAL_1:226;
assume (east_halfline |[(((E-bound C) + (q1 `1)) / 2),(q1 `2)]|) /\ C <> {} ; :: according to XBOOLE_0:def 7 :: thesis: contradiction
then consider y being object such that
A9: y in (east_halfline |[(((E-bound C) + (q1 `1)) / 2),(q1 `2)]|) /\ C by XBOOLE_0:def 1;
A10: y in C by A9, XBOOLE_0:def 4;
A11: y in east_halfline |[(((E-bound C) + (q1 `1)) / 2),(q1 `2)]| by A9, XBOOLE_0:def 4;
reconsider y = y as Point of (TOP-REAL 2) by A9;
y `1 >= |[(((E-bound C) + (q1 `1)) / 2),(q1 `2)]| `1 by A11, TOPREAL1:def 11;
then y `1 > E-bound C by A8, XXREAL_0:2;
hence contradiction by A10, PSCOMP_1:24; :: thesis: verum

end;

then A12: east_halfline |[(((E-bound C) + (q1 `1)) / 2),(q1 `2)]| c= UBD C by JORDAN2C:127;
A13: q1 `2 = |[(((E-bound C) + (q1 `1)) / 2),(q1 `2)]| `2 by EUCLID:52;
q1 `1 > |[(((E-bound C) + (q1 `1)) / 2),(q1 `2)]| `1 by A7, A6, XREAL_1:226;
then q1 in east_halfline |[(((E-bound C) + (q1 `1)) / 2),(q1 `2)]| by A13, TOPREAL1:def 11;
then q1 in (BDD C) /\ (UBD C) by A4, A12, XBOOLE_0:def 4;
then BDD C meets UBD C ;
hence contradiction by JORDAN2C:24; :: thesis: verum

end;

end;

end;

hence contradiction ; :: thesis: verum