C is Jordan by Lm92;
then BDD C is_inside_component_of C by JORDAN2C:108;
then BDD C is_a_component_of C ` ;
then ex B1 being Subset of ((TOP-REAL 2) | (C `)) st
( B1 = BDD C & B1 is a_component ) ;
then BDD C <> {} ((TOP-REAL 2) | (C `)) by CONNSP_1:32;
hence not BDD C is empty ; :: thesis: verum