C is Jordan by Lm92;
then BDD C is_inside_component_of C by JORDAN2C:116;
then BDD C is_a_component_of C ` by JORDAN2C:def 3;
then ex B1 being Subset of ((TOP-REAL 2) | (C `)) st
( B1 = BDD C & B1 is a_component ) by CONNSP_1:def 6;
then BDD C <> {} ((TOP-REAL 2) | (C `)) by CONNSP_1:34;
hence not BDD C is empty ; :: thesis: verum