let GX be TopSpace; :: thesis: for A, C being Subset of GX st GX is connected & A is connected & C is_a_component_of ([#] GX) \ A holds
([#] GX) \ C is connected
let A, C be Subset of GX; :: thesis: ( GX is connected & A is connected & C is_a_component_of ([#] GX) \ A implies ([#] GX) \ C is connected )
assume that
A1: GX is connected and
A2: A is connected and
A3: C is_a_component_of ([#] GX) \ A ; :: thesis: ([#] GX) \ C is connected
consider C1 being Subset of (GX | (([#] GX) \ A)) such that
A4: C1 = C and
A5: C1 is a_component by A3;
reconsider C2 = C1 as Subset of GX by A4;
C1 c= [#] (GX | (([#] GX) \ A)) ;
then C1 c= ([#] GX) \ A by PRE_TOPC:def 5;
then (([#] GX) \ A) ` c= C2 ` by SUBSET_1:12;
then A6: A c= C2 ` by PRE_TOPC:3;

now :: thesis: for P, Q being Subset of GX st ([#] GX) \ C = P \/ Q & P,Q are_separated & not P = {} GX holds
Q = {} GX

A misses C1 by A6, SUBSET_1:23;
then A7: A /\ C1 = {} ;
A8: C is connected by A4, A5, Th23;
let P, Q be Subset of GX; :: thesis: ( ([#] GX) \ C = P \/ Q & P,Q are_separated & not P = {} GX implies Q = {} GX )
assume that
A9: ([#] GX) \ C = P \/ Q and
A10: P,Q are_separated ; :: thesis: ( P = {} GX or Q = {} GX )
A11: P misses P ` by XBOOLE_1:79;
A12: P misses Q by A10, Th1;

A13: now :: thesis: ( A c= Q implies P = {} GX )

A14: Q misses Q ` by XBOOLE_1:79;
assume A15: A c= Q ; :: thesis: P = {} GX
P c= Q ` by A12, SUBSET_1:23;
then A /\ P c= Q /\ (Q `) by A15, XBOOLE_1:27;
then A16: A /\ P c= {} by A14;
(C \/ P) /\ A = (A /\ C) \/ (A /\ P) by XBOOLE_1:23
.= {} by A4, A7, A16 ;
then C \/ P misses A ;
then C \/ P c= A ` by SUBSET_1:23;
then C \/ P c= [#] (GX | (([#] GX) \ A)) by PRE_TOPC:def 5;
then reconsider C1P1 = C \/ P as Subset of (GX | (([#] GX) \ A)) ;
A17: C misses C ` by XBOOLE_1:79;
C \/ P is connected by A1, A9, A10, A8, Th20;
then A18: C1P1 is connected by Th23;
C c= C1 \/ P by A4, XBOOLE_1:7;
then C1P1 = C1 by A4, A5, A18;
then A19: P c= C by A4, XBOOLE_1:7;
P c= ([#] GX) \ C by A9, XBOOLE_1:7;
then P c= C /\ (([#] GX) \ C) by A19, XBOOLE_1:19;
then P c= {} by A17;
hence P = {} GX ; :: thesis: verum

end;

A20: Q c= ([#] GX) \ C by A9, XBOOLE_1:7;

now :: thesis: ( A c= P implies Q = {} GX )

assume A21: A c= P ; :: thesis: Q = {} GX
Q c= P ` by A12, SUBSET_1:23;
then A /\ Q c= P /\ (P `) by A21, XBOOLE_1:27;
then A22: A /\ Q c= {} by A11;
(C \/ Q) /\ A = (A /\ C) \/ (A /\ Q) by XBOOLE_1:23
.= {} by A4, A7, A22 ;
then C \/ Q misses A ;
then C \/ Q c= A ` by SUBSET_1:23;
then C \/ Q c= [#] (GX | (([#] GX) \ A)) by PRE_TOPC:def 5;
then reconsider C1Q1 = C \/ Q as Subset of (GX | (([#] GX) \ A)) ;
C \/ Q is connected by A1, A9, A10, A8, Th20;
then A23: C1Q1 is connected by Th23;
C1 c= C1 \/ Q by XBOOLE_1:7;
then C1Q1 = C1 by A4, A5, A23;
then Q c= C by A4, XBOOLE_1:7;
then A24: Q c= C /\ (([#] GX) \ C) by A20, XBOOLE_1:19;
C misses C ` by XBOOLE_1:79;
then Q c= {} by A24;
hence Q = {} GX ; :: thesis: verum

end;

hence ( P = {} GX or Q = {} GX ) by A2, A4, A6, A9, A10, A13, Th16; :: thesis: verum

end;

hence ([#] GX) \ C is connected by Th15; :: thesis: verum