let FT be non empty RelStr ; :: thesis: for A, C being Subset of
for D being non empty Subset of st FT is reflexive & FT is symmetric & D = ([#] FT) \ A & FT is connected & A is connected & C is_a_component_of D holds
([#] FT) \ C is connected
let A, C be Subset of ; :: thesis: for D being non empty Subset of st FT is reflexive & FT is symmetric & D = ([#] FT) \ A & FT is connected & A is connected & C is_a_component_of D holds
([#] FT) \ C is connected
let D be non empty Subset of ; :: thesis: ( FT is reflexive & FT is symmetric & D = ([#] FT) \ A & FT is connected & A is connected & C is_a_component_of D implies ([#] FT) \ C is connected )
assume that
A1: ( FT is reflexive & FT is symmetric ) and
A2: D = ([#] FT) \ A and
A3: FT is connected and
A4: A is connected and
A5: C is_a_component_of D ; :: thesis: ([#] FT) \ C is connected
consider C1 being Subset of such that
A6: C1 = C and
A7: C1 is_a_component_of FT | D by A5, Def5;
reconsider C2 = C1 as Subset of by A6;
C1 c= [#] (FT | D) ;
then C1 c= ([#] FT) \ A by A2, Def3;
then A8: (([#] FT) \ A) ` c= C2 ` by SUBSET_1:31;
then A9: A c= C2 ` by PRE_TOPC:22;
A10: A c= ([#] FT) \ C2 by A8, PRE_TOPC:22;
A11: C1 is connected by A7, Def4;

now

A misses C1 by A9, SUBSET_1:43;
then A12: A /\ C1 = {} by XBOOLE_0:def 7;
let P, Q be Subset of ; :: thesis: ( ([#] FT) \ C = P \/ Q & P misses Q & P,Q are_separated & not P = {} FT implies Q = {} FT )
assume that
A13: ([#] FT) \ C = P \/ Q and
A14: P misses Q and
A15: P,Q are_separated ; :: thesis: ( P = {} FT or Q = {} FT )
A16: C is connected by A6, A11, Th38;

A17: now

assume A18: A c= Q ; :: thesis: P = {} FT
P c= Q ` by A14, SUBSET_1:43;
then ( Q misses Q ` & A /\ P c= Q /\ (Q ` ) ) by A18, XBOOLE_1:27, XBOOLE_1:79;
then A19: A /\ P c= {} by XBOOLE_0:def 7;
(C \/ P) /\ A = (A /\ C) \/ (A /\ P) by XBOOLE_1:23
.= {} by A6, A12, A19, XBOOLE_1:3 ;
then C \/ P misses A by XBOOLE_0:def 7;
then C \/ P c= A ` by SUBSET_1:43;
then C \/ P c= [#] (FT | D) by A2, Def3;
then reconsider C1P1 = C \/ P as Subset of ;
C \/ P is connected by A1, A3, A13, A15, A16, Th37;
then A20: C1P1 is connected by Th38;
C c= C1 \/ P by A6, XBOOLE_1:7;
then C1P1 = C1 by A6, A7, A20, Def4;
then A21: P c= C by A6, XBOOLE_1:7;
P c= ([#] FT) \ C by A13, XBOOLE_1:7;
then ( C misses C ` & P c= C /\ (([#] FT) \ C) ) by A21, XBOOLE_1:19, XBOOLE_1:79;
then P c= {} by XBOOLE_0:def 7;
hence P = {} FT by XBOOLE_1:3; :: thesis: verum

end;

A22: P misses P ` by XBOOLE_1:79;
A23: Q c= ([#] FT) \ C by A13, XBOOLE_1:7;

now

assume A24: A c= P ; :: thesis: Q = {} FT
Q c= P ` by A14, SUBSET_1:43;
then A /\ Q c= P /\ (P ` ) by A24, XBOOLE_1:27;
then A25: A /\ Q c= {} by A22, XBOOLE_0:def 7;
(C \/ Q) /\ A = (A /\ C) \/ (A /\ Q) by XBOOLE_1:23
.= {} by A6, A12, A25, XBOOLE_1:3 ;
then C \/ Q misses A by XBOOLE_0:def 7;
then C \/ Q c= A ` by SUBSET_1:43;
then C \/ Q c= [#] (FT | D) by A2, Def3;
then reconsider C1Q1 = C \/ Q as Subset of ;
C \/ Q is connected by A1, A3, A13, A15, A16, Th37;
then A26: C1Q1 is connected by Th38;
C1 c= C1 \/ Q by XBOOLE_1:7;
then C1Q1 = C1 by A6, A7, A26, Def4;
then Q c= C by A6, XBOOLE_1:7;
then ( C misses C ` & Q c= C /\ (([#] FT) \ C) ) by A23, XBOOLE_1:19, XBOOLE_1:79;
then Q c= {} by XBOOLE_0:def 7;
hence Q = {} FT by XBOOLE_1:3; :: thesis: verum

end;

hence ( P = {} FT or Q = {} FT ) by A1, A4, A6, A10, A13, A15, A17, Th33; :: thesis: verum

end;

hence ([#] FT) \ C is connected by A1, Th7; :: thesis: verum