let Y be set ; :: thesis: for H being covering T_3 Hierarchy of Y st H is lower-bounded & not {} in H holds
for A being Subset of Y st A in H holds
ex P being a_partition of Y st
( A in P & P c= H )

let H be covering T_3 Hierarchy of Y; :: thesis: ( H is lower-bounded & not {} in H implies for A being Subset of Y st A in H holds
ex P being a_partition of Y st
( A in P & P c= H ) )

assume that
A1: H is lower-bounded and
A2: not {} in H ; :: thesis: for A being Subset of Y st A in H holds
ex P being a_partition of Y st
( A in P & P c= H )

let A be Subset of Y; :: thesis: ( A in H implies ex P being a_partition of Y st
( A in P & P c= H ) )

assume A3: A in H ; :: thesis: ex P being a_partition of Y st
( A in P & P c= H )

set k1 = {A};
A4: {A} c= H by ;
A5: A in {A} by TARSKI:def 1;
A6: {A} c= bool Y by ZFMISC_1:31;
defpred S1[ set ] means ( A in \$1 & \$1 is mutually-disjoint & \$1 c= H );
consider K being set such that
A7: for S being set holds
( S in K iff ( S in bool (bool Y) & S1[S] ) ) from {A} is mutually-disjoint by Th9;
then A8: {A} in K by A7, A5, A4, A6;
for Z being set st Z c= K & Z is c=-linear holds
ex X3 being set st
( X3 in K & ( for X1 being set st X1 in Z holds
X1 c= X3 ) )
proof
let Z be set ; :: thesis: ( Z c= K & Z is c=-linear implies ex X3 being set st
( X3 in K & ( for X1 being set st X1 in Z holds
X1 c= X3 ) ) )

assume that
A9: Z c= K and
A10: Z is c=-linear ; :: thesis: ex X3 being set st
( X3 in K & ( for X1 being set st X1 in Z holds
X1 c= X3 ) )

A11: for X1, X2 being set st X1 in Z & X2 in Z & not X1 c= X2 holds
X2 c= X1 by ;
per cases ( Z <> {} or Z = {} ) ;
suppose A12: Z <> {} ; :: thesis: ex X3 being set st
( X3 in K & ( for X1 being set st X1 in Z holds
X1 c= X3 ) )

set X3 = union Z;
take union Z ; :: thesis: ( union Z in K & ( for X1 being set st X1 in Z holds
X1 c= union Z ) )

now :: thesis: ( A in union Z & union Z in bool (bool Y) & union Z is mutually-disjoint & union Z c= H )
consider z being object such that
A13: z in Z by ;
reconsider z = z as set by TARSKI:1;
A in z by A7, A9, A13;
hence A in union Z by ; :: thesis: ( union Z in bool (bool Y) & union Z is mutually-disjoint & union Z c= H )
A14: for a being set st a in Z holds
a c= H by A7, A9;
then union Z c= H by ZFMISC_1:76;
then union Z c= bool Y by XBOOLE_1:1;
hence union Z in bool (bool Y) ; :: thesis: ( union Z is mutually-disjoint & union Z c= H )
thus union Z is mutually-disjoint :: thesis: union Z c= H
proof
let t1, t2 be set ; :: according to TAXONOM2:def 5 :: thesis: ( t1 in union Z & t2 in union Z & t1 <> t2 implies t1 misses t2 )
assume that
A15: t1 in union Z and
A16: t2 in union Z and
A17: t1 <> t2 ; :: thesis: t1 misses t2
consider v1 being set such that
A18: t1 in v1 and
A19: v1 in Z by ;
A20: v1 is mutually-disjoint by A7, A9, A19;
consider v2 being set such that
A21: t2 in v2 and
A22: v2 in Z by ;
A23: v2 is mutually-disjoint by A7, A9, A22;
per cases ( v1 c= v2 or v2 c= v1 ) by ;
end;
end;
thus union Z c= H by ; :: thesis: verum
end;
hence union Z in K by A7; :: thesis: for X1 being set st X1 in Z holds
X1 c= union Z

thus for X1 being set st X1 in Z holds
X1 c= union Z by ZFMISC_1:74; :: thesis: verum
end;
suppose A24: Z = {} ; :: thesis: ex X3 being set st
( X3 in K & ( for X1 being set st X1 in Z holds
X1 c= X3 ) )

consider a being set such that
A25: a in K by A8;
take a ; :: thesis: ( a in K & ( for X1 being set st X1 in Z holds
X1 c= a ) )

thus a in K by A25; :: thesis: for X1 being set st X1 in Z holds
X1 c= a

thus for X1 being set st X1 in Z holds
X1 c= a by A24; :: thesis: verum
end;
end;
end;
then consider M being set such that
A26: M in K and
A27: for Z being set st Z in K & Z <> M holds
not M c= Z by ;
A28: M is mutually-disjoint Subset-Family of Y by ;
A29: for C being mutually-disjoint Subset-Family of Y st A in C & C c= H & M c= C holds
M = C by ;
A30: A in M by ;
take M ; :: thesis: ( M is Element of bool (bool Y) & M is a_partition of Y & A in M & M c= H )
M c= H by ;
hence ( M is Element of bool (bool Y) & M is a_partition of Y & A in M & M c= H ) by A1, A2, A28, A30, A29, Th15; :: thesis: verum