defpred S₁[ set ] means ( W is Chain of W & ( for p being FinSequence of NAT st p in W holds
ProperPrefixes p c= W ) );
consider X being set such that
A1: for Y being set holds
( Y in X iff ( Y in bool W & S₁[Y] ) ) from XBOOLE_0:sch 1();
( {} is Chain of W & ( for p being FinSequence of NAT st p in {} holds
ProperPrefixes p c= {} ) ) by Th21;
then A2: X <> {} by A1;

now

let Z be set ; :: thesis: ( Z <> {} & Z c= X & Z is c=-linear implies union Z in X )
assume that
A3: ( Z <> {} & Z c= X ) and
A4: Z is c=-linear ; :: thesis: union Z in X
union Z c= W

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in union Z or x in W )
assume x in union Z ; :: thesis: x in W
then consider Y being set such that
A5: ( x in Y & Y in Z ) by TARSKI:def 4;
Y in bool W by A1, A3, A5;
hence x in W by A5; :: thesis: verum

end;

then reconsider Z' = union Z as Subset of W ;
A6: Z' is Chain of W

proof

let p be FinSequence of NAT ; :: according to TREES_2:def 3 :: thesis: for q being FinSequence of NAT st p in Z' & q in Z' holds
p,q are_c=-comparable
let q be FinSequence of NAT ; :: thesis: ( p in Z' & q in Z' implies p,q are_c=-comparable )
assume p in Z' ; :: thesis: ( not q in Z' or p,q are_c=-comparable )
then consider X1 being set such that
A7: ( p in X1 & X1 in Z ) by TARSKI:def 4;
assume q in Z' ; :: thesis: p,q are_c=-comparable
then consider X2 being set such that
A8: ( q in X2 & X2 in Z ) by TARSKI:def 4;
( X1,X2 are_c=-comparable & X1 in X & X2 in X ) by A3, A4, A7, A8, ORDINAL1:def 9;
then ( ( X1 c= X2 or X2 c= X1 ) & X1 in X & X2 in X ) by XBOOLE_0:def 9;
then ( ( p in X2 or q in X1 ) & X1 is Chain of W & X2 is Chain of W ) by A1, A7, A8;
hence p,q are_c=-comparable by A7, A8, Def3; :: thesis: verum

end;

now

let p be FinSequence of NAT ; :: thesis: ( p in union Z implies ProperPrefixes p c= union Z )
assume p in union Z ; :: thesis: ProperPrefixes p c= union Z
then consider X1 being set such that
A9: ( p in X1 & X1 in Z ) by TARSKI:def 4;
( ProperPrefixes p c= X1 & X1 c= union Z ) by A1, A3, A9, ZFMISC_1:92;
hence ProperPrefixes p c= union Z by XBOOLE_1:1; :: thesis: verum

end;

hence union Z in X by A1, A6; :: thesis: verum

end;

then consider Y being set such that
A10: Y in X and
A11: for Z being set st Z in X & Z <> Y holds
not Y c= Z by A2, ORDERS_1:177;
reconsider Y = Y as Chain of W by A1, A10;
take Y ; :: thesis: Y is Branch-like
thus for p being FinSequence of NAT st p in Y holds
ProperPrefixes p c= Y by A1, A10; :: according to TREES_2:def 7 :: thesis: for p being FinSequence of NAT holds
( not p in W or ex q being FinSequence of NAT st
( q in Y & not q is_a_proper_prefix_of p ) )
given p being FinSequence of NAT such that A12: p in W and
A13: for q being FinSequence of NAT st q in Y holds
q is_a_proper_prefix_of p ; :: thesis: contradiction
set Z = (ProperPrefixes p) \/ {p};
( ProperPrefixes p c= W & {p} c= W ) by A12, TREES_1:def 5, ZFMISC_1:37;
then reconsider Z' = (ProperPrefixes p) \/ {p} as Subset of W by XBOOLE_1:8;
A14: Z' is Chain of W

proof

let q be FinSequence of NAT ; :: according to TREES_2:def 3 :: thesis: for q being FinSequence of NAT st q in Z' & q in Z' holds
q,q are_c=-comparable
let r be FinSequence of NAT ; :: thesis: ( q in Z' & r in Z' implies q,r are_c=-comparable )
assume ( q in Z' & r in Z' ) ; :: thesis: q,r are_c=-comparable
then ( ( q in ProperPrefixes p or q in {p} ) & ( r in ProperPrefixes p or r in {p} ) ) by XBOOLE_0:def 3;
then ( ( q is_a_proper_prefix_of p or q = p ) & ( r is_a_proper_prefix_of p or r = p ) ) by TARSKI:def 1, TREES_1:36;
then ( q is_a_prefix_of p & r is_a_prefix_of p ) by XBOOLE_0:def 8;
hence q,r are_c=-comparable by Th1; :: thesis: verum

end;

now

let q be FinSequence of NAT ; :: thesis: ( q in (ProperPrefixes p) \/ {p} implies ProperPrefixes q c= (ProperPrefixes p) \/ {p} )
assume q in (ProperPrefixes p) \/ {p} ; :: thesis: ProperPrefixes q c= (ProperPrefixes p) \/ {p}
then ( q in ProperPrefixes p or q in {p} ) by XBOOLE_0:def 3;
then ( q is_a_proper_prefix_of p or q = p ) by TARSKI:def 1, TREES_1:36;
then q is_a_prefix_of p by XBOOLE_0:def 8;
then ( ProperPrefixes q c= ProperPrefixes p & ProperPrefixes p c= (ProperPrefixes p) \/ {p} ) by TREES_1:41, XBOOLE_1:7;
hence ProperPrefixes q c= (ProperPrefixes p) \/ {p} by XBOOLE_1:1; :: thesis: verum

end;

then A15: (ProperPrefixes p) \/ {p} in X by A1, A14;
( not p is_a_proper_prefix_of p & p in {p} ) by TARSKI:def 1;
then A16: ( not p in Y & p in (ProperPrefixes p) \/ {p} ) by A13, XBOOLE_0:def 3;
Y c= (ProperPrefixes p) \/ {p}

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in Y or x in (ProperPrefixes p) \/ {p} )
assume A17: x in Y ; :: thesis: x in (ProperPrefixes p) \/ {p}
then reconsider t = x as Element of W ;
t is_a_proper_prefix_of p by A13, A17;
then t in ProperPrefixes p by TREES_1:36;
hence x in (ProperPrefixes p) \/ {p} by XBOOLE_0:def 3; :: thesis: verum

end;

hence contradiction by A11, A15, A16; :: thesis: verum