let T, T1 be Tree; :: thesis: for P being AntiChain_of_Prefixes of T
for p being FinSequence of NAT st p in P holds
T1 = (tree (T,P,T1)) | p

let P be AntiChain_of_Prefixes of T; :: thesis: for p being FinSequence of NAT st p in P holds
T1 = (tree (T,P,T1)) | p

let p be FinSequence of NAT ; :: thesis: ( p in P implies T1 = (tree (T,P,T1)) | p )
assume A1: p in P ; :: thesis: T1 = (tree (T,P,T1)) | p
ex q, r being FinSequence of NAT st
( q in P & r in T1 & p = q ^ r )
proof
consider q being FinSequence of NAT such that
A2: q = p ;
consider r being FinSequence of NAT such that
A3: r = <*> NAT ;
A4: r in T1 by ;
p = q ^ r by ;
hence ex q, r being FinSequence of NAT st
( q in P & r in T1 & p = q ^ r ) by A1, A2, A4; :: thesis: verum
end;
then A5: p in tree (T,P,T1) by Def1;
let x be FinSequence of NAT ; :: according to TREES_2:def 1 :: thesis: ( ( not x in T1 or x in (tree (T,P,T1)) | p ) & ( not x in (tree (T,P,T1)) | p or x in T1 ) )
thus ( x in T1 implies x in (tree (T,P,T1)) | p ) :: thesis: ( not x in (tree (T,P,T1)) | p or x in T1 )
proof
assume x in T1 ; :: thesis: x in (tree (T,P,T1)) | p
then p ^ x in tree (T,P,T1) by ;
hence x in (tree (T,P,T1)) | p by ; :: thesis: verum
end;
thus ( x in (tree (T,P,T1)) | p implies x in T1 ) :: thesis: verum
proof
assume x in (tree (T,P,T1)) | p ; :: thesis: x in T1
then A6: p ^ x in tree (T,P,T1) by ;
A7: now :: thesis: ( p ^ x in T & ( for r being FinSequence of NAT st r in P holds
not r is_a_proper_prefix_of p ^ x ) implies x in T1 )
assume that
p ^ x in T and
A8: for r being FinSequence of NAT st r in P holds
not r is_a_proper_prefix_of p ^ x ; :: thesis: x in T1
A9: not p is_a_proper_prefix_of p ^ x by A1, A8;
p is_a_prefix_of p ^ x by TREES_1:1;
then p ^ x = p by A9
.= p ^ () by FINSEQ_1:34 ;
then x = {} by FINSEQ_1:33;
hence x in T1 by TREES_1:22; :: thesis: verum
end;
now :: thesis: ( ex s, r being FinSequence of NAT st
( s in P & r in T1 & p ^ x = s ^ r ) implies x in T1 )
given s, r being FinSequence of NAT such that A10: s in P and
A11: r in T1 and
A12: p ^ x = s ^ r ; :: thesis: x in T1
hence x in T1 by ; :: thesis: verum
end;
hence x in T1 by A1, A6, A7, Def1; :: thesis: verum
end;