let T, T1 be DecoratedTree; :: thesis: for P being AntiChain_of_Prefixes of dom T st P <> {} holds
for q being FinSequence of NAT holds
( not q in dom (tree T,P,T1) or for p being FinSequence of NAT st p in P holds
( not p is_a_prefix_of q & (tree T,P,T1) . q = T . q ) or ex p, r being FinSequence of NAT st
( p in P & r in dom T1 & q = p ^ r & (tree T,P,T1) . q = T1 . r ) )
let P be AntiChain_of_Prefixes of dom T; :: thesis: ( P <> {} implies for q being FinSequence of NAT holds
( not q in dom (tree T,P,T1) or for p being FinSequence of NAT st p in P holds
( not p is_a_prefix_of q & (tree T,P,T1) . q = T . q ) or ex p, r being FinSequence of NAT st
( p in P & r in dom T1 & q = p ^ r & (tree T,P,T1) . q = T1 . r ) ) )
assume A1: P <> {} ; :: thesis: for q being FinSequence of NAT holds
( not q in dom (tree T,P,T1) or for p being FinSequence of NAT st p in P holds
( not p is_a_prefix_of q & (tree T,P,T1) . q = T . q ) or ex p, r being FinSequence of NAT st
( p in P & r in dom T1 & q = p ^ r & (tree T,P,T1) . q = T1 . r ) )
let q be FinSequence of NAT ; :: thesis: ( not q in dom (tree T,P,T1) or for p being FinSequence of NAT st p in P holds
( not p is_a_prefix_of q & (tree T,P,T1) . q = T . q ) or ex p, r being FinSequence of NAT st
( p in P & r in dom T1 & q = p ^ r & (tree T,P,T1) . q = T1 . r ) )
assume A2: q in dom (tree T,P,T1) ; :: thesis: ( for p being FinSequence of NAT st p in P holds
( not p is_a_prefix_of q & (tree T,P,T1) . q = T . q ) or ex p, r being FinSequence of NAT st
( p in P & r in dom T1 & q = p ^ r & (tree T,P,T1) . q = T1 . r ) )
A3: q in tree (dom T),P,(dom T1) by A1, A2, Def2;
thus ( for p being FinSequence of NAT st p in P holds
( not p is_a_prefix_of q & (tree T,P,T1) . q = T . q ) or ex p, r being FinSequence of NAT st
( p in P & r in dom T1 & q = p ^ r & (tree T,P,T1) . q = T1 . r ) ) by A3, Def2; :: thesis: verum