let p be FinSequence of NAT ; :: thesis: for T, T1 being DecoratedTree st p in dom T holds
for q being FinSequence of NAT holds
( not q in dom (T with-replacement p,T1) or ( not p is_a_prefix_of q & (T with-replacement p,T1) . q = T . q ) or ex r being FinSequence of NAT st
( r in dom T1 & q = p ^ r & (T with-replacement p,T1) . q = T1 . r ) )
let T, T1 be DecoratedTree; :: thesis: ( p in dom T implies for q being FinSequence of NAT holds
( not q in dom (T with-replacement p,T1) or ( not p is_a_prefix_of q & (T with-replacement p,T1) . q = T . q ) or ex r being FinSequence of NAT st
( r in dom T1 & q = p ^ r & (T with-replacement p,T1) . q = T1 . r ) ) )
assume A1: p in dom T ; :: thesis: for q being FinSequence of NAT holds
( not q in dom (T with-replacement p,T1) or ( not p is_a_prefix_of q & (T with-replacement p,T1) . q = T . q ) or ex r being FinSequence of NAT st
( r in dom T1 & q = p ^ r & (T with-replacement p,T1) . q = T1 . r ) )
let q be FinSequence of NAT ; :: thesis: ( not q in dom (T with-replacement p,T1) or ( not p is_a_prefix_of q & (T with-replacement p,T1) . q = T . q ) or ex r being FinSequence of NAT st
( r in dom T1 & q = p ^ r & (T with-replacement p,T1) . q = T1 . r ) )
assume q in dom (T with-replacement p,T1) ; :: thesis: ( ( not p is_a_prefix_of q & (T with-replacement p,T1) . q = T . q ) or ex r being FinSequence of NAT st
( r in dom T1 & q = p ^ r & (T with-replacement p,T1) . q = T1 . r ) )
then q in (dom T) with-replacement p,(dom T1) by A1, TREES_2:def 12;
hence ( ( not p is_a_prefix_of q & (T with-replacement p,T1) . q = T . q ) or ex r being FinSequence of NAT st
( r in dom T1 & q = p ^ r & (T with-replacement p,T1) . q = T1 . r ) ) by A1, TREES_2:def 12; :: thesis: verum