let p be FinSequence of NAT ; :: thesis: for T, T1 being DecoratedTree st p in dom T holds
for q being FinSequence of NAT st q in dom (T with-replacement (p,T1)) & q in { (p ^ s) where s is Element of dom T1 : verum } holds
ex r being Element of dom T1 st
( 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 st q in dom (T with-replacement (p,T1)) & q in { (p ^ s) where s is Element of dom T1 : verum } holds
ex r being Element of dom T1 st
( q = p ^ r & (T with-replacement (p,T1)) . q = T1 . r ) )
assume A1: p in dom T ; :: thesis: for q being FinSequence of NAT st q in dom (T with-replacement (p,T1)) & q in { (p ^ s) where s is Element of dom T1 : verum } holds
ex r being Element of dom T1 st
( q = p ^ r & (T with-replacement (p,T1)) . q = T1 . r )
let q be FinSequence of NAT ; :: thesis: ( q in dom (T with-replacement (p,T1)) & q in { (p ^ s) where s is Element of dom T1 : verum } implies ex r being Element of dom T1 st
( q = p ^ r & (T with-replacement (p,T1)) . q = T1 . r ) )
assume that
A2: q in dom (T with-replacement (p,T1)) and
A3: q in { (p ^ s) where s is Element of dom T1 : verum } ; :: thesis: ex r being Element of dom T1 st
( q = p ^ r & (T with-replacement (p,T1)) . q = T1 . r )

per cases ( ( 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, A2, Th11;

suppose A4: ( not p is_a_prefix_of q & (T with-replacement (p,T1)) . q = T . q ) ; :: thesis: ex r being Element of dom T1 st
( q = p ^ r & (T with-replacement (p,T1)) . q = T1 . r )

ex r being Element of dom T1 st q = p ^ r by A3;
hence ex r being Element of dom T1 st
( q = p ^ r & (T with-replacement (p,T1)) . q = T1 . r ) by A4, TREES_1:1; :: thesis: verum

end;

suppose ex r being FinSequence of NAT st
( r in dom T1 & q = p ^ r & (T with-replacement (p,T1)) . q = T1 . r ) ; :: thesis: ex r being Element of dom T1 st
( q = p ^ r & (T with-replacement (p,T1)) . q = T1 . r )

hence ex r being Element of dom T1 st
( q = p ^ r & (T with-replacement (p,T1)) . q = T1 . r ) ; :: thesis: verum

end;

end;