set T2 = tree T,P,T1;
tree T,P,T1 is D -valued

proof

let y be set ; :: according to TARSKI:def 3,RELAT_1:def 19 :: thesis: ( not y in proj2 (tree T,P,T1) or y in D )
assume y in rng (tree T,P,T1) ; :: thesis: y in D
then consider x being set such that
A4: x in dom (tree T,P,T1) and
A5: y = (tree T,P,T1) . x by FUNCT_1:def 5;
x is Element of dom (tree T,P,T1) by A4;
then reconsider q = x as FinSequence of NAT ;
dom (tree T,P,T1) = tree (dom T),P,(dom T1) by A1, Def2;
then A8: dom (tree T,P,T1) = { t1 where t1 is Element of dom T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } \/ { (p ^ s) where p is Element of dom T, s is Element of dom T1 : p in P } by A1, Th7;

per cases ( q in { t1 where t1 is Element of dom T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } or q in { (p ^ s) where p is Element of dom T, s is Element of dom T1 : p in P } ) by A4, A8, XBOOLE_0:def 3;

suppose A9: q in { t1 where t1 is Element of dom T : for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t1 } ; :: thesis: y in D

then A10: (tree T,P,T1) . q = T . q by A1, A4, Th15;

now

ex t9 being Element of dom T st
( q = t9 & ( for p being FinSequence of NAT st p in P holds
not p is_a_prefix_of t9 ) ) by A9;
hence q in dom T ; :: thesis: verum

end;

then A13: y in rng T by A5, A10, FUNCT_1:def 5;
rng T c= D by RELAT_1:def 19;
hence y in D by A13; :: thesis: verum

end;

suppose q in { (p ^ s) where p is Element of dom T, s is Element of dom T1 : p in P } ; :: thesis: y in D

then ex p being Element of dom T ex r being Element of dom T1 st
( q = p ^ r & p in P & (tree T,P,T1) . q = T1 . r ) by A4, Th17;
hence y in D by A5; :: thesis: verum

end;

end;

end;

hence tree T,P,T1 is DecoratedTree of D ; :: thesis: verum