let D1, D2 be non empty set ; :: thesis: for d1 being Element of D1
for d2 being Element of D2
for p being FinSequence of FinTrees [:D1,D2:] ex p2 being FinSequence of FinTrees D2 st
( dom p2 = dom p & ( for i being Nat st i in dom p holds
ex T being Element of FinTrees [:D1,D2:] st
( T = p . i & p2 . i = T `2 ) ) & ([d1,d2] -tree p) `2 = d2 -tree p2 )
let d1 be Element of D1; :: thesis: for d2 being Element of D2
for p being FinSequence of FinTrees [:D1,D2:] ex p2 being FinSequence of FinTrees D2 st
( dom p2 = dom p & ( for i being Nat st i in dom p holds
ex T being Element of FinTrees [:D1,D2:] st
( T = p . i & p2 . i = T `2 ) ) & ([d1,d2] -tree p) `2 = d2 -tree p2 )
let d2 be Element of D2; :: thesis: for p being FinSequence of FinTrees [:D1,D2:] ex p2 being FinSequence of FinTrees D2 st
( dom p2 = dom p & ( for i being Nat st i in dom p holds
ex T being Element of FinTrees [:D1,D2:] st
( T = p . i & p2 . i = T `2 ) ) & ([d1,d2] -tree p) `2 = d2 -tree p2 )
let p be FinSequence of FinTrees [:D1,D2:]; :: thesis: ex p2 being FinSequence of FinTrees D2 st
( dom p2 = dom p & ( for i being Nat st i in dom p holds
ex T being Element of FinTrees [:D1,D2:] st
( T = p . i & p2 . i = T `2 ) ) & ([d1,d2] -tree p) `2 = d2 -tree p2 )
consider p2 being FinSequence of Trees D2 such that
A1: ( dom p2 = dom p & ( for i being Nat st i in dom p holds
ex T being Element of FinTrees [:D1,D2:] st
( T = p . i & p2 . i = T `2 ) ) ) and
A2: ([d1,d2] -tree p) `2 = d2 -tree p2 by Th30;
rng p2 c= FinTrees D2

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng p2 or x in FinTrees D2 )
assume x in rng p2 ; :: thesis: x in FinTrees D2
then consider y being object such that
A3: y in dom p2 and
A4: x = p2 . y by FUNCT_1:def 3;
reconsider y = y as Nat by A3;
consider T being Element of FinTrees [:D1,D2:] such that
T = p . y and
A5: p2 . y = T `2 by A1, A3;
dom (T `2) = dom T by Th24;
hence x in FinTrees D2 by A4, A5, TREES_3:def 8; :: thesis: verum

end;

then p2 is FinSequence of FinTrees D2 by FINSEQ_1:def 4;
hence ex p2 being FinSequence of FinTrees D2 st
( dom p2 = dom p & ( for i being Nat st i in dom p holds
ex T being Element of FinTrees [:D1,D2:] st
( T = p . i & p2 . i = T `2 ) ) & ([d1,d2] -tree p) `2 = d2 -tree p2 ) by A1, A2; :: thesis: verum