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 Element of 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 Element of 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 Element of 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 Element of 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 Element of 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;
A3: rng p2 c= FinTrees D2

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng p2 or x in FinTrees D2 )
assume A4: x in rng p2 ; :: thesis: x in FinTrees D2
consider y being set such that
A5: y in dom p2 and
A6: x = p2 . y by A4, FUNCT_1:def 5;
reconsider y = y as Element of NAT by A5;
consider T being Element of FinTrees [:D1,D2:] such that
T = p . y and
A7: p2 . y = T `2 by A1, A5;
A8: dom (T `2 ) = dom T by Th24;
thus x in FinTrees D2 by A6, A7, A8, TREES_3:def 8; :: thesis: verum

end;

A9: p2 is FinSequence of FinTrees D2 by A3, FINSEQ_1:def 4;
thus ex p2 being FinSequence of FinTrees D2 st
( dom p2 = dom p & ( for i being Element of 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, A9; :: thesis: verum