let T1, T2 be DecoratedTree; :: thesis: ( ex q being DTree-yielding FinSequence st
( p = q & dom T1 = tree (doms q) ) & T1 . {} = x & ( for n being Nat st n < len p holds
T1 | <*n*> = p . (n + 1) ) & ex q being DTree-yielding FinSequence st
( p = q & dom T2 = tree (doms q) ) & T2 . {} = x & ( for n being Nat st n < len p holds
T2 | <*n*> = p . (n + 1) ) implies T1 = T2 )

given q1 being DTree-yielding FinSequence such that A16: p = q1 and
A17: dom T1 = tree (doms q1) ; :: thesis: ( not T1 . {} = x or ex n being Nat st
( n < len p & not T1 | <*n*> = p . (n + 1) ) or for q being DTree-yielding FinSequence holds
( not p = q or not dom T2 = tree (doms q) ) or not T2 . {} = x or ex n being Nat st
( n < len p & not T2 | <*n*> = p . (n + 1) ) or T1 = T2 )

assume that
A18: T1 . {} = x and
A19: for n being Nat st n < len p holds
T1 | <*n*> = p . (n + 1) ; :: thesis: ( for q being DTree-yielding FinSequence holds
( not p = q or not dom T2 = tree (doms q) ) or not T2 . {} = x or ex n being Nat st
( n < len p & not T2 | <*n*> = p . (n + 1) ) or T1 = T2 )

given q2 being DTree-yielding FinSequence such that A20: ( p = q2 & dom T2 = tree (doms q2) ) ; :: thesis: ( not T2 . {} = x or ex n being Nat st
( n < len p & not T2 | <*n*> = p . (n + 1) ) or T1 = T2 )

assume that
A21: T2 . {} = x and
A22: for n being Nat st n < len p holds
T2 | <*n*> = p . (n + 1) ; :: thesis: T1 = T2
now :: thesis: for q being FinSequence of NAT st q in dom T1 holds
T1 . q = T2 . q
let q be FinSequence of NAT ; :: thesis: ( q in dom T1 implies T1 . q = T2 . q )
assume A23: q in dom T1 ; :: thesis: T1 . q = T2 . q
now :: thesis: ( q <> {} implies T1 . q = T2 . q )
assume q <> {} ; :: thesis: T1 . q = T2 . q
then consider s being FinSequence of NAT , n being Element of NAT such that
A24: q = <*n*> ^ s by FINSEQ_2:130;
A25: <*n*> in dom T1 by ;
A26: n < len (doms q1) by ;
len (doms q1) = len p by ;
then A27: ( T1 | <*n*> = p . (n + 1) & T2 | <*n*> = p . (n + 1) ) by ;
A28: s in (dom T1) | <*n*> by ;
then T1 . q = (T1 | <*n*>) . s by ;
hence T1 . q = T2 . q by ; :: thesis: verum
end;
hence T1 . q = T2 . q by ; :: thesis: verum
end;
hence T1 = T2 by ; :: thesis: verum