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 Element of 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 Element of NAT st n < len p holds
T2 | <*n*> = p . (n + 1) ) implies T1 = T2 )
given q1 being DTree-yielding FinSequence such that A14: ( p = q1 & dom T1 = tree (doms q1) ) ; :: thesis: ( not T1 . {} = x or ex n being Element of 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 Element of NAT st
( n < len p & not T2 | <*n*> = p . (n + 1) ) or T1 = T2 )
assume A15: ( T1 . {} = x & ( for n being Element of 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 Element of NAT st
( n < len p & not T2 | <*n*> = p . (n + 1) ) or T1 = T2 )
given q2 being DTree-yielding FinSequence such that A16: ( p = q2 & dom T2 = tree (doms q2) ) ; :: thesis: ( not T2 . {} = x or ex n being Element of NAT st
( n < len p & not T2 | <*n*> = p . (n + 1) ) or T1 = T2 )
assume A17: ( T2 . {} = x & ( for n being Element of NAT st n < len p holds
T2 | <*n*> = p . (n + 1) ) ) ; :: thesis: T1 = T2

now

let q be FinSequence of NAT ; :: thesis: ( q in dom T1 implies T1 . q = T2 . q )
assume A18: q in dom T1 ; :: thesis: T1 . q = T2 . q

now

assume q <> {} ; :: thesis: T1 . q = T2 . q
then consider s being FinSequence of NAT , n being Nat such that
A19: q = <*n*> ^ s by FINSEQ_2:150;
reconsider n = n as Element of NAT by ORDINAL1:def 13;
A20: ( <*n*> in dom T1 & n < len (doms q1) & len (doms q1) = len p ) by A14, A18, A19, TREES_1:46, TREES_3:40, TREES_3:51;
then A21: ( T1 | <*n*> = p . (n + 1) & T2 | <*n*> = p . (n + 1) ) by A15, A17;
s in (dom T1) | <*n*> by A18, A19, A20, TREES_1:def 9;
then ( T1 . q = (T1 | <*n*>) . s & T2 . q = (T2 | <*n*>) . s ) by A14, A16, A19, TREES_2:def 11;
hence T1 . q = T2 . q by A21; :: thesis: verum

end;

hence T1 . q = T2 . q by A15, A17; :: thesis: verum

end;

hence T1 = T2 by A14, A16, TREES_2:33; :: thesis: verum