let x, y be object ; :: thesis: for p being DTree-yielding FinSequence holds
( y in dom (x -tree p) iff ( y = {} or ex i being Nat ex T being DecoratedTree ex q being Node of T st
( i < len p & T = p . (i + 1) & y = <*i*> ^ q ) ) )

let p be DTree-yielding FinSequence; :: thesis: ( y in dom (x -tree p) iff ( y = {} or ex i being Nat ex T being DecoratedTree ex q being Node of T st
( i < len p & T = p . (i + 1) & y = <*i*> ^ q ) ) )

A1: dom (x -tree p) = tree (doms p) by Th10;
A2: now :: thesis: ( ex i being Nat ex q being FinSequence st
( i < len (doms p) & q in (doms p) . (i + 1) & y = <*i*> ^ q ) implies ex i being Nat ex T being DecoratedTree ex q being Node of T st
( i < len p & T = p . (i + 1) & y = <*i*> ^ q ) )
given i being Nat, q being FinSequence such that A3: i < len (doms p) and
A4: q in (doms p) . (i + 1) and
A5: y = <*i*> ^ q ; :: thesis: ex i being Nat ex T being DecoratedTree ex q being Node of T st
( i < len p & T = p . (i + 1) & y = <*i*> ^ q )

len (doms p) = len p by TREES_3:38;
then A6: i + 1 in dom p by ;
then reconsider T = p . (i + 1) as DecoratedTree by TREES_3:24;
take i = i; :: thesis: ex T being DecoratedTree ex q being Node of T st
( i < len p & T = p . (i + 1) & y = <*i*> ^ q )

take T = T; :: thesis: ex q being Node of T st
( i < len p & T = p . (i + 1) & y = <*i*> ^ q )

reconsider q = q as Node of T by ;
take q = q; :: thesis: ( i < len p & T = p . (i + 1) & y = <*i*> ^ q )
thus ( i < len p & T = p . (i + 1) & y = <*i*> ^ q ) by ; :: thesis: verum
end;
now :: thesis: ( ex i being Nat ex T being DecoratedTree ex q being Node of T st
( i < len p & T = p . (i + 1) & y = <*i*> ^ q ) implies ex i being Nat ex q being FinSequence st
( i < len (doms p) & q in (doms p) . (i + 1) & y = <*i*> ^ q ) )
given i being Nat, T being DecoratedTree, q being Node of T such that A7: i < len p and
A8: T = p . (i + 1) and
A9: y = <*i*> ^ q ; :: thesis: ex i being Nat ex q being FinSequence st
( i < len (doms p) & q in (doms p) . (i + 1) & y = <*i*> ^ q )

reconsider q = q as FinSequence ;
take i = i; :: thesis: ex q being FinSequence st
( i < len (doms p) & q in (doms p) . (i + 1) & y = <*i*> ^ q )

take q = q; :: thesis: ( i < len (doms p) & q in (doms p) . (i + 1) & y = <*i*> ^ q )
i + 1 in dom p by ;
then (doms p) . (i + 1) = dom T by ;
hence ( i < len (doms p) & q in (doms p) . (i + 1) & y = <*i*> ^ q ) by ; :: thesis: verum
end;
hence ( y in dom (x -tree p) iff ( y = {} or ex i being Nat ex T being DecoratedTree ex q being Node of T st
( i < len p & T = p . (i + 1) & y = <*i*> ^ q ) ) ) by ; :: thesis: verum