let S be non void Signature; :: thesis: for X being ManySortedSet of the carrier of S
for o being OperSymbol of S st the_arity_of o = {} holds
root-tree [o, the carrier of S] in the Sorts of (Free (S,X)) .

let X be ManySortedSet of the carrier of S; :: thesis: for o being OperSymbol of S st the_arity_of o = {} holds
root-tree [o, the carrier of S] in the Sorts of (Free (S,X)) .

let o be OperSymbol of S; :: thesis: ( the_arity_of o = {} implies root-tree [o, the carrier of S] in the Sorts of (Free (S,X)) . )
assume A1: the_arity_of o = {} ; :: thesis: root-tree [o, the carrier of S] in the Sorts of (Free (S,X)) .
set Y = X (\/) ( the carrier of S --> );
A2: Args (o,(FreeMSA (X (\/) ( the carrier of S --> )))) = (( the Sorts of (FreeMSA (X (\/) ( the carrier of S --> ))) #) * the Arity of S) . o by MSUALG_1:def 4
.= ( the Sorts of (FreeMSA (X (\/) ( the carrier of S --> ))) #) . ( the Arity of S . o) by FUNCT_2:15
.= ( the Sorts of (FreeMSA (X (\/) ( the carrier of S --> ))) #) . (<*> the carrier of S) by
.= by PRE_CIRC:2 ;
then A3: dom (Den (o,(FreeMSA (X (\/) ( the carrier of S --> ))))) c= ;
A4: ex A being MSSubset of (FreeMSA (X (\/) ( the carrier of S --> ))) st
( Free (S,X) = GenMSAlg A & A = (Reverse (X (\/) ( the carrier of S --> ))) "" X ) by Def1;
then reconsider FX = the Sorts of (Free (S,X)) as MSSubset of (FreeMSA (X (\/) ( the carrier of S --> ))) by MSUALG_2:def 9;
((FX #) * the Arity of S) . o = (FX #) . ( the Arity of S . o) by FUNCT_2:15
.= (FX #) . (<*> the carrier of S) by
.= by PRE_CIRC:2 ;
then A5: (Den (o,(FreeMSA (X (\/) ( the carrier of S --> ))))) | (((FX #) * the Arity of S) . o) = Den (o,(FreeMSA (X (\/) ( the carrier of S --> )))) by ;
set a = the ArgumentSeq of Sym (o,(X (\/) ( the carrier of S --> )));
reconsider a = the ArgumentSeq of Sym (o,(X (\/) ( the carrier of S --> ))) as Element of Args (o,(FreeMSA (X (\/) ( the carrier of S --> )))) by INSTALG1:1;
a = {} by ;
then root-tree [o, the carrier of S] = [o, the carrier of S] -tree a by TREES_4:20;
then (Den (o,(FreeMSA (X (\/) ( the carrier of S --> ))))) . a = root-tree [o, the carrier of S] by INSTALG1:3;
then A6: root-tree [o, the carrier of S] in rng (Den (o,(FreeMSA (X (\/) ( the carrier of S --> ))))) by FUNCT_2:4;
FX is opers_closed by ;
then FX is_closed_on o ;
then A7: rng ((Den (o,(FreeMSA (X (\/) ( the carrier of S --> ))))) | (((FX #) * the Arity of S) . o)) c= (FX * the ResultSort of S) . o ;
(FX * the ResultSort of S) . o = FX . ( the ResultSort of S . o) by FUNCT_2:15
.= FX . by MSUALG_1:def 2 ;
hence root-tree [o, the carrier of S] in the Sorts of (Free (S,X)) . by A7, A5, A6; :: thesis: verum