let S be non empty non void ManySortedSign ; :: thesis: for o being OperSymbol of S
for V being non-empty ManySortedSet of the carrier of S
for x being set holds
( x is ArgumentSeq of Sym (o,V) iff x is Element of Args (o,(FreeMSA V)) )
let o be OperSymbol of S; :: thesis: for V being non-empty ManySortedSet of the carrier of S
for x being set holds
( x is ArgumentSeq of Sym (o,V) iff x is Element of Args (o,(FreeMSA V)) )
let V be non-empty ManySortedSet of the carrier of S; :: thesis: for x being set holds
( x is ArgumentSeq of Sym (o,V) iff x is Element of Args (o,(FreeMSA V)) )
let x be set ; :: thesis: ( x is ArgumentSeq of Sym (o,V) iff x is Element of Args (o,(FreeMSA V)) )
A1: TS (DTConMSA V) = S -Terms V by MSATERM:def 1;
A2: FreeMSA V = MSAlgebra(# (FreeSort V),(FreeOper V) #) by MSAFREE:def 14;

hereby :: thesis: ( x is Element of Args (o,(FreeMSA V)) implies x is ArgumentSeq of Sym (o,V) )

assume x is ArgumentSeq of Sym (o,V) ; :: thesis: x is Element of Args (o,(FreeMSA V))
then reconsider p = x as ArgumentSeq of Sym (o,V) ;
reconsider p = p as FinSequence of TS (DTConMSA V) by MSATERM:def 1;
Sym (o,V) ==> roots p by MSATERM:21;
hence x is Element of Args (o,(FreeMSA V)) by A2, MSAFREE:10; :: thesis: verum

end;

assume x is Element of Args (o,(FreeMSA V)) ; :: thesis: x is ArgumentSeq of Sym (o,V)
then reconsider x = x as Element of Args (o,(FreeMSA V)) ;
rng x c= TS (DTConMSA V)

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng x or y in TS (DTConMSA V) )
assume y in rng x ; :: thesis: y in TS (DTConMSA V)
then consider z being object such that
A3: z in dom x and
A4: y = x . z by FUNCT_1:def 3;
reconsider z = z as Element of NAT by A3;
A5: (FreeSort V) . ((the_arity_of o) /. z) = FreeSort (V,((the_arity_of o) /. z)) by MSAFREE:def 11;
dom x = dom (the_arity_of o) by MSUALG_6:2;
then y in (FreeSort V) . ((the_arity_of o) /. z) by A2, A3, A4, MSUALG_6:2;
hence y in TS (DTConMSA V) by A5; :: thesis: verum

end;

then reconsider x = x as FinSequence of TS (DTConMSA V) by FINSEQ_1:def 4;
Sym (o,V) ==> roots x by A2, MSAFREE:10;
hence x is ArgumentSeq of Sym (o,V) by A1, MSATERM:21; :: thesis: verum