let S be non empty non void ManySortedSign ; :: thesis: for o being OperSymbol of S
for V being V5() 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 V5() 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 V5() 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 16;

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 set ; :: 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 set such that
A3: z in dom x and
A4: y = x . z by FUNCT_1:def 5;
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 13;
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