
theorem Th1:
  for S being non empty non void ManySortedSign 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)
proof
  let S be non empty non void ManySortedSign;
  let o be OperSymbol of S;
  let V be non-empty ManySortedSet of the carrier of S;
  let x be set;
A1: TS DTConMSA V = S-Terms V by MSATERM:def 1;
A2: FreeMSA V = MSAlgebra(#FreeSort V, FreeOper V#) by MSAFREE:def 14;
  hereby
    assume x is ArgumentSeq of Sym(o,V);
    then reconsider p = x as ArgumentSeq of Sym(o,V);
    reconsider 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;
  end;
  assume x is Element of Args(o, FreeMSA V);
  then reconsider x as Element of Args(o, FreeMSA V);
  rng x c= TS DTConMSA V
  proof
    let y be object;
    assume y in rng x;
    then consider z being object such that
A3: z in dom x and
A4: y = x.z by FUNCT_1:def 3;
    reconsider 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 thesis by A5;
  end;
  then reconsider x as FinSequence of TS DTConMSA V by FINSEQ_1:def 4;
  Sym(o, V) ==> roots x by A2,MSAFREE:10;
  hence thesis by A1,MSATERM:21;
end;
