
theorem Th3:
  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 Element of Args(o, FreeMSA V) holds Den(o,FreeMSA V).x = [o, the carrier
  of S]-tree x
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 Element of Args(o, FreeMSA V);
A1: FreeMSA V = MSAlgebra(#FreeSort(V), FreeOper(V)#) by MSAFREE:def 14;
  reconsider p = x as ArgumentSeq of Sym(o,V) by Th1;
A2: Sym(o, V) = [o, the carrier of S] by MSAFREE:def 9;
  p is FinSequence of TS DTConMSA V & Sym(o, V) ==> roots p by MSATERM:21,def 1
;
  then DenOp(o,V).x = [o, the carrier of S]-tree x by A2,MSAFREE:def 12;
  hence thesis by A1,MSAFREE:def 13;
end;
