reserve X,Y for set, x,y,z for object, i,j,n for natural number;

theorem Th5:
  for S being non empty non void ManySortedSign
  for o being OperSymbol of S, s,r being SortSymbol of S,
      T being MSAlgebra over S holds
  o is_of_type <*s*>,r & x in (the Sorts of T).s implies <*x*> in Args(o,T)
  proof
    let S be non empty non void ManySortedSign;
    let o be OperSymbol of S;
    let s,r be SortSymbol of S;
    let T be MSAlgebra over S;
    assume A1: (the Arity of S).o = <*s*> & (the ResultSort of S).o = r;
    assume A2: x in (the Sorts of T).s;
A3: dom the Sorts of T = the carrier of S by PARTFUN1:def 2;
    Args(o,T) = product ((the Sorts of T)*the_arity_of o) by PRALG_2:3
    .= product <*(the Sorts of T).s*> by A1,A3,FINSEQ_2:34;
    hence <*x*> in Args(o,T) by A2,FINSEQ_3:123;
  end;
