
theorem Th7:
  for S being non empty non void ManySortedSign
  for A being non-empty MSAlgebra over S
  for o being OperSymbol of S, a being set, r being SortSymbol of S
  st o is_of_type a,r
  holds Den(o,A) is Function of (the Sorts of A)#.a, (the Sorts of A).r &
  Args(o,A) = (the Sorts of A)#.a & Result(o,A) = (the Sorts of A).r
  proof
    let S be non empty non void ManySortedSign;
    let A be non-empty MSAlgebra over S;
    let o be OperSymbol of S;
    let a be set;
    let r be SortSymbol of S;
    assume A1: (the Arity of S).o = a & (the ResultSort of S).o = r;
    then
A2: ((the Sorts of A)#*the Arity of S).o = (the Sorts of A)#.a by FUNCT_2:15;
    ((the Sorts of A)*the ResultSort of S).o = (the Sorts of A).r
    by A1,FUNCT_2:15;
    hence Den(o,A) is Function of (the Sorts of A)#.a, (the Sorts of A).r
    by A2;
    thus thesis by A1,FUNCT_2:15;
  end;
