reserve i for Nat, x,y for set;
reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;

theorem Th45:
  for S being non empty non void ManySortedSign
  for A being non-empty ManySortedSet of the carrier of S
  for f being ManySortedFunction of A#*the Arity of S, A*the ResultSort of S
  for o being OperSymbol of S
  for d being Function of (A#*the Arity of S).o, (A*the ResultSort of S).o
  holds
  f+*(o,d) is ManySortedFunction of A#*the Arity of S, A*the ResultSort of S
  proof
    let S be non empty non void ManySortedSign;
    let A be non-empty ManySortedSet of the carrier of S;
    let f be ManySortedFunction of A#*the Arity of S, A*the ResultSort of S;
    let o be OperSymbol of S;
    let d be Function of (A#*the Arity of S).o, (A*the ResultSort of S).o;
    let x be object; assume x in the carrier' of S; then
    reconsider x as OperSymbol of S;
    dom f = the carrier' of S by PARTFUN1:def 2; then
    (x = o implies f+*(o,d).x = d) & (x <> o implies f+*(o,d).x = f.x)
    by FUNCT_7:31,32;
    hence thesis;
  end;
