reserve I,J for set,i,j,x for object,
  S for non empty ManySortedSign;

theorem Th11:
  for I be non empty set, S be non void non empty ManySortedSign,
A be MSAlgebra-Family of I,S, o be OperSymbol of S holds dom doms (A?.o) = I &
  for i be Element of I holds (doms (A?.o)).i = Args(o,A.i)
proof
  let I be non empty set, S be non void non empty ManySortedSign, A be
  MSAlgebra-Family of I,S, o be OperSymbol of S;
A1: dom(A?.o) = I by PARTFUN1:def 2;
  for i be Element of I holds (doms (A?.o)).i = Args(o,A.i)
  proof
    let i be Element of I;
    (A?.o).i = Den(o,A.i) by Th7;
    then (doms (A?.o)).i = dom (Den(o,A.i)) by A1,FUNCT_6:22;
    hence thesis by FUNCT_2:def 1;
  end;
  hence thesis by A1,FUNCT_6:def 2;
end;
