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

theorem Th7:
  for I be non empty set, i be Element of I, S be non void non
empty ManySortedSign, A be MSAlgebra-Family of I,S, o be OperSymbol of S holds
  (A?.o).i = Den(o,A.i)
proof
  let I be non empty set, i be Element of I, S be non void non empty
  ManySortedSign, A be MSAlgebra-Family of I,S, o be OperSymbol of S;
  set O = the carrier' of S, f = uncurry (OPER A);
A1: [i,o]`1 = i & [i,o]`2 = o;
A2: ex U0 being MSAlgebra over S st U0 = A.i & (OPER A).i = the Charact of
  U0 by Def11;
A3: dom f = [:I,O:] by Th5;
  then
A4: [i,o] in dom f by ZFMISC_1:87;
   consider g be Function such that
A5: (curry' f).o = g and
  dom g = I and
  rng g c= rng f and
A6: for x st x in I holds g.x = f.(x,o) by A3,FUNCT_5:32,ZFMISC_1:90;
  g.i = f.(i,o) by A6;
  then g.i = (the Charact of A.i).o by A4,A1,A2,FUNCT_5:def 2
    .= Den(o,A.i) by MSUALG_1:def 6;
  hence thesis by A5;
end;
