reserve U1,U2,U3 for Universal_Algebra,
  n,m for Nat,
  x,y,z for object,
  A,B for non empty set,
  h1 for FinSequence of [:A,B:];
reserve h1 for homogeneous quasi_total non empty PartFunc of
    (the carrier of U1)*,the carrier of U1,
  h2 for homogeneous quasi_total non empty PartFunc of
    (the carrier of U2)*,the carrier of U2;

theorem Th11:
  for J be non empty set, B be non-empty ManySortedSet of J, O be
  ManySortedOperation of B holds O is equal-arity iff for i,j be Element of J
  holds arity (O.i) = arity (O.j)
proof
  let J be non empty set, B be non-empty ManySortedSet of J, O be
  ManySortedOperation of B;
  thus O is equal-arity implies for i,j be Element of J holds arity (O.i) =
  arity (O.j)
  proof
    assume
A1: O is equal-arity;
    let i,j be Element of J;
A2: dom (O.j) = arity(O.j)-tuples_on (B.j) by MARGREL1:22;
    dom O = J & dom (O.i) = arity(O.i)-tuples_on (B.i)
      by MARGREL1:22,PARTFUN1:def 2;
    hence thesis by A1,A2;
  end;
  assume
A3: for i,j be Element of J holds arity (O.i) = arity (O.j);
  let x,y be set;
  assume x in dom O & y in dom O;
  then reconsider x1 = x, y1 = y as Element of J by PARTFUN1:def 2;
  let f,g be Function;
  assume that
A4: O.x = f and
A5: O.y = g;
  arity (O.x1) = arity (O.y1) by A3;
  then
A6: dom g = (arity (O.x1))-tuples_on (B.y1) by A5,MARGREL1:22;
  let n,m be Nat;
  let X,Y be non empty set;
  assume that
A7: dom f = n-tuples_on X and
A8: dom g = m-tuples_on Y;
  dom f = (arity (O.x1))-tuples_on (B.x1) by A4,MARGREL1:22;
  then
A9: n = arity (O.x1) by A7,FINSEQ_2:110;
  set p = the Element of n-tuples_on X;
  set q = the Element of m-tuples_on Y;
  let o1 be homogeneous quasi_total non empty PartFunc of X*,X, o2 be
  homogeneous quasi_total non empty PartFunc of Y*,Y;
  assume that
A10: f = o1 and
A11: g = o2;
A12: arity o2 = len q by A8,A11,MARGREL1:def 25
    .= m by CARD_1:def 7;
  arity o1 = len p by A7,A10,MARGREL1:def 25
    .= n by CARD_1:def 7;
  hence thesis by A8,A6,A9,A12,FINSEQ_2:110;
end;
