reserve A for preIfWhileAlgebra;
reserve A for Euclidean preIfWhileAlgebra;
reserve X for non empty countable set;
reserve T for Subset of Funcs(X, INT);
reserve f for Euclidean ExecutionFunction of A, Funcs(X, INT), T;
reserve A for Euclidean preIfWhileAlgebra,
  X for non empty countable set,
   z for (Element of X),
  s,s9 for (Element of Funcs(X, INT)),
  T for Subset of Funcs(X, INT),
  f for Euclidean ExecutionFunction of A, Funcs(X, INT), T,
  v for INT-Variable of A,f,
  t for INT-Expression of A,f;
reserve i for Integer;

theorem Th33:
  for x,y being Variable of f holds f.(s, x*=y).x = s.x*s.y & for
  z st z <> x holds f.(s, x*=y).z = s.z
proof
  let x,y be Variable of f;
A1: dom (.x(#).y) = Funcs(X, INT) by FUNCT_2:def 1;
  (^x).s = x;
  hence f.(s, x*=y).x = (.x(#).y).s by Th24
    .= ((.x).s)*(.y.s) by A1,VALUED_1:def 4
    .= (s.x)*(.y.s) by Th22
    .= (s.x)*(s.y) by Th22;
  thus thesis by Th26;
end;
