reserve R for non empty Poset,
  S1 for OrderSortedSign;

theorem
  for U1 being OSAlgebra of S1 holds (U1 is monotone iff MSAlgebra(# the
    Sorts of U1, the Charact of U1 #) is monotone)
proof
  let U1 be OSAlgebra of S1;
  set U2 = MSAlgebra(# the Sorts of U1, the Charact of U1 #);
A1: now
    let o1 being OperSymbol of S1;
    thus Den(o1,U1) = (the Charact of U2).o1 by MSUALG_1:def 6
      .= Den(o1,U2) by MSUALG_1:def 6;
    thus Args(o1,U1) = ((the Sorts of U2)# * the Arity of S1).o1 by
MSUALG_1:def 4
      .= Args(o1,U2) by MSUALG_1:def 4;
  end;
  thus U1 is monotone implies U2 is monotone
  proof
    assume
A2: U1 is monotone;
    let o1,o2 be OperSymbol of S1;
    assume o1 <= o2;
    then
A3: Den(o2,U1)|Args(o1,U1) = Den(o1,U1) by A2;
    thus Den(o2,U2)|Args(o1,U2) = Den(o2,U1)|Args(o1,U2) by A1
      .= Den(o1,U1) by A1,A3
      .= Den(o1,U2) by A1;
  end;
  assume
A4: U2 is monotone;
  let o1,o2 be OperSymbol of S1;
  assume o1 <= o2;
  then
A5: Den(o2,U2)|Args(o1,U2) = Den(o1,U2) by A4;
  thus Den(o2,U1)|Args(o1,U1) = Den(o2,U2)|Args(o1,U1) by A1
    .= Den(o1,U2) by A1,A5
    .= Den(o1,U1) by A1;
end;
