reserve x for set,
  R for non empty Poset;
reserve S1 for OrderSortedSign,
  OU0 for OSAlgebra of S1;
reserve s,s1,s2,s3,s4 for SortSymbol of S1;

theorem
  for A being OSSubset of OU0 holds OSConstants(OU0) = OSCl Constants( OU0)
proof
  let A be OSSubset of OU0;
A1: now
    let i be set;
    assume i in the carrier of S1;
    then reconsider s = i as SortSymbol of S1;
    set c1 = { (Constants(OU0)).s1: s1 <= s}, c2 = { Constants(OU0,s1): s1 <=
    s};
    for x being object holds (x in c1 iff x in c2)
    proof
      let x be object;
      hereby
        assume x in c1;
        then consider s1 such that
A2:     x = (Constants(OU0)).s1 and
A3:     s1 <= s;
        x = Constants(OU0,s1) by A2,MSUALG_2:def 4;
        hence x in c2 by A3;
      end;
      assume x in c2;
      then consider s1 such that
A4:   x = Constants(OU0,s1) and
A5:   s1 <= s;
      x = (Constants(OU0)).s1 by A4,MSUALG_2:def 4;
      hence thesis by A5;
    end;
    then
A6: c1 = c2 by TARSKI:2;
    (OSConstants(OU0)).s = OSConstants(OU0,s) by Def5
      .= (OSCl Constants(OU0)).s by A6,Def4;
    hence (OSConstants(OU0)).i = (OSCl Constants(OU0)).i;
  end;
  then
  for i being object st i in the carrier of S1 holds (OSCl Constants(OU0)).i
  c= (OSConstants(OU0)).i;
  then
A7: OSCl Constants(OU0) c= OSConstants(OU0);
  for i being object st i in the carrier of S1 holds (OSConstants(OU0)).i c=
  (OSCl Constants(OU0)).i by A1;
  then OSConstants(OU0) c= OSCl Constants(OU0);
  hence thesis by A7,PBOOLE:146;
end;
