reserve x for set,
  R for non empty Poset;

theorem Th5:
  for S being OrderSortedSign for U0 being OSAlgebra of S for U1
  being MSAlgebra over S holds U1 is OSSubAlgebra of U0 iff the Sorts of U1 is
  OSSubset of U0 & for B be OSSubset of U0 st B = the Sorts of U1 holds B is
  opers_closed & the Charact of U1 = Opers(U0,B)
proof
  let S be OrderSortedSign;
  let U0 be OSAlgebra of S;
  let U1 be MSAlgebra over S;
  hereby
    assume
A1: U1 is OSSubAlgebra of U0;
    then the Sorts of U1 is OrderSortedSet of S & the Sorts of U1 is MSSubset
    of U0 by MSUALG_2:def 9,OSALG_1:17;
    hence the Sorts of U1 is OSSubset of U0 by Def2;
    let B be OSSubset of U0;
    assume B = the Sorts of U1;
    hence B is opers_closed & the Charact of U1 = Opers(U0,B) by A1,
MSUALG_2:def 9;
  end;
  assume
A2: the Sorts of U1 is OSSubset of U0;
  assume
A3: for B be OSSubset of U0 st B = the Sorts of U1 holds B is
  opers_closed & the Charact of U1 = Opers(U0,B);
A4: U1 is MSSubAlgebra of U0
  proof
    thus the Sorts of U1 is MSSubset of U0 by A2;
    let B be MSSubset of U0 such that
A5: B = the Sorts of U1;
    reconsider B1=B as OSSubset of U0 by A2,A5;
    B1 is opers_closed by A3,A5;
    hence thesis by A3,A5;
  end;
  the Sorts of U1 is OrderSortedSet of S by A2,Def2;
  hence thesis by A4,OSALG_1:17;
end;
