reserve x,y for object;
reserve S for non void non empty ManySortedSign,
  o for OperSymbol of S,
  U0,U1, U2 for MSAlgebra over S;

theorem Th8:
  for U1,U2 be MSSubAlgebra of U0 st the Sorts of U1 c= the Sorts
  of U2 holds U1 is MSSubAlgebra of U2
proof
  let U1, U2 be MSSubAlgebra of U0;
  reconsider B1 = the Sorts of U1, B2 = the Sorts of U2 as MSSubset of U0 by
Def9;
  assume
A1: the Sorts of U1 c= the Sorts of U2;
  hence the Sorts of U1 is MSSubset of U2 by PBOOLE:def 18;
  let B be MSSubset of U2;
A2: B1 is opers_closed by Def9;
  set O = the Charact of U1, P = Opers(U2,B);
A3: the Charact of U1 = Opers(U0,B1) by Def9;
A4: B2 is opers_closed by Def9;
A5: the Charact of U2 = Opers(U0,B2) by Def9;
  assume
A6: B = the Sorts of U1;
A7: for o be OperSymbol of S holds B is_closed_on o
  proof
    let o be OperSymbol of S;
A8: B1 is_closed_on o by A2;
A9: B2 is_closed_on o by A4;
A10: Den(o,U2) = Opers(U0,B2).o by A5,MSUALG_1:def 6
      .= o/.B2 by Def8
      .= (Den(o,U0))|((B2# * the Arity of S).o) by A9,Def7;
    Den(o,U1) = Opers(U0,B1).o by A3,MSUALG_1:def 6
      .= o/.B1 by Def8
      .= (Den(o,U0))|((B1# * the Arity of S).o) by A8,Def7
      .= (Den(o,U0))|(((B2# * the Arity of S).o) /\ ((B1# * the Arity of S).
    o)) by A1,Th2,XBOOLE_1:28
      .= (Den(o,U2))|((B1# * the Arity of S).o) by A10,RELAT_1:71;
    then rng ((Den(o,U2))|((B1# * the Arity of S).o)) c= Result(o,U1) by
RELAT_1:def 19;
    then rng ((Den(o,U2))|((B1# * the Arity of S).o)) c= ((the Sorts of U1) *
    the ResultSort of S).o by MSUALG_1:def 5;
    hence thesis by A6;
  end;
  hence B is opers_closed;
  for x being object st x in the carrier' of S holds O.x =P.x
  proof
    let x be object;
    assume x in the carrier' of S;
    then reconsider o = x as OperSymbol of S;
A11: B1 is_closed_on o by A2;
A12: B2 is_closed_on o by A4;
A13: Den(o,U2) = Opers(U0,B2).o by A5,MSUALG_1:def 6
      .= o/.B2 by Def8
      .= (Den(o,U0))|((B2# * the Arity of S).o) by A12,Def7;
    thus O.x = o/.B1 by A3,Def8
      .= (Den(o,U0))|((B1# * the Arity of S).o) by A11,Def7
      .= (Den(o,U0))|(((B2# * the Arity of S).o) /\ ((B1# * the Arity of S).
    o)) by A1,Th2,XBOOLE_1:28
      .= (Den(o,U2))|((B1# * the Arity of S).o) by A13,RELAT_1:71
      .= o/.B by A6,A7,Def7
      .= P.x by Def8;
  end;
  hence thesis;
end;
