
theorem Th10:
  for S being locally_directed OrderSortedSign, A being OSAlgebra
  of S, E being MSEquivalence-like (OrderSortedRelation of A), s1,s2 being
  Element of S st s1 <= s2 holds OSClass(E,s1) c= OSClass(E,s2)
proof
  let S be locally_directed OrderSortedSign;
  let A be OSAlgebra of S;
  let E be MSEquivalence-like OrderSortedRelation of A;
  let s1,s2 be Element of S;
  reconsider s3 = s1, s4 = s2 as Element of S;
  assume
A1: s1 <= s2;
  then
A2: CComp(s1) = CComp(s2) by Th4;
  thus OSClass(E,s1) c= OSClass(E,s2)
  proof
    reconsider SO = the Sorts of A as OrderSortedSet of S by OSALG_1:17;
    let z be object;
    assume z in OSClass(E,s1);
    then
A3: ex x being set st x in (the Sorts of A).s1 & z = Class( CompClass(E,
    CComp(s1)), x) by Def10;
    SO.s3 c= SO.s4 by A1,OSALG_1:def 16;
    hence thesis by A2,A3,Def10;
  end;
end;
