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 Th8:
  for M being ManySortedSet of the carrier of S1, A being
  OrderSortedSet of S1 holds M c= A implies OSCl M c= A
proof
  let M be ManySortedSet of the carrier of S1, A be OrderSortedSet of S1;
  assume
A1: M c= A;
  assume not OSCl M c= A;
  then consider i being object such that
A2: i in the carrier of S1 and
A3: not (OSCl M).i c= A.i;
  reconsider s = i as SortSymbol of S1 by A2;
  consider x being object such that
A4: x in (OSCl M).i and
A5: not x in A.i by A3;
  (OSCl M).s = union { M.s2 : s2 <= s} by Def4;
  then consider X1 being set such that
A6: x in X1 and
A7: X1 in { M.s2 : s2 <= s} by A4,TARSKI:def 4;
  consider s1 being SortSymbol of S1 such that
A8: X1 = M.s1 and
A9: s1 <= s by A7;
  M.s1 c= A.s1 by A1;
  then
A10: x in A.s1 by A6,A8;
  A.s1 c= A.s by A9,OSALG_1:def 16;
  hence contradiction by A5,A10;
end;
