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 Th11:
  for A being OSSubset of OU0 holds Constants(OU0) c= A implies
  OSConstants(OU0) c= A
proof
  let A be OSSubset of OU0;
  assume
A1: Constants(OU0) c= A;
  assume not OSConstants(OU0) c= A;
  then consider i being object such that
A2: i in the carrier of S1 and
A3: not (OSConstants(OU0)).i c= A.i;
  reconsider s = i as SortSymbol of S1 by A2;
  consider x being object such that
A4: x in (OSConstants(OU0)).i and
A5: not x in A.i by A3;
  (OSConstants(OU0)).s = OSConstants(OU0,s) by Def5
    .= union {Constants(OU0,s2) : s2 <= s};
  then consider X1 being set such that
A6: x in X1 and
A7: X1 in {Constants(OU0,s2) : s2 <= s} by A4,TARSKI:def 4;
  consider s1 being SortSymbol of S1 such that
A8: X1 = Constants(OU0,s1) and
A9: s1 <= s by A7;
A10: (Constants(OU0)).s1 c= A.s1 by A1;
  x in (Constants(OU0)).s1 by A6,A8,MSUALG_2:def 4;
  then
A11: x in A.s1 by A10;
  A is OrderSortedSet of S1 by Def2;
  then A.s1 c= A.s by A9,OSALG_1:def 16;
  hence contradiction by A5,A11;
end;
