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 Th17:
  for A be MSSubset of U0 for B be MSSubset of U0 st B in SubSort(
  A) holds ((MSSubSort A)# * (the Arity of S)).o c= (B# * (the Arity of S)).o
proof
  let A be MSSubset of U0, B be MSSubset of U0;
  assume
A1: B in SubSort(A);
  MSSubSort (A) c= B
  proof
    let i be object;
    assume i in the carrier of S;
    then reconsider s = i as SortSymbol of S;
    (MSSubSort A).s = meet (SubSort(A,s)) & B.s in (SubSort(A,s)) by A1,Def13
,Def14;
    hence thesis by SETFAM_1:3;
  end;
  hence thesis by Th2;
end;
