reserve S for non void non empty ManySortedSign,
  U1, U2, U3 for non-empty MSAlgebra over S,
  I for set,
  A for ManySortedSet of I,
  B, C for non-empty ManySortedSet of I;

theorem
  for F be ManySortedFunction of A, B for X be ManySortedSubset of A
  holds doms (F || X) c= doms F
proof
  let F be ManySortedFunction of A, B, X be ManySortedSubset of A;
  let i be object;
A1: dom (F||X) = I by PARTFUN1:def 2;
  assume
A2: i in I;
  then reconsider f = F.i as Function of A.i, B.i by PBOOLE:def 15;
  dom F = I by PARTFUN1:def 2;
  then
A3: (doms F).i = dom f by A2,FUNCT_6:22;
  (F||X).i = f|(X.i) by A2,MSAFREE:def 1;
  then (doms (F||X)).i = dom (f|(X.i)) by A2,A1,FUNCT_6:22;
  hence thesis by A3,RELAT_1:60;
end;
