reserve i, x, I for set,
  A, M for ManySortedSet of I,
  f for Function,
  F for ManySortedFunction of I;

theorem
  for A, X being ManySortedSet of I
  for B being non-empty ManySortedSet of I
  for F being ManySortedFunction of A, B st X in A holds F..X in B
proof
  let A, X be ManySortedSet of I;
  let B be non-empty ManySortedSet of I;
  let F be ManySortedFunction of A, B such that
A1: X in A;
  let i be object such that
A2: i in I;
  reconsider g = F.i as Function;
A3: g is Function of A.i, B.i by A2,PBOOLE:def 15;
  X.i in A.i by A1,A2;
  then dom F = I & g.(X.i) in B.i by A2,A3,FUNCT_2:5,PARTFUN1:def 2;
  hence thesis by A2,PRALG_1:def 20;
end;
