
theorem
  for f being Function, I being set for A,B being ManySortedSet of I st
for i being set st i in I holds A.i c= dom f & f.:(A.i) c= B.i holds f-MSF(I,A)
  is ManySortedFunction of A,B
proof
  let f be Function, I be set;
  let A,B be ManySortedSet of I such that
A1: for i being set st i in I holds A.i c= dom f & f.:(A.i) c= B.i;
  let i be object;
  assume
A2: i in I;
  then
A3: (f-MSF(I,A)).i = f|(A.i) by Def1;
  f.:(A.i) c= B.i by A1,A2;
  then
A4: rng ((f-MSF(I,A)).i) c= B.i by A3,RELAT_1:115;
  dom ((f-MSF(I,A)).i) = A.i by A1,A2,A3,RELAT_1:62;
  hence thesis by A4,FUNCT_2:2;
end;
