
theorem Th39:
  for I being set, A,B being ManySortedSet of I for C being
ManySortedSubset of A for F being ManySortedFunction of A,B for i being set st
i in I for f,g being Function st f = F.i & g = (F||C).i for x being set st x in
  C.i holds g.x = f.x
proof
  let I be set, A,B be ManySortedSet of I;
  let C be ManySortedSubset of A;
  let F be ManySortedFunction of A,B;
  let i be set;
  assume
A1: i in I;
  then reconsider Fi = F.i as Function of A.i, B.i by PBOOLE:def 15;
  let f,g be Function;
  assume that
A2: f = F.i and
A3: g = (F||C).i;
  let x be set;
  g = Fi|(C.i) by A1,A3,MSAFREE:def 1;
  hence thesis by A2,FUNCT_1:49;
end;
