reserve i, x, I for set,
  A, M for ManySortedSet of I,
  f for Function,
  F for ManySortedFunction of I;
reserve P, R for MSSetOp of M,
  E, T for Element of bool M;

theorem Th24:
  P is reflexive & i in I & f = P.i implies
  for x being Element of bool (M.i) holds x c= f.x
proof
  assume that
A1: P is reflexive and
A2: i in I and
A3: f = P.i;
  let x be Element of bool (M.i);
  dom (EmptyMS I +* (i .--> x)) = I by A2,PZFMISC1:1;
  then reconsider X = EmptyMS I +* (i .--> x) as ManySortedSet of I
    by PARTFUN1:def 2,RELAT_1:def 18;
  X is Element of bool M by Lm2,MSSUBFAM:11;
  then X c= P..X by A1; then
A4: X.i c= (P..X).i by A2;
  dom (i .--> x) = {i} & i in {i} by TARSKI:def 1; then
A5: X.i = (i .--> x).i by FUNCT_4:13
    .= x by FUNCOP_1:72;
  i in dom X & i in dom P by A2,PARTFUN1:def 2; then
  i in dom P /\ dom X by XBOOLE_0:def 4; then
  i in dom (P..X) by PRALG_1:def 19;
  hence thesis by A3,A5,A4,PRALG_1:def 19;
end;
