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
  P is topological & i in I & f = P.i implies
  for x, y being Element of bool (M.i) holds f.(x \/ y) = (f.x) \/ (f.y)
proof
  assume that
A1: P is topological and
A2: i in I and
A3: f = P.i;
A4: i in dom P by A2,PARTFUN1:def 2;
  let x, y be Element of bool (M.i);
  dom (EmptyMS I +* (i .--> y)) = I by A2,PZFMISC1:1;
  then reconsider Y = EmptyMS I +* (i .--> y) as ManySortedSet of I
   by PARTFUN1:def 2,RELAT_1:def 18;
  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;
A5: i in {i} by TARSKI:def 1;
  dom (i .--> y) = {i}; then
A6: Y.i = (i .--> y).i by A5,FUNCT_4:13
    .= y by FUNCOP_1:72;
A7: X is Element of bool M & Y is Element of bool M by Lm2,MSSUBFAM:11;
    i in dom (X (\/) Y) by A2,PARTFUN1:def 2; then
    i in (dom P) /\ dom (X (\/) Y) by A4,XBOOLE_0:def 4; then  
B1: i in dom (P..(X (\/) Y)) by PRALG_1:def 19;
    i in dom X by A2,PARTFUN1:def 2; then
    i in dom P /\ dom X by XBOOLE_0:def 4,A4; then
B2: i in dom (P..X) by PRALG_1:def 19;
    i in dom Y by A2,PARTFUN1:def 2; then
    i in dom P /\ dom Y by XBOOLE_0:def 4,A4; then
B3: i in dom (P..Y) by PRALG_1:def 19;
  dom (i .--> x) = {i};
  then
A8: X.i = (i .--> x).i by A5,FUNCT_4:13
    .= x by FUNCOP_1:72;
  hence f.(x \/ y) = f.((X (\/) Y).i) by A2,A6,PBOOLE:def 4
    .= (P..(X (\/) Y)).i by A3,PRALG_1:def 19,B1
    .= ((P..X) (\/) (P..Y)).i by A1,A7
    .= (P..X).i \/ (P..Y).i by A2,PBOOLE:def 4
    .= f.(X.i) \/ (P..Y).i by A3,PRALG_1:def 19,B2
    .= (f.x) \/ (f.y) by A3,A8,A6,PRALG_1:def 19,B3;
end;
