let i, I be set ; :: thesis: for M being ManySortedSet of I
for f being Function
for P being MSSetOp of M st P is idempotent & i in I & f = P . i holds
for x being Element of bool (M . i) holds f . x = f . (f . x)

let M be ManySortedSet of I; :: thesis: for f being Function
for P being MSSetOp of M st P is idempotent & i in I & f = P . i holds
for x being Element of bool (M . i) holds f . x = f . (f . x)

let f be Function; :: thesis: for P being MSSetOp of M st P is idempotent & i in I & f = P . i holds
for x being Element of bool (M . i) holds f . x = f . (f . x)

let P be MSSetOp of M; :: thesis: ( P is idempotent & i in I & f = P . i implies for x being Element of bool (M . i) holds f . x = f . (f . x) )
assume that
A1: P is idempotent and
A2: i in I and
A3: f = P . i ; :: thesis: for x being Element of bool (M . i) holds f . x = f . (f . x)
A4: i in dom P by ;
let x be Element of bool (M . i); :: thesis: f . x = f . (f . x)
dom (() +* (i .--> x)) = I by ;
then reconsider X = () +* (i .--> x) as ManySortedSet of I by ;
A5: X is Element of bool M by ;
( dom (i .--> x) = {i} & i in {i} ) by TARSKI:def 1;
then A6: X . i = (i .--> x) . i by FUNCT_4:13
.= x by FUNCOP_1:72 ;
hence f . x = (P .. X) . i by
.= (P .. (P .. X)) . i by A1, A5
.= f . ((P .. X) . i) by
.= f . (f . x) by ;
:: thesis: verum