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 A2, PARTFUN1:def 2;
let x be Element of bool (M . i); :: thesis: f . x = f . (f . x)
dom (([[0]] I) +* (i .--> x)) = I by A2, PZFMISC1:1;
then reconsider X = ([[0]] I) +* (i .--> x) as ManySortedSet of I by PARTFUN1:def 2, RELAT_1:def 18;
A5: X is Element of bool M by Lm2, MSSUBFAM:11;
( dom (i .--> x) = {i} & i in {i} ) by FUNCOP_1:13, 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 A3, A4, PRALG_1:def 17
.= (P .. (P .. X)) . i by A1, A5, Def4
.= f . ((P .. X) . i) by A3, A4, PRALG_1:def 17
.= f . (f . x) by A3, A6, A4, PRALG_1:def 17 ;
:: thesis: verum