let f be Function; :: thesis: for O being non empty connected Poset
for T being non empty array of O
for r, p, q being Element of dom T st r in p & f = Swap ((id (dom T)),p,q) holds
( ((T,p,q) incl) . (r,q) = [(f . r),(f . q)] & ((T,p,q) incl) . (r,p) = [(f . r),(f . p)] )
let O be non empty connected Poset; :: thesis: for T being non empty array of O
for r, p, q being Element of dom T st r in p & f = Swap ((id (dom T)),p,q) holds
( ((T,p,q) incl) . (r,q) = [(f . r),(f . q)] & ((T,p,q) incl) . (r,p) = [(f . r),(f . p)] )
let T be non empty array of O; :: thesis: for r, p, q being Element of dom T st r in p & f = Swap ((id (dom T)),p,q) holds
( ((T,p,q) incl) . (r,q) = [(f . r),(f . q)] & ((T,p,q) incl) . (r,p) = [(f . r),(f . p)] )
let r, p, q be Element of dom T; :: thesis: ( r in p & f = Swap ((id (dom T)),p,q) implies ( ((T,p,q) incl) . (r,q) = [(f . r),(f . q)] & ((T,p,q) incl) . (r,p) = [(f . r),(f . p)] ) )
assume that
Z0: r in p and
Z1: f = Swap ((id (dom T)),p,q) ; :: thesis: ( ((T,p,q) incl) . (r,q) = [(f . r),(f . q)] & ((T,p,q) incl) . (r,p) = [(f . r),(f . p)] )
set X = dom T;
set i = id (dom T);
set s = Swap ((id (dom T)),p,q);
set h = [:(Swap ((id (dom T)),p,q)),(Swap ((id (dom T)),p,q)):];
set Y = (succ q) \ p;
set Z1 = [:{p},((succ q) \ p):];
set Z2 = [:((succ q) \ p),{q}:];
set g = id ([:{p},((succ q) \ p):] \/ [:((succ q) \ p),{q}:]);
dom (id (dom T)) = dom T by RELAT_1:45;
then A4: dom (Swap ((id (dom T)),p,q)) = dom T by FUNCT_7:99;
not p c= r by Z0, ORDINAL6:4;
then ( r nin {p} & r nin (succ q) \ p ) by TT, TARSKI:def 1;
then ( [r,p] nin [:{p},((succ q) \ p):] & [r,p] nin [:((succ q) \ p),{q}:] & [r,q] nin [:{p},((succ q) \ p):] & [r,q] nin [:((succ q) \ p),{q}:] ) by ZFMISC_1:87;
then A3: ( [r,q] nin dom (id ([:{p},((succ q) \ p):] \/ [:((succ q) \ p),{q}:])) & [r,p] nin dom (id ([:{p},((succ q) \ p):] \/ [:((succ q) \ p),{q}:])) ) by XBOOLE_0:def 3;
hence ((T,p,q) incl) . (r,q) = [:(Swap ((id (dom T)),p,q)),(Swap ((id (dom T)),p,q)):] . (r,q) by FUNCT_4:11
.= [(f . r),(f . q)] by Z1, A4, FUNCT_3:def 8 ;
:: thesis: ((T,p,q) incl) . (r,p) = [(f . r),(f . p)]
thus ((T,p,q) incl) . (r,p) = [:(Swap ((id (dom T)),p,q)),(Swap ((id (dom T)),p,q)):] . (r,p) by A3, FUNCT_4:11
.= [(f . r),(f . p)] by Z1, A4, FUNCT_3:def 8 ; :: thesis: verum