let O be non empty connected Poset; :: thesis: for T being non empty array of O
for p, q, r being Element of dom T st r in p & p in q holds
( ((T,p,q) incl) . (r,p) = [r,q] & ((T,p,q) incl) . (r,q) = [r,p] )
let T be non empty array of O; :: thesis: for p, q, r being Element of dom T st r in p & p in q holds
( ((T,p,q) incl) . (r,p) = [r,q] & ((T,p,q) incl) . (r,q) = [r,p] )
let p, q, r be Element of dom T; :: thesis: ( r in p & p in q implies ( ((T,p,q) incl) . (r,p) = [r,q] & ((T,p,q) incl) . (r,q) = [r,p] ) )
assume A1: ( r in p & p in q ) ; :: thesis: ( ((T,p,q) incl) . (r,p) = [r,q] & ((T,p,q) incl) . (r,q) = [r,p] )
set X = dom T;
set i = id (dom T);
set f = 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;
A2: dom (id (dom T)) = dom T ;
A3: ( r <> p & r <> q ) by A1;
thus ((T,p,q) incl) . (r,p) = [((Swap ((id (dom T)),p,q)) . r),((Swap ((id (dom T)),p,q)) . p)] by A1, Th64
.= [((id (dom T)) . r),((Swap ((id (dom T)),p,q)) . p)] by A3, Th33
.= [r,((Swap ((id (dom T)),p,q)) . p)]
.= [r,((id (dom T)) . q)] by A2, Th29
.= [r,q] ; :: thesis: ((T,p,q) incl) . (r,q) = [r,p]
thus ((T,p,q) incl) . (r,q) = [((Swap ((id (dom T)),p,q)) . r),((Swap ((id (dom T)),p,q)) . q)] by A1, Th64
.= [((id (dom T)) . r),((Swap ((id (dom T)),p,q)) . q)] by A3, Th33
.= [r,((Swap ((id (dom T)),p,q)) . q)]
.= [r,((id (dom T)) . p)] by A2, Th31
.= [r,p] ; :: thesis: verum