let D be non empty set ; for p1, p2, p3 being Element of D holds Swap (<*p1,p2,p3*>,1,2) = <*p2,p1,p3*>
let p1, p2, p3 be Element of D; Swap (<*p1,p2,p3*>,1,2) = <*p2,p1,p3*>
set f = <*p1,p2,p3*>;
A1:
len <*p1,p2,p3*> = 3
by FINSEQ_1:45;
then
1 in dom <*p1,p2,p3*>
by FINSEQ_3:25;
then A2: <*p1,p2,p3*> /. 1 =
<*p1,p2,p3*> . 1
by PARTFUN1:def 6
.=
p1
;
2 in dom <*p1,p2,p3*>
by A1, FINSEQ_3:25;
then A3: <*p1,p2,p3*> /. 2 =
<*p1,p2,p3*> . 2
by PARTFUN1:def 6
.=
p2
;
Swap (<*p1,p2,p3*>,1,2) =
Replace ((Replace (<*p1,p2,p3*>,1,(<*p1,p2,p3*> /. 2))),2,(<*p1,p2,p3*> /. 1))
by A1, Def2
.=
Replace (<*p2,p2,p3*>,2,(<*p1,p2,p3*> /. 1))
by A3, Th15
;
hence
Swap (<*p1,p2,p3*>,1,2) = <*p2,p1,p3*>
by A2, Th16; verum