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_1:62;
then
1 in dom <*p1,p2,p3*>
by FINSEQ_3:27;
then A2: <*p1,p2,p3*> /. 1 =
<*p1,p2,p3*> . 1
by PARTFUN1:def 8
.=
p1
by FINSEQ_1:62
;
2 in dom <*p1,p2,p3*>
by A1, FINSEQ_3:27;
then A3: <*p1,p2,p3*> /. 2 =
<*p1,p2,p3*> . 2
by PARTFUN1:def 8
.=
p2
by FINSEQ_1:62
;
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, Th17
;
hence
Swap <*p1,p2,p3*>,1,2 = <*p2,p1,p3*>
by A2, Th18; verum