let I be set ; :: thesis: for A, B being V2() ManySortedSet of I
for F being ManySortedFunction of A,B st F is "1-1" & F is "onto" holds
(F "" ) "" = F
let A, B be V2() ManySortedSet of I; :: thesis: for F being ManySortedFunction of A,B st F is "1-1" & F is "onto" holds
(F "" ) "" = F
let F be ManySortedFunction of A,B; :: thesis: ( F is "1-1" & F is "onto" implies (F "" ) "" = F )
assume A1: ( F is "1-1" & F is "onto" ) ; :: thesis: (F "" ) "" = F

now

let i be set ; :: thesis: ( i in I implies ((F "" ) "" ) . i = F . i )
assume A2: i in I ; :: thesis: ((F "" ) "" ) . i = F . i
then reconsider f = F . i as Function of (A . i),(B . i) by PBOOLE:def 18;
reconsider f9 = (F "" ) . i as Function of (B . i),(A . i) by A2, PBOOLE:def 18;
f is one-to-one by A1, A2, MSUALG_3:1;
then A3: (f " ) " = f by FUNCT_1:65;
( F "" is "1-1" & F "" is "onto" ) by A1, Th14;
then ((F "" ) "" ) . i = f9 " by A2, MSUALG_3:def 5;
hence ((F "" ) "" ) . i = F . i by A1, A2, A3, MSUALG_3:def 5; :: thesis: verum

end;

hence (F "" ) "" = F by PBOOLE:3; :: thesis: verum