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 "" is "1-1" & F "" is "onto" )
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 "" is "1-1" & F "" is "onto" )
let F be ManySortedFunction of A,B; :: thesis: ( F is "1-1" & F is "onto" implies ( F "" is "1-1" & F "" is "onto" ) )
assume A1: ( F is "1-1" & F is "onto" ) ; :: thesis: ( F "" is "1-1" & F "" is "onto" )

now

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

end;

hence F "" is "1-1" by MSUALG_3:1; :: thesis: F "" is "onto"
thus F "" is "onto" :: thesis: verum

proof

let i be set ; :: according to MSUALG_3:def 3 :: thesis: ( not i in I or proj2 ((F "" ) . i) = A . i )
assume A3: i in I ; :: thesis: proj2 ((F "" ) . i) = A . i
then reconsider g = F . i as Function of (A . i),(B . i) by PBOOLE:def 18;
A4: g is one-to-one by A1, A3, MSUALG_3:1;
A . i = dom g by A3, FUNCT_2:def 1
.= rng (g " ) by A4, FUNCT_1:55 ;
hence proj2 ((F "" ) . i) = A . i by A1, A3, MSUALG_3:def 5; :: thesis: verum

end;