let I be set ; :: thesis: for A, B being non-empty 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 non-empty 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 :: thesis: for i being set st i in I holds
(F "") . i is one-to-one

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 15;
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 4; :: 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 rng ((F "") . i) = A . i )
assume A3: i in I ; :: thesis: rng ((F "") . i) = A . i
then reconsider g = F . i as Function of (A . i),(B . i) by PBOOLE:def 15;
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_1:33 ;
hence rng ((F "") . i) = A . i by A1, A3, MSUALG_3:def 4; :: thesis: verum

end;