reserve a, I for set,
  S for non empty non void ManySortedSign;

theorem Th10:
  for A being ManySortedSet of I, B being non-empty ManySortedSet
  of I for F being ManySortedFunction of {A}, B holds F is "1-1"
proof
  let A be ManySortedSet of I, B be non-empty ManySortedSet of I, F be
  ManySortedFunction of {A}, B;
  now
    let i be set;
    assume i in I;
    then {A}.i = {A.i} & F.i is Function of {A}.i, B.i by PBOOLE:def 15
,PZFMISC1:def 1;
    hence F.i is one-to-one by PARTFUN1:17;
  end;
  hence thesis by MSUALG_3:1;
end;
