reserve S for non void non empty ManySortedSign,
  U1, U2, U3 for non-empty MSAlgebra over S,
  I for set,
  A for ManySortedSet of I,
  B, C for non-empty ManySortedSet of I;

theorem
  for F be ManySortedFunction of A, B for X be ManySortedSubset of A
  holds F is "1-1" implies F || X is "1-1"
proof
  let F be ManySortedFunction of A, B, X be ManySortedSubset of A;
  assume
A1: F is "1-1";
  now
    let i be set;
    assume
A2: i in I;
    then reconsider f = F.i as Function of A.i, B.i by PBOOLE:def 15;
    f is one-to-one by A1,A2,MSUALG_3:1;
    then f | (X.i) is one-to-one by FUNCT_1:52;
    hence (F||X).i is one-to-one by A2,MSAFREE:def 1;
  end;
  hence thesis by MSUALG_3:1;
end;
