reserve S for non void non empty ManySortedSign,
  U1,U2 for MSAlgebra over S,
  o for OperSymbol of S,
  n for Nat;

theorem Th1:
  for I be set,F be ManySortedFunction of I holds F is "1-1" iff
  for i be set st i in I holds F.i is one-to-one
proof
  let I be set;
  let F be ManySortedFunction of I;
A1: dom F = I by PARTFUN1:def 2;
  hence F is "1-1" implies for i be set st i in I holds F.i is one-to-one;
  assume for i be set st i in I holds F.i is one-to-one;
  then for i be set, f being Function st i in dom F & f = F.i holds f is
  one-to-one by A1;
  hence thesis;
end;
