reserve UA for Universal_Algebra,
  f, g for Function of UA, UA;
reserve I for set,
  A, B, C for ManySortedSet of I;
reserve S for non void non empty ManySortedSign,
  U1, U2 for non-empty MSAlgebra over S;

theorem Th27:
  for F be ManySortedFunction of MSAlg UA, MSAlg UA
  for f be Element of UAAut UA st F = 0 .--> f holds F in MSAAut MSAlg UA
proof
  let F be ManySortedFunction of MSAlg UA, MSAlg UA;
  let f be Element of UAAut UA;
  assume F = 0 .--> f;
  then
A1: F = MSAlg f by MSUHOM_1:def 3;
  f is_isomorphism by Def1;
  then MSAlg f is_isomorphism MSAlg UA, MSAlg UA Over MSSign UA by MSUHOM_1:20;
  then F is_isomorphism MSAlg UA, MSAlg UA by A1,MSUHOM_1:9;
  hence thesis by Def5;
end;
