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 Th31:
  for UA1, UA2 be Universal_Algebra st UA1, UA2 are_similar
  for F be ManySortedFunction of MSAlg UA1, (MSAlg UA2 Over MSSign UA1) holds
  F.0 is Function of UA1, UA2
proof
  let UA1, UA2 be Universal_Algebra;
A1: 0 in {0} by TARSKI:def 1;
  assume UA1, UA2 are_similar;
  then MSSign UA1 = MSSign UA2 by MSUHOM_1:10;
  then
A2: MSAlg UA2 = MSAlgebra (#MSSorts UA2, MSCharact UA2#) & MSAlg UA2 Over
  MSSign UA1 = MSAlg UA2 by MSUALG_1:def 11,MSUHOM_1:9;
  let F be ManySortedFunction of MSAlg UA1, (MSAlg UA2 Over MSSign UA1);
A3: the carrier of MSSign UA1 = {0} & MSAlg UA1 = MSAlgebra (#MSSorts UA1,
    MSCharact UA1#) by MSUALG_1:def 8,def 11;
A4: (MSSorts UA2).0 = (0 .--> the carrier of UA2).0 by MSUALG_1:def 9
    .= the carrier of UA2 by A1,FUNCOP_1:7;
  (MSSorts UA1).0 = (0 .--> the carrier of UA1).0 by MSUALG_1:def 9
    .= the carrier of UA1 by A1,FUNCOP_1:7;
  hence thesis by A1,A3,A4,A2,PBOOLE:def 15;
end;
