reserve U1,U2,U3 for Universal_Algebra,
  m,n for Nat,
  a for set,
  A for non empty set,
  h for Function of U1,U2;

theorem Th23:
  for U1, U2, h st U1, U2 are_similar holds MSAlg h
  is_monomorphism MSAlg U1, (MSAlg U2 Over MSSign U1) implies h is_monomorphism
proof
  let U1,U2,h;
  assume
A1: U1,U2 are_similar;
  assume
A2: MSAlg h is_monomorphism MSAlg U1,(MSAlg U2 Over MSSign U1);
  then
A3: MSAlg h is "1-1";
  MSAlg h is_homomorphism MSAlg U1,(MSAlg U2 Over MSSign U1) by A2;
  then
A4: h is_homomorphism by A1,Th21;
A5: the carrier of MSSign U1 = {0} by MSUALG_1:def 8;
A6: 0 in {0} by TARSKI:def 1;
  (MSAlg h).0 = (0.--> h).0 by A1,Def3,Th10
    .= h by A6,FUNCOP_1:7;
  then h is one-to-one by A3,A5,A6,MSUALG_3:1;
  hence thesis by A4;
end;
