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 Th19:
  h is_monomorphism implies MSAlg h is_monomorphism MSAlg
  U1,(MSAlg U2 Over MSSign U1)
proof
  set f = MSAlg h;
  the carrier of MSSign U1 = {0} by MSUALG_1:def 8;
  then
A1: dom f = {0} by PARTFUN1:def 2;
  assume
A2: h is_monomorphism;
  then
A3: h is_homomorphism;
  hence MSAlg h is_homomorphism MSAlg U1,(MSAlg U2 Over MSSign U1) by Th16;
  U1,U2 are_similar by A3;
  then f = 0 .--> h by Def3,Th10;
  then
A4: f.0 = h by FUNCOP_1:72;
  let i be set, h9 be Function;
  assume i in dom f & f.i = h9;
  then h = h9 by A4,A1,TARSKI:def 1;
  hence thesis by A2;
end;
