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 Th18:
  h is_epimorphism implies MSAlg h is_epimorphism MSAlg U1,
  (MSAlg U2 Over MSSign U1)
proof
  set f = MSAlg h;
  set A = MSAlg U2 Over MSSign U1;
A1: 0 in {0} by TARSKI:def 1;
  MSSorts U2 = 0 .--> the carrier of U2 by MSUALG_1:def 9;
  then
A2: the carrier of MSSign U1 = {0} & (MSSorts U2).0 = the carrier of U2 by A1,
FUNCOP_1:7,MSUALG_1:def 8;
A3: MSAlg U2 = MSAlgebra(#MSSorts U2,MSCharact U2#) by MSUALG_1:def 11;
  assume
A4: h is_epimorphism;
  then
A5: rng h = the carrier of U2;
A6: h is_homomorphism by A4;
  then
A7: U1,U2 are_similar;
  then MSSign U1 = MSSign U2 by Th10;
  then
A8: the Sorts of A = MSSorts U2 by A3,Th9;
  thus f is_homomorphism MSAlg U1,A by A6,Th16;
  let i be set;
  assume i in the carrier of MSSign U1;
  then reconsider i9=i as Element of MSSign U1;
  reconsider h9 = f.i as Function of (the Sorts of MSAlg U1).i9, (the Sorts of
  A).i9 by PBOOLE:def 15;
  f.0 = (0.--> h).0 by A7,Def3,Th10
    .= h by A1,FUNCOP_1:7;
  then the carrier of MSSign U1 = {0} & rng h9 = (the Sorts of A).0 by A5,A8,A2
,TARSKI:def 1;
  hence thesis by TARSKI:def 1;
end;
