reserve I for set;
reserve S for non empty non void ManySortedSign,
  U0, U1 for non-empty MSAlgebra over S;

theorem
  for B being strict non-empty MSAlgebra over S for G being GeneratorSet
of U0, H being non-empty GeneratorSet of B for f being ManySortedFunction of U0
  , B st H c= f.:.:G & f is_homomorphism U0, B holds f is_epimorphism U0, B
proof
  let B be strict non-empty MSAlgebra over S, G be GeneratorSet of U0, H be
  non-empty GeneratorSet of B, f be ManySortedFunction of U0, B such that
A1: H c= f.:.:G and
A2: f is_homomorphism U0, B;
  reconsider I = the Sorts of Image f as non-empty MSSubset of B by
MSUALG_2:def 9;
  f.:.:G c= f.:.:the Sorts of U0 by EXTENS_1:2;
  then H c= f.:.:the Sorts of U0 by A1,PBOOLE:13;
  then H c= the Sorts of Image f by A2,MSUALG_3:def 12;
  then H is ManySortedSubset of the Sorts of Image f by PBOOLE:def 18;
  then
A3: GenMSAlg H is MSSubAlgebra of GenMSAlg I by EXTENS_1:17;
  reconsider I1 = the Sorts of Image f as non-empty MSSubset of Image f by
PBOOLE:def 18;
  I is GeneratorSet of Image f & GenMSAlg I = GenMSAlg I1 by EXTENS_1:18
,MSAFREE2:6;
  then GenMSAlg I = Image f by MSAFREE:3;
  then B is MSSubAlgebra of Image f by A3,MSAFREE:3;
  then Image f = B by MSUALG_2:7;
  hence thesis by A2,MSUALG_3:19;
end;
