reserve S for non void non empty ManySortedSign,
  U0 for MSAlgebra over S;
reserve S for non void non empty ManySortedSign,
  X for ManySortedSet of the carrier of S,
  o for OperSymbol of S,
  b for Element of ([:the carrier' of S,{the
  carrier of S}:] \/ Union (coprod X))*;
reserve x for set;

theorem
  for S be non void non empty ManySortedSign, U1 be strict non-empty
  MSAlgebra over S ex U0 be strict free non-empty MSAlgebra over S, F be
  ManySortedFunction of U0,U1 st F is_epimorphism U0,U1 & Image F = U1
proof
  let S be non void non empty ManySortedSign, U1 be strict non-empty MSAlgebra
  over S;
  consider U0 be strict free non-empty MSAlgebra over S, F be
  ManySortedFunction of U0,U1 such that
A1: F is_epimorphism U0,U1 by Th18;
  Image F = U1 by A1,MSUALG_3:19;
  hence thesis by A1;
end;
