reserve S for non empty non void ManySortedSign,
  V for non-empty ManySortedSet of the carrier of S,
  A for non-empty MSAlgebra over S,
  X for non empty Subset of S-Terms V,
  t for Element of X;
reserve S for non empty non void ManySortedSign,
  A for non-empty finite-yielding MSAlgebra over S,
  V for Variables of A,
  X for SetWithCompoundTerm of S,V;

theorem Th30:
  for S1, S2 being non empty ManySortedSign
  for f, g being Function st f, g form_morphism_between S1, S2
  for v being Vertex of S1 holds f.v is Vertex of S2
proof
  let S1, S2 be non empty ManySortedSign;
  let f, g be Function;
  assume that
A1: dom f = the carrier of S1 and dom g = the carrier' of S1 and
A2: rng f c= the carrier of S2;
  now
    let v be Vertex of S1;
    f.v in rng f by A1,FUNCT_1:def 3;
    hence f.v in the carrier of S2 by A2;
  end;
  hence thesis;
end;
