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 Th31:
  for S1, S2 being non empty non void ManySortedSign
  for f, g being Function st f, g form_morphism_between S1, S2
  for v being Gate of S1 holds g.v is Gate of S2
proof
  let S1, S2 be non empty non void ManySortedSign;
  let f, g be Function;
  assume that
  dom f = the carrier of S1 and
A1: dom g = the carrier' of S1 and rng f c= the carrier of S2 and
A2: rng g c= the carrier' of S2;
  now
    let v be Gate of S1;
    g.v in rng g by A1,FUNCT_1:def 3;
    hence g.v in the carrier' of S2 by A2;
  end;
  hence thesis;
end;
