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 Th28:
  for S1, S2 being non empty ManySortedSign, f,g being Function
  st S1, S2 are_equivalent_wrt f, g holds
  f.:InputVertices S1 = InputVertices S2 &
  f.:InnerVertices S1 = InnerVertices S2
proof
  let S1, S2 be non empty ManySortedSign, f,g be Function such that
A1: S1, S2 are_equivalent_wrt f, g;
A2: f is one-to-one by A1;
A3: f, g form_morphism_between S1, S2 by A1;
A4: rng g = the carrier' of S2 by A1,Th24;
A5: dom the ResultSort of S2 = the carrier' of S2 by FUNCT_2:def 1;
A6: f.:rng the ResultSort of S1 = rng (f*the ResultSort of S1) by RELAT_1:127
    .= rng ((the ResultSort of S2)*g) by A3
    .= rng the ResultSort of S2 by A4,A5,RELAT_1:28;
  thus f.:InputVertices S1 = f.:(the carrier of S1) \ f.:
  rng the ResultSort of S1 by A2,FUNCT_1:64
    .= InputVertices S2 by A1,A6,Th23;
  thus thesis by A6;
end;
