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 Th33:
  for S1, S2 being Circuit-like non void non empty ManySortedSign
  for f, g being Function st f, g form_morphism_between S1, S2
  for v1 being Vertex of S1 st v1 in InnerVertices S1
  for v2 being Vertex of S2 st v2 = f.v1 holds action_at v2 = g.action_at v1
proof
  let S1, S2 be Circuit-like non void non empty ManySortedSign;
  let f,g be Function such that
A1: f, g form_morphism_between S1, S2;
  let v1 be Vertex of S1 such that
A2: v1 in InnerVertices S1;
  let v2 be Vertex of S2 such that
A3: v2 = f.v1;
  reconsider g1 = g.action_at v1 as Gate of S2 by A1,Th31;
A4: dom g = the carrier' of S1 by A1;
A5: dom f = the carrier of S1 by A1;
A6: dom the ResultSort of S1 = the carrier' of S1 by FUNCT_2:def 1;
A7: f.:InnerVertices S1 c= InnerVertices S2 by A1,Th32;
A8: v2 in f.:InnerVertices S1 by A2,A3,A5,FUNCT_1:def 6;
  the_result_sort_of g1 =
  ((the ResultSort of S2)*g).action_at v1 by A4,FUNCT_1:13
    .= (f*the ResultSort of S1).action_at v1 by A1
    .= f.(the_result_sort_of action_at v1) by A6,FUNCT_1:13
    .= v2 by A2,A3,MSAFREE2:def 7;
  hence thesis by A7,A8,MSAFREE2:def 7;
end;
