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;
reserve G1, G2 for Circuit-like non void non empty ManySortedSign,
  f, g for Function,
  C1 for non-empty Circuit of G1,
  C2 for non-empty Circuit of G2;

theorem
  C1, C2 are_similar_wrt f, g implies
  for s1 being State of C1, s2 being State of C2 st s1 = s2*f
  holds s1 is stable iff s2 is stable
proof
  assume
A1: C1, C2 are_similar_wrt f, g;
  then
A2: C2, C1 are_similar_wrt f", g" by Th39;
  let s1 be State of C1, s2 be State of C2 such that
A3: s1 = s2*f;
A4: s2 = s1*f" by A1,A3,Th51;
  thus s1 is stable implies s2 is stable
  by A2,A4,Th54;
  assume s2 = Following s2;
  hence s1 = Following s1 by A1,A3,Th54;
end;
