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 Th49:
  f, g form_embedding_of C1, C2 & f preserves_inputs_of G1, G2 implies
  for s2 being State of C2, s1 being State of C1 st s1 = s2*f
  for v1 being Vertex of G1 holds s1 is_stable_at v1 iff s2 is_stable_at f.v1
proof
  assume that
A1: f, g form_embedding_of C1, C2 and
A2: f preserves_inputs_of G1, G2;
  let s2 be State of C2, s1 be State of C1 such that
A3: s1 = s2*f;
  let v1 be Vertex of G1;
A4: f,g form_morphism_between G1,G2 by A1;
  then
A5: dom f = the carrier of G1;
  reconsider v2 = f.v1 as Vertex of G2 by A4,Th30;
  thus s1 is_stable_at v1 implies s2 is_stable_at f.v1
  proof
    assume
A6: for n being Nat holds (Following(s1,n)).v1 = s1.v1;
    let n be Nat;
    Following(s1,n) = Following(s2,n)*f by A1,A2,A3,Th47;
    hence (Following(s2,n)).(f.v1) = (Following(s1,n)).v1 by A5,FUNCT_1:13
      .= s1.v1 by A6
      .= s2.(f.v1) by A3,A5,FUNCT_1:13;
  end;
  assume
A7: for n being Nat holds (Following(s2,n)).(f.v1) = s2.(f.v1);
  let n be Nat;
  Following(s1,n) = Following(s2,n)*f by A1,A2,A3,Th47;
  hence (Following(s1,n)).v1 = (Following(s2,n)).v2 by A5,FUNCT_1:13
    .= s2.v2 by A7
    .= s1.v1 by A3,A5,FUNCT_1:13;
end;
