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 Th46:
  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
  holds Following s1 = (Following s2)*f
proof
  assume that
A1: f, g form_embedding_of C1, C2 and
A2: f.:InputVertices G1 c= InputVertices G2;
  let s2 be State of C2, s1 be State of C1 such that
A3: s1 = s2*f;
A4: dom f = the carrier of G1 by A1,Th41;
  now
    let v be Vertex of G1;
    assume v in InputVertices G1;
    then f.v in f.:InputVertices G1 by A4,FUNCT_1:def 6;
    hence s2 is_stable_at f.v by A2,FACIRC_1:18;
  end;
  hence thesis by A1,A3,Th45;
end;
