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;
