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 Th47:
  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 n being Nat holds Following(s1,n) = Following(s2,n)*f
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;
  defpred P[Nat] means Following(s1,$1) = Following(s2,$1)*f;
  Following(s1,0) = s1 by FACIRC_1:11;
  then
A4: P[ 0 ] by A3,FACIRC_1:11;
A5: now
    let n be Nat;
    assume P[n];
    then Following Following(s1,n) = (Following Following(s2,n))*f
    by A1,A2,Th46;
    then Following(s1,n+1) = (Following Following(s2,n))*f by FACIRC_1:12
      .= Following(s2,n+1)*f by FACIRC_1:12;
    hence P[n+1];
  end;
  thus for n be Nat holds P[n] from NAT_1:sch 2(A4,A5);
end;
