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 Th42:
  f, g form_embedding_of C1, C2 implies
  for o1 being Gate of G1, o2 being Gate of G2 st o2 = g.o1
  holds Den(o2, C2) = Den(o1, C1)
proof
  assume that f is one-to-one and g is one-to-one and
A1: f, g form_morphism_between G1, G2 and
  the Sorts of C1 = (the Sorts of C2)*f and
A2: the Charact of C1 = (the Charact of C2)*g;
  let o1 be Gate of G1, o2 be Gate of G2 such that
A3: o2 = g.o1;
  dom g = the carrier' of G1 by A1;
  hence thesis by A2,A3,FUNCT_1:13;
end;
