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 Th44:
  f, g form_embedding_of C1, C2 implies
  for s being State of C2 holds s*f is State of C1
proof
  set S1 = G1, S2 = G2;
  assume that f is one-to-one and g is one-to-one and
A1: f, g form_morphism_between S1, S2 and
A2: the Sorts of C1 = (the Sorts of C2)*f and
  the Charact of C1 = (the Charact of C2)*g;
  let s be State of C2;
A3: dom the Sorts of C2 = the carrier of S2 by PARTFUN1:def 2;
A4: dom the Sorts of C1 = the carrier of S1 by PARTFUN1:def 2;
A5: dom s = dom the Sorts of C2 by CARD_3:9;
A6: rng f c= the carrier of S2 by A1;
A7: dom f = the carrier of S1 by A1;
  then
A8: dom (s*f) = the carrier of S1 by A3,A5,A6,RELAT_1:27;
  now
    let x be object;
    assume
A9: x in the carrier of S1;
    then
A10: f.x in rng f by A7,FUNCT_1:def 3;
    (s*f).x = s.(f.x) by A7,A9,FUNCT_1:13;
    then (s*f).x in (the Sorts of C2).(f.x) by A3,A6,A10,CARD_3:9;
    hence (s*f).x in (the Sorts of C1).x by A2,A7,A9,FUNCT_1:13;
  end;
  hence thesis by A4,A8,CARD_3:9;
end;
