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;

theorem Th23:
  for S1, S2 being non empty ManySortedSign, f,g being Function
  st S1, S2 are_equivalent_wrt f, g
  holds the carrier of S2 = f.:the carrier of S1 &
  the carrier' of S2 = g.:the carrier' of S1
proof
  let S1, S2 be non empty ManySortedSign, f,g be Function such that
A1: f is one-to-one and
A2: g is one-to-one and
A3: f, g form_morphism_between S1, S2 and
A4: f", g" form_morphism_between S2, S1;
  thus the carrier of S2 = dom (f") by A4
    .= rng f by A1,FUNCT_1:33
    .= f.:dom f by RELAT_1:113
    .= f.:the carrier of S1 by A3;
  thus the carrier' of S2 = dom (g") by A4
    .= rng g by A2,FUNCT_1:33
    .= g.:dom g by RELAT_1:113
    .= g.:the carrier' of S1 by A3;
end;
