reserve U0,U1,U2,U3 for Universal_Algebra,
  n for Nat,
  x,y for set;
reserve A for non empty Subset of U0,
  o for operation of U0,
  x1,y1 for FinSequence of A;

theorem Th12:
  for U1,U2 be strict SubAlgebra of U0 st the carrier of U1 = the
  carrier of U2 holds U1 = U2
proof
  let U1,U2 be strict SubAlgebra of U0;
  assume the carrier of U1 = the carrier of U2;
  then U1 is strict SubAlgebra of U2 & U2 is strict SubAlgebra of U1 by Th11;
  hence thesis by Th10;
end;
