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 Th19:
  for U0 be Universal_Algebra, U1 be strict SubAlgebra of U0, B be
  non empty Subset of U0 st B = the carrier of U1 holds GenUnivAlg(B) = U1
proof
  let U0 be Universal_Algebra,U1 be strict SubAlgebra of U0, B be non empty
  Subset of U0;
  set W = GenUnivAlg(B);
  assume
A1: B = the carrier of U1;
  then GenUnivAlg(B) is SubAlgebra of U1 by Def12;
  then
A2: the carrier of W is non empty Subset of U1 by Def7;
  the carrier of U1 c= the carrier of W by A1,Def12;
  then the carrier of U1 = the carrier of W by A2;
  hence thesis by Th12;
end;
