
theorem
  for L be non empty multMagma for B be non empty AlgebraStr over L for
A be non empty Subalgebra of B ex C being Subset of B st the carrier of A = C &
  C is opers_closed
proof
  let L be non empty multMagma;
  let B be non empty AlgebraStr over L;
  let A be non empty Subalgebra of B;
  take C = the carrier of A;
A1: 1.B = 1.A & 0.B = 0.A by Def3;
  reconsider C as Subset of B by Def3;
A2: for a being Element of L, v being Element of B st v in C holds a * v in C
  proof
    let a be Element of L, v be Element of B;
    assume v in C;
    then reconsider x = v as Element of A;
    a * v = a * x by Th17;
    hence thesis;
  end;
A3: for x,y being Element of B st x in C & y in C holds x*y in C
  proof
    let x,y be Element of B such that
A4: x in C & y in C;
    reconsider x9 = x, y9 = y as Element of B;
    reconsider x1 = x9, y1 = y9 as Element of A by A4;
    x*y = x1 * y1 by Th16;
    hence thesis;
  end;
  for v,u being Element of B st v in C & u in C holds v + u in C
  proof
    let v,u be Element of B;
    assume v in C & u in C;
    then reconsider x = u, y = v as Element of A;
    v + u = y + x by Th15;
    hence thesis;
  end;
  then C is linearly-closed by A2,VECTSP_4:def 1;
  hence thesis by A1,A3,Def4;
end;
