
theorem Th14:
  for W being Normed_Complex_AlgebraStr,
      V being ComplexAlgebra st
      ComplexAlgebraStr(# the carrier of W,the multF of W,
                  the addF of W,the Mult of W,
                  the OneF of W,the ZeroF of W #) = V holds
     W is ComplexAlgebra
proof
  let W be Normed_Complex_AlgebraStr,
      V be ComplexAlgebra;
  assume
A1:  ComplexAlgebraStr(# the carrier of W,the multF of W,
                        the addF of W,the Mult of W,
                        the OneF of W,the ZeroF of W #) = V;
  reconsider W as non empty ComplexAlgebraStr by A1;
A2: W is right_unital
  proof
    let x be Element of W;
    reconsider x1 = x as Element of V by A1;
    x * (1.W) = x1 * (1.V) by A1;
    hence x * (1.W) = x;
  end;
A3: W is right-distributive
  proof
    let x, y, z be Element of W;
    reconsider x1 = x, y1 = y, z1 = z as Element of V by A1;
    x * (y + z) = x1 * (y1 + z1) by A1;
    then x * (y + z) = (x1 * y1) + (x1 * z1) by VECTSP_1:def 2;
    hence x * (y + z) = (x * y) + (x * z) by A1;
  end;
A4: for x being Element of W holds x is right_complementable
  proof
    let x be Element of W;
    reconsider x1 = x as Element of V by A1;
    consider v being Element of V such that
A5: x1 + v = 0.V by ALGSTR_0:def 11;
    reconsider y = v as Element of W by A1;
    x + y = 0.W by A1,A5;
    hence x is right_complementable;
  end;
A6: for a, b, x being Element of W holds (a * b) * x = a * (b * x)
  proof
    let a, b, x be Element of W;
    reconsider y = x, a1 = a, b1 = b as Element of V by A1;
    (a * b) * x = (a1 * b1) * y by A1;
    then (a * b) * x = a1 * (b1 * y) by GROUP_1:def 3;
    hence (a * b) * x = a * (b * x) by A1;
  end;
A7: for v, w being Element of W holds v * w = w * v
  proof
    let v, w be Element of W;
    reconsider v1 = v, w1 = w as Element of V by A1;
    v * w = v1 * w1 by A1;
    then v * w = w1 * v1;
    hence v * w = w * v by A1;
  end;
A8: for x, y, z being VECTOR of W holds (x + y) + z = x + (y + z)
  proof
    let x, y, z be VECTOR of W;
    reconsider x1 = x, y1 = y, z1 = z as VECTOR of V by A1;
    (x + y) + z = (x1 + y1) + z1 by A1;
    then (x + y) + z = x1 + (y1 + z1) by RLVECT_1:def 3;
    hence (x + y) + z = x + (y + z) by A1;
  end;
A9: for x, y being VECTOR of W holds x + y = y + x
  proof
    let x, y be VECTOR of W;
    reconsider x1 = x, y1 = y as VECTOR of V by A1;
    x + y = x1 + y1 by A1;
    then x + y = y1 + x1;
    hence x + y = y + x by A1;
  end;
A10: W is vector-distributive
  proof
    let a be Complex;
    let x, y be VECTOR of W;
    reconsider x1 = x, y1 = y as Element of V by A1;
    a * (x + y) = a * (x1 + y1) by A1;
    then a * (x + y) = (a * x1) + (a * y1) by CLVECT_1:def 2;
    hence a * (x + y) = (a * x) + (a * y) by A1;
  end;
A11: W is scalar-distributive
  proof
    let a, b be Complex;
    let x be VECTOR of W;
    reconsider x1 = x as Element of V by A1;
    (a + b) * x = (a + b) * x1 by A1;
    then (a + b) * x = (a * x1) + (b * x1) by CLVECT_1:def 3;
    hence (a + b) * x = (a * x) + (b * x) by A1;
  end;
A12: W is scalar-associative
  proof
    let a, b be Complex;
    let x be VECTOR of W;
    reconsider x1 = x as Element of V by A1;
    (a * b) * x = (a * b) * x1 by A1;
    then (a * b) * x = a * (b * x1) by CLVECT_1:def 4;
    hence (a * b) * x = a * (b * x) by A1;
  end;
A13: W is vector-associative
  proof
    let x, y be VECTOR of W;
    reconsider x1 = x, y1 = y as Element of V by A1;
    let a be Complex;
    a * (x * y) = a * (x1 * y1) by A1;
    then a * (x * y) = (a * x1) * y1 by CFUNCDOM:def 9;
    hence a * (x * y) = (a * x) * y by A1;
  end;
  for x being VECTOR of W holds x + (0.W) = x
  proof
    let x be VECTOR of W;
    reconsider y = x, z = 0.W as VECTOR of V by A1;
    x + (0.W) = y + (0.V) by A1;
    hence x + (0.W) = x by RLVECT_1:4;
  end;
  hence thesis by A9,A8,A4,A7,A6,A2,A3,A10,A11,A12,A13,ALGSTR_0:def 16
,GROUP_1:def 3,def 12,RLVECT_1:def 2,def 3,def 4;
end;
