If $a,b,c,d$ are Distinct Real no. such that 

$a = \sqrt{4+\sqrt{5+a}}$

$b = \sqrt{4-\sqrt{5+b}}$

$c = \sqrt{4+\sqrt{5-c}}$

$d = \sqrt{4-\sqrt{5-d}}$

Then $abcd = $
1. **Given Equations:**
   \[
   a = \sqrt{4 + \sqrt{5 + a}}
   \]
   \[
   b = \sqrt{4 - \sqrt{5 + b}}
   \]
   \[
   c = \sqrt{4 + \sqrt{5 - c}}
   \]
   \[
   d = \sqrt{4 - \sqrt{5 - d}}
   \]

2. **Forming the Polynomial:**
   Let's consider the polynomial \( f(x) = (x^2 - 4)^2 - 5 \). We need to verify that \( a, b, c, d \) are roots of this polynomial.

3. **Verification for \( a \):**
   \[
   a = \sqrt{4 + \sqrt{5 + a}}
   \]
   Squaring both sides:
   \[
   a^2 = 4 + \sqrt{5 + a}
   \]
   Isolating the square root term:
   \[
   a^2 - 4 = \sqrt{5 + a}
   \]
   Squaring both sides again:
   \[
   (a^2 - 4)^2 = 5 + a
   \]
   Rearranging:
   \[
   (a^2 - 4)^2 - 5 = a
   \]
   Thus, \( a \) satisfies the polynomial \( f(x) = (x^2 - 4)^2 - 5 - x = 0 \).

4. **Verification for \( b \):**
   \[
   b = \sqrt{4 - \sqrt{5 + b}}
   \]
   Squaring both sides:
   \[
   b^2 = 4 - \sqrt{5 + b}
   \]
   Isolating the square root term:
   \[
   b^2 - 4 = -\sqrt{5 + b}
   \]
   Squaring both sides again:
   \[
   (b^2 - 4)^2 = 5 + b
   \]
   Rearranging:
   \[
   (b^2 - 4)^2 - 5 = b
   \]
   Thus, \( b \) satisfies the polynomial \( f(x) = (x^2 - 4)^2 - 5 - x = 0 \).

5. **Verification for \( c \):**
   \[
   c = \sqrt{4 + \sqrt{5 - c}}
   \]
   Squaring both sides:
   \[
   c^2 = 4 + \sqrt{5 - c}
   \]
   Isolating the square root term:
   \[
   c^2 - 4 = \sqrt{5 - c}
   \]
   Squaring both sides again:
   \[
   (c^2 - 4)^2 = 5 - c
   \]
   Rearranging:
   \[
   (c^2 - 4)^2 - 5 = -c
   \]
   Thus, \( c \) satisfies the polynomial \( f(x) = (x^2 - 4)^2 - 5 + x = 0 \).

6. **Verification for \( d \):**
   \[
   d = \sqrt{4 - \sqrt{5 - d}}
   \]
   Squaring both sides:
   \[
   d^2 = 4 - \sqrt{5 - d}
   \]
   Isolating the square root term:
   \[
   d^2 - 4 = -\sqrt{5 - d}
   \]
   Squaring both sides again:
   \[
   (d^2 - 4)^2 = 5 - d
   \]
   Rearranging:
   \[
   (d^2 - 4)^2 - 5 = -d
   \]
   Thus, \( d \) satisfies the polynomial \( f(x) = (x^2 - 4)^2 - 5 + x = 0 \).

7. **Roots and Product of Roots:**
   The polynomial \( f(x) = (x^2 - 4)^2 - 5 \pm x = 0 \) has roots \( a, b, c, d \). By Vieta's formulas, the product of the roots of a polynomial \( ax^n + bx^{n-1} + \cdots + k = 0 \) is given by \( (-1)^n \frac{k}{a} \).

   For the polynomial \( f(x) = (x^2 - 4)^2 - 5 \pm x = 0 \), the product of the roots is:
   \[
   abcd = 11
   \]

The final answer is \(\boxed{11}\)