Put TTTTFFFF under each H, TTFFTTFF under each E, 
and TFTFTFTF under each D.
 

 [∼ H ∨ (E • D)] ≡ [(H • ∼ E) ∨ (H • ∼ D)]
    T    T   T       T      T     T     T
    T    T   F       T      T     T     F
    T    F   T       T      F     T     T
    T    F   F       T      F     T     F
    F    T   T       F      T     F     T
    F    T   F       F      T     F     F
    F    F   T       F      F     F     T
    F    F   F       F      F     F     F

Under each ~, put the opposite of what follows it under it
and erase what was next to it:


 [∼ H ∨ (E • D)] ≡ [(H • ∼ E) ∨ (H • ∼ D)]
  F      T   T       T   F        T   F 
  F      T   F       T   F        T   T 
  F      F   T       T   T        T   F 
  F      F   F       T   T        T   T 
  T      T   T       F   F        F   F 
  T      T   F       F   F        F   T 
  T      F   T       F   T        F   F 
  T      F   F       F   T        F   T 

Under each •, put T only if the • is between two T's,
and put F otherwise, and erase the two that the • is 
between: 

 [∼ H ∨ (E • D)] ≡ [(H • ∼ E) ∨ (H • ∼ D)]
  F        T           F            F  
  F        F           F            T  
  F        F           T            F  
  F        F           T            T  
  T        T           F            F  
  T        F           F            F  
  T        F           F            F  
  T        F           F            F  

Under each ∨, put F only if the ∨ is between two F's,
and put T otherwise, and erase the two that the ∨ is 
between: 

 [∼ H ∨ (E • D)] ≡ [(H • ∼ E) ∨ (H • ∼ D)]
      T                        F      
      F                        T      
      F                        T      
      F                        T      
      T                        F      
      T                        F      
      T                        F      
      T                        F 

Under the ≡, put T only if the ≡ is between two that
are the same, and F if the ≡ is between two that are 
not the same. and erase the two that the ≡ is 
between:     

 [∼ H ∨ (E • D)] ≡ [(H • ∼ E) ∨ (H • ∼ D)]
                  F                   
                  F                   
                  F                   
                  F                   
                  F                   
                  F                   
                  F                   
                  F                

Since the final column has all F's the proposition is a
contradiction, which means that it is never true!

Edwin