YES

Problem:
 f(h(x)) -> f(i(x))
 g(i(x)) -> g(h(x))
 h(a()) -> b()
 i(a()) -> b()

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
         [1]
   [b] = [0]
         [0],
   
         [1]
   [a] = [0]
         [1],
   
             [1 0 0]  
   [g](x0) = [0 0 0]x0
             [0 0 0]  ,
   
             [1 0 0]  
   [i](x0) = [0 0 0]x0
             [0 0 0]  ,
   
             [1 1 0]     [0]
   [f](x0) = [0 0 0]x0 + [0]
             [0 0 1]     [1],
   
             [1 0 0]     [0]
   [h](x0) = [0 0 0]x0 + [1]
             [0 1 1]     [0]
  orientation:
             [1 0 0]    [1]    [1 0 0]    [0]          
   f(h(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = f(i(x))
             [0 1 1]    [1]    [0 0 0]    [1]          
   
             [1 0 0]     [1 0 0]           
   g(i(x)) = [0 0 0]x >= [0 0 0]x = g(h(x))
             [0 0 0]     [0 0 0]           
   
            [1]    [1]      
   h(a()) = [1] >= [0] = b()
            [1]    [0]      
   
            [1]    [1]      
   i(a()) = [0] >= [0] = b()
            [0]    [0]      
  problem:
   g(i(x)) -> g(h(x))
   h(a()) -> b()
   i(a()) -> b()
  Matrix Interpretation Processor: dim=3
   
   interpretation:
          [0]
    [b] = [1]
          [0],
    
          [0]
    [a] = [1]
          [0],
    
              [1 0 0]  
    [g](x0) = [0 0 1]x0
              [0 0 0]  ,
    
              [1 0 0]     [1]
    [i](x0) = [0 1 0]x0 + [0]
              [1 0 0]     [0],
    
              [1 0 0]     [0]
    [h](x0) = [0 0 0]x0 + [1]
              [0 0 0]     [0]
   orientation:
              [1 0 0]    [1]    [1 0 0]           
    g(i(x)) = [1 0 0]x + [0] >= [0 0 0]x = g(h(x))
              [0 0 0]    [0]    [0 0 0]           
    
             [0]    [0]      
    h(a()) = [1] >= [1] = b()
             [0]    [0]      
    
             [1]    [0]      
    i(a()) = [1] >= [1] = b()
             [0]    [0]      
   problem:
    h(a()) -> b()
   Matrix Interpretation Processor: dim=3
    
    interpretation:
           [0]
     [b] = [0]
           [0],
     
           [0]
     [a] = [0]
           [0],
     
               [1 0 0]     [1]
     [h](x0) = [0 0 0]x0 + [0]
               [0 0 0]     [0]
    orientation:
              [1]    [0]      
     h(a()) = [0] >= [0] = b()
              [0]    [0]      
    problem:
     
    Qed