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<title>Explanation of homomorphism</title>
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 <p><strong>Theorem</strong>:  <em>The composition of group homomorphims is a homomorphism.</em></p>
 <p>This example illustrates that beta reduction works well with paramodulation.  </p>
 <p>The composition of two functions f and g is expressed as the lambda term lambda(x,f(g(x))).
  The verification that the composition of homomorphisms is a homomorphism is a 
  straightforward piece of equational reasoning.    However, we also need to prove
  that if  Phi maps G to H and Psi maps H to K then the composition maps G to K.  
  
In this file,  the groups G, H, and K  are given by "propositional functions", that is,
  G, H, and K are treated as constants, and "x belongs to G"  is represented by Ap(G,x) = $T.
  ($T means "true").</p>
 <p>This theorem illustrates how lambda logic can sometimes make it very easy to formulate
    a precise version of a theorem that is rather difficult to formulate in set theory or 
    in a first-order algebraic theory.   Wick and McCune [1] showed how to formulate and prove
    this theorem in set theory.  It was quite difficult.   </p>
 <p>[1] Wick, C., and McCune, W.,  Automated reasoning about elementary point-set topology, 
     <em>J. Automated Reasoning</em> <strong>5(2)</strong> 239-255, 1989. </p>
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