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<div>:: <span class="kw">deftheorem </span>   defines <a href="proofs_1.html#V19" title="PROOFS_1:attr.19">-closed</a> <a onclick="hs(this)" href="javascript:()">PROOFS_1:def 36 : <br/></a><span> for <font color="Olive" title="b1">P</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="proofs_1.html#L1" title="PROOFS_1:struct.1">ProofSystem</a> <br/>  for <font color="Olive" title="b2">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b2">X</font> is <font color="Olive" title="b1">P</font> <a href="proofs_1.html#V19" title="PROOFS_1:attr.19">-closed</a>  iff ( <font color="Olive" title="b2">X</font> is  the <a href="proofs_1.html#U2" title="PROOFS_1:sel.2">Rules</a> of <font color="Olive" title="b1">P</font> <a href="proofs_1.html#V18" title="PROOFS_1:attr.18">-closed</a>  &amp; <font color="Olive" title="b2">X</font> is  the <a href="proofs_1.html#U1" title="PROOFS_1:sel.1">Axioms</a> of <font color="Olive" title="b1">P</font> <a href="proofs_1.html#V6" title="PROOFS_1:attr.6">-extending</a>  ) );<br/></span></div>
