<?xml version="1.0"?>
<div>:: <span class="kw">deftheorem </span><span class="lab"><font color="Green" title="E31">defsgn</font></span>   defines <a href="realalg3.html#K7" title="REALALG3:func.7">signum</a> <a onclick="hs(this)" href="javascript:()">REALALG3:def 9 : <br/></a><span> for <font color="Olive" title="b1">R</font> being   <a href="realalg1.html#V14" title="REALALG1:attr.14">ordered</a>  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</a><br/>  for <font color="Olive" title="b2">P</font> being   <a href="realalg1.html#NM2" title="REALALG1:NM.2">Ordering</a> of <font color="Olive" title="b1">R</font><br/>  for <font color="Olive" title="b3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">R</font> holds <br/> ( ( <font color="Olive" title="b3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">R</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> implies  <a href="realalg3.html#K7" title="REALALG3:func.7">signum</a> (<font color="Olive" title="b2">P</font>,<font color="Olive" title="b3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">R</font> implies  <a href="realalg3.html#K7" title="REALALG3:func.7">signum</a> (<font color="Olive" title="b2">P</font>,<font color="Olive" title="b3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp; (  not <font color="Olive" title="b3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">R</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> &amp;  not <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">R</font> implies  <a href="realalg3.html#K7" title="REALALG3:func.7">signum</a> (<font color="Olive" title="b2">P</font>,<font color="Olive" title="b3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 ) );<br/></span></div>
