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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E46">thEucl1</font></span>: <a NAME="T30"><span class="comment"><font color="firebrick">:: RING_2:30</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">L</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a><br/>  for <font color="Olive" title="b2">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Olive" title="b1">L</font></span>)</span><br/>  for <font color="Olive" title="b3">q</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Olive" title="b1">L</font></span>)</span>  ex <font color="Olive" title="b4">u</font>, <font color="Olive" title="b5">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Olive" title="b1">L</font></span>)</span> st <br/>( <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">u</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">q</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b5">r</font> &amp; ( <font color="Olive" title="b5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Olive" title="b1">L</font></span>)</span> or <span class="p1">(<span class="default"><a href="ring_2.html#K9" title="RING_2:func.9">deg*</a> <font color="Olive" title="b1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="ring_2.html#K9" title="RING_2:func.9">deg*</a> <font color="Olive" title="b1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">q</font> ) )</div></div>
