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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_2.html#V2">domRing-like</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:51"/><i><font color="Green">E42</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E2:51"/><i><font color="Green">E43</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E3:51"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:51"/><b>then </b>
( <font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> &amp; <font color="Maroon">q</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:51"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/>
<a NAME="E6:51"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="polynom4.html#T13">POLYNOM4:13</a></i>;<br/>
<a NAME="E7:51"/>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E8:51"/><b>then </b>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E9:51"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i/>;<br/>


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