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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">L</font>;<br/>





<a NAME="E1:23"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom3.html#K9" title="POLYNOM3:func.9">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom3.html#K9" title="POLYNOM3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom3.html#K9" title="POLYNOM3:func.9">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> = 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#D6" title="POLYNOM3:def.6">POLYNOM3:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#D3" title="POLYNOM5:def.3">POLYNOM5:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#D3" title="POLYNOM5:def.3">POLYNOM5:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom3.html#K9" title="POLYNOM3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="ref" href="polynom3.html#D6" title="POLYNOM3:def.6">POLYNOM3:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E2:23_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom3.html#K9" title="POLYNOM3:func.9">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom3.html#K9" title="POLYNOM3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="ref" href="polynom5.html#D3" title="POLYNOM5:def.3">POLYNOM5:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:23"/>
<font color="Maroon" title="c2">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom3.html#K9" title="POLYNOM3:func.9">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom3.html#K9" title="POLYNOM3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
