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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:49"/><i><font color="Green">E42</font></i>: 
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font> is <a href="mod_4.html#V2" target="_self">monomorphism</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<a NAME="E1:49_1"/><i><font color="Green">E43</font></i>: 
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font> is <a href="ringcat1.html#V1">linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:49_1_1"/>
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> ) &amp; <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="mod_4.html#E52"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:49_1_1"/>
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font> is <a href="ringcat1.html#V1">linear</a>
 <b>by </b><i><a class="ref" href="ringcat1.html#D2">RINGCAT1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:49_1"/>
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="mod_4.html#E52"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:49_1"/>
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font> is <a href="mod_4.html#V2" target="_self">monomorphism</a>
 <b>by </b><i><a class="txt" href="mod_4.html#E1:49_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mod_4.html#D7" target="_self">Def7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:49"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="mod_4.html#E52"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<a NAME="E3:49"/><b>then </b>
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font> is <a href="mod_4.html#V6" target="_self">isomorphism</a>
 
<b>by </b><i><a class="txt" href="mod_4.html#E1:49"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mod_4.html#D11" target="_self">Def11</a></i>;<br/>
<b>hence </b><a NAME="E4:49"/>
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font> is <a href="mod_4.html#V10" target="_self">automorphism</a>
 <b>by </b><i><a class="ref" href="mod_4.html#D15" target="_self">Def15</a></i>;<br/>


</div>
