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<b>let </b><font color="Maroon" title="c1">K</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/>

<b>set </b><font color="Maroon" title="c2">J</font> =  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font>;<br/>
<a NAME="E1:49"/><i><font color="Green" title="E42">A1</font></i>: 
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font> is <a href="mod_4.html#V2" title="MOD_4:attr.2">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" title="E43">A2</font></i>: 
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font> is <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">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" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Scalar</a> of <font color="Maroon" title="c1">K</font> holds  <span class="p1">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Scalar</a> of <font color="Maroon" title="c1">K</font> holds  <span class="p1">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span> ) &amp; <span class="p1">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">K</font> )
 
<b>by </b><i><a class="txt" href="mod_4.html#E55"><i><font color="Green" title="E41">Lm11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:49_1_1"/>
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font> is <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a>
 <b>by </b><i><a class="ref" href="ringcat1.html#D2" title="RINGCAT1:def.2">RINGCAT1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:49_1"/>
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="mod_4.html#E55"><i><font color="Green" title="E41">Lm11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:49_1"/>
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font> is <a href="mod_4.html#V2" title="MOD_4:attr.2">monomorphism</a>
 <b>by </b><i><a class="txt" href="mod_4.html#E1:49_1"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="mod_4.html#D7" target="_self" title="MOD_4:def.7">Def7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:49"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font>
 
<b>by </b><i><a class="txt" href="mod_4.html#E55"><i><font color="Green" title="E41">Lm11</font></i></a></i>;<br/>
<a NAME="E3:49"/><b>then </b>
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font> is <a href="mod_4.html#V6" title="MOD_4:attr.6">isomorphism</a>
 
<b>by </b><i><a class="txt" href="mod_4.html#E1:49"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="mod_4.html#D11" target="_self" title="MOD_4:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E4:49"/>
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">K</font> is <a href="mod_4.html#V10" title="MOD_4:attr.10">automorphism</a>
 <b>by </b><i><a class="ref" href="mod_4.html#D15" target="_self" title="MOD_4:def.15">Def15</a></i>;<br/>


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