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<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">H</font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">R</font>;<br/>





<b>set </b><font color="Maroon">f</font> =  <a href="grcat_1.html#K8">ZeroMap</a> <font color="Maroon">G</font>,<font color="Maroon">H</font>;<br/>
<b>reconsider </b><font color="Maroon">G'</font> = <font color="Maroon">G</font>, <font color="Maroon">H'</font> = <font color="Maroon">H</font> as   <a href="vectsp_1.html#NM1">AbGroup</a> ;<br/>
<a NAME="E2:7"/>
 <a href="grcat_1.html#K8">ZeroMap</a> <font color="Maroon">G'</font>,<font color="Maroon">H'</font> is <a href="grcat_1.html#V1">additive</a>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T16">GRCAT_1:16</a></i>;<br/>
<a NAME="E3:7"/><b>then </b><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">G</font> holds  <span class="p1">(<span class="default"><a href="grcat_1.html#K8">ZeroMap</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</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#K8">ZeroMap</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K8">ZeroMap</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="grcat_1.html#D13">GRCAT_1:def 13</a></i>;<br/>
<a NAME="E4:7"/>
for <font color="Olive">a</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font><br/> for <font color="Olive">x</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">G</font> holds  <span class="p1">(<span class="default"><a href="grcat_1.html#K8">ZeroMap</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K8">ZeroMap</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">G</font>;<br/>



<a NAME="E1:7_1"/>
( <span class="p1">(<span class="default"><a href="grcat_1.html#K8">ZeroMap</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">H</font> &amp; <span class="p1">(<span class="default"><a href="grcat_1.html#K8">ZeroMap</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E2:7_1"/>
<span class="p1">(<span class="default"><a href="grcat_1.html#K8">ZeroMap</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K8">ZeroMap</a> <font color="Maroon">G</font>,<font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T59">VECTSP_1:59</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:7"/>
 <a href="grcat_1.html#K8">ZeroMap</a> <font color="Maroon">G</font>,<font color="Maroon">H</font> is <a href="mod_2.html#V1" target="_self">linear</a>
 <b>by </b><i>, </i>;<br/>


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