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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">H</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">R</font>;<br/>





<b>set </b><font color="Maroon" title="c4">f</font> =  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">ZeroMap</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">G'</font> = <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c6">H'</font> = <font color="Maroon" title="c3">H</font> as   <a href="vectsp_1.html#NM1" title="VECTSP_1:NM.1">AbGroup</a> ;<br/>
<a NAME="E2:7"/>
 <a href="grcat_1.html#K8" title="GRCAT_1:func.8">ZeroMap</a> <font color="Maroon" title="c5">G'</font>,<font color="Maroon" title="c6">H'</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T16" title="GRCAT_1:th.16">GRCAT_1:16</a></i>;<br/>
<a NAME="E3:7"/><b>then </b><i><font color="Green" title="E7">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">G</font> holds  <span class="p1">(<span class="default"><a href="grcat_1.html#K8" title="GRCAT_1:func.8">ZeroMap</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</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#K4" title="RLVECT_1:func.4">+</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#K8" title="GRCAT_1:func.8">ZeroMap</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</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#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K8" title="GRCAT_1:func.8">ZeroMap</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a></i>;<br/>
<a NAME="E4:7"/>
for <font color="Olive" title="b1">a</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Scalar</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">G</font> holds  <span class="p1">(<span class="default"><a href="grcat_1.html#K8" title="GRCAT_1:func.8">ZeroMap</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</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">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K8" title="GRCAT_1:func.8">ZeroMap</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">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" title="c7">a</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Scalar</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c8">x</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">G</font>;<br/>



<a NAME="E1:7_1"/>
( <span class="p1">(<span class="default"><a href="grcat_1.html#K8" title="GRCAT_1:func.8">ZeroMap</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c3">H</font> &amp; <span class="p1">(<span class="default"><a href="grcat_1.html#K8" title="GRCAT_1:func.8">ZeroMap</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c3">H</font> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">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" title="GRCAT_1:func.8">ZeroMap</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K8" title="GRCAT_1:func.8">ZeroMap</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T59" title="VECTSP_1:th.59">VECTSP_1:59</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:7"/>
 <a href="grcat_1.html#K8" title="GRCAT_1:func.8">ZeroMap</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> is <a href="mod_2.html#V1" title="MOD_2:attr.1">linear</a>
 <b>by </b><i><a class="txt" href="mod_2.html#E3:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="mod_2.html#D5" target="_self" title="MOD_2:def.5">Def5</a></i>;<br/>


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