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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  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">H</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">H</font>,<font color="Maroon">S</font>;<br/>











<b>assume </b><a NAME="E1:6"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">f</font> is <a href="mod_2.html#V1" target="_self">linear</a> &amp; <font color="Maroon">g</font> is <a href="mod_2.html#V1" target="_self">linear</a> )
 ;<br/>

<a NAME="E2:6"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">f</font> is <a href="grcat_1.html#V1">additive</a> &amp; <font color="Maroon">g</font> is <a href="grcat_1.html#V1">additive</a> )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">h</font> = <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>;<br/>
<a NAME="E3:6"/><i><font color="Green">E34</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"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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#K2">+</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"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:6_1"/>
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> is <a href="grcat_1.html#V1">additive</a>
 
<b>by </b><i>, <a class="ref" href="grcat_1.html#T14">GRCAT_1:14</a></i>;<br/>
<b>hence </b><a NAME="E2:6_1"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="grcat_1.html#D13">GRCAT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>now </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/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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> = 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">g</font> <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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</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="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:6"/>
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> is <a href="mod_2.html#V1" target="_self">linear</a>
 <b>by </b><i>, </i>;<br/>


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