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<b>let </b><font color="Maroon">L</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">J</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">K</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">K</font>;<br/>









<b>assume </b><a NAME="E1:52"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">J</font> is <a href="ringcat1.html#V1">linear</a>
 ;<br/>

<b>thus </b><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E10"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="ringcat1.html#D2">RINGCAT1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E10"><i><font color="Green">Lemma45</font></i></a>, </i>
;<br/>


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