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<div><span class="kw">theorem </span><a NAME="T23"><span class="comment"><font color="firebrick">:: MOD_4:23</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">K</font>, <font color="Olive" title="b2">L</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</a><br/>  for <font color="Olive" title="b3">J</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">L</font><br/>  for <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">y</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Scalar</a> of <font color="Olive" title="b1">K</font>  st <font color="Olive" title="b3">J</font> is  <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a>  holds <br/>( <font color="Olive" title="b3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b2">L</font> &amp; <font color="Olive" title="b3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font></span>)</span> &amp; <font color="Olive" title="b3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">y</font></span>)</span> ) <span class="kw">by</span> <span class="lab"><a class="txt" href="mod_4.html#E43"><span class="lab"><font color="Green" title="E21">Lm6</font></span></a>, <a class="txt" href="mod_4.html#E44"><span class="lab"><font color="Green" title="E22">Lm7</font></span></a>, <a class="txt" href="mod_4.html#E45"><span class="lab"><font color="Green" title="E23">Lm8</font></span></a></span>;<br/></div></div>
