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<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">b'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/>







<a NAME="E1:4"/><i><font color="Green">E52</font></i>: 
(  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">n</font>,<font color="Maroon">n</font> <a href="matrix_2.html#K2" target="_self">--&gt;</a> <font color="Maroon">a'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">n</font>,<font color="Maroon">n</font> <a href="matrix_2.html#K2" target="_self">--&gt;</a> <font color="Maroon">b'</font></span>)</span> &amp;  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">n</font>,<font color="Maroon">n</font> <a href="matrix_2.html#K2" target="_self">--&gt;</a> <font color="Maroon">a'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">n</font>,<font color="Maroon">n</font> <a href="matrix_2.html#K2" target="_self">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b'</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T27">MATRIX_1:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">n</font>,<font color="Maroon">n</font> <a href="matrix_2.html#K2" target="_self">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b'</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:4_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font>,<font color="Maroon">n</font> <a href="matrix_2.html#K2" target="_self">--&gt;</a> <font color="Maroon">a'</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:4_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font>,<font color="Maroon">n</font> <a href="matrix_2.html#K2" target="_self">--&gt;</a> <font color="Maroon">a'</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font>,<font color="Maroon">n</font> <a href="matrix_2.html#K2" target="_self">--&gt;</a> <font color="Maroon">b'</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b'</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:4_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font>,<font color="Maroon">n</font> <a href="matrix_2.html#K2" target="_self">--&gt;</a> <font color="Maroon">a'</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font>,<font color="Maroon">n</font> <a href="matrix_2.html#K2" target="_self">--&gt;</a> <font color="Maroon">b'</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font>,<font color="Maroon">n</font> <a href="matrix_2.html#K2" target="_self">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b'</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:4"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font>,<font color="Maroon">n</font> <a href="matrix_2.html#K2" target="_self">--&gt;</a> <font color="Maroon">a'</font></span>)</span> <a href="matrix_1.html#K14">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font>,<font color="Maroon">n</font> <a href="matrix_2.html#K2" target="_self">--&gt;</a> <font color="Maroon">b'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>,<font color="Maroon">n</font> <a href="matrix_2.html#K2" target="_self">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D14">MATRIX_1:def 14</a></i>;<br/>


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