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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">F</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">F</font>;<br/>





<a NAME="E1:58"/><i><font color="Green">E49</font></i>: 
(  <a href="matrix_1.html#K2" target="_self">Indices</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2" target="_self">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K13" target="_self">-</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="matrix_1.html#K2" target="_self">Indices</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2" target="_self">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K11" target="_self">0.</a> <font color="Maroon">F</font>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:58"/><i><font color="Green">E52</font></i>: 
 <a href="matrix_1.html#K2" target="_self">Indices</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2" target="_self">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K14" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K13" target="_self">-</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_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="E1:58_1"/><i><font color="Green">E57</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" target="_self">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K14" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K13" target="_self">-</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K14" target="_self">+</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K13" target="_self">-</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3" target="_self">*</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"><a href="matrix_1.html#K13" target="_self">-</a> <font color="Maroon">A</font></span>)</span> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3" target="_self">*</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"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D10">RLVECT_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K11" target="_self">0.</a> <font color="Maroon">F</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i>, , , </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:58"/>
<font color="Maroon">A</font> <a href="matrix_1.html#K14" target="_self">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K13" target="_self">-</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K11" target="_self">0.</a> <font color="Maroon">F</font>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


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