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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">F</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">C</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">F</font>;<br/>









<a NAME="E1:55"/><i><font color="Green" title="E34">A1</font></i>: 
(  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">B</font> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">C</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T27" target="_self" title="MATRIX_1:th.27">Th27</a></i>;<br/>
<a NAME="E2:55"/><i><font color="Green" title="E35">A2</font></i>: 
(  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c5">C</font></span>)</span> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c5">C</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T27" target="_self" title="MATRIX_1:th.27">Th27</a></i>;<br/>
<a NAME="E3:55"/><i><font color="Green" title="E36">A3</font></i>: 
(  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c5">C</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T27" target="_self" title="MATRIX_1:th.27">Th27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:55_1"/><i><font color="Green" title="E37">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c5">C</font></span>)</span>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_1.html#E2:55"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="matrix_1.html#E3:55"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="matrix_1.html#D14" target="_self" title="MATRIX_1:def.14">Def14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_1.html#E2:55"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="matrix_1.html#E1:55_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="matrix_1.html#D14" target="_self" title="MATRIX_1:def.14">Def14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_1.html#E1:55"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix_1.html#E2:55"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="matrix_1.html#E1:55_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="matrix_1.html#D14" target="_self" title="MATRIX_1:def.14">Def14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c5">C</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="txt" href="matrix_1.html#E2:55"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="matrix_1.html#E1:55_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="matrix_1.html#D14" target="_self" title="MATRIX_1:def.14">Def14</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:55"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c5">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T28" target="_self" title="MATRIX_1:th.28">Th28</a></i>;<br/>


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