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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   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/>  for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">F</font> holds  <font color="Olive" title="b2">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Olive" title="b2">A</font></span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <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> holds  <font color="Olive" title="b1">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Olive" title="b1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c3">A</font></span><br/>







<a NAME="E1:54"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T26" target="_self" title="MATRIX_1:th.26">Th27</a></span>;<br/>
<a NAME="E2:54"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T26" target="_self" title="MATRIX_1:th.26">Th27</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">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"><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> implies <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#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">A</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:54_1"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">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"><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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">A</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>)</span><br/><span class="kw">hence </span><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#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>) = 
<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="c5">i</font>,<font color="Maroon" title="c6">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="c4">B</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="matrix_1.html#D13" target="_self" title="MATRIX_1:def.13">Def14</a></span>
<br/>.= 
<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="c3">A</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E1:54_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="matrix_1.html#D13" target="_self" title="MATRIX_1:def.13">Def14</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:54"/>
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T27" target="_self" title="MATRIX_1:th.27">Th28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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