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<div class="add">

<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">K</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/>  for <font color="Olive" title="b2">M1</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">K</font>  st <font color="Olive" title="b2">M1</font> is   <a href="matrix_2.html#NM1" title="MATRIX_2:NM.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b2">M1</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font> holds <br/><font color="Olive" title="b2">M1</font> is  <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</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">M1</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">K</font>  st <font color="Olive" title="b1">M1</font> is   <a href="matrix_2.html#NM1" title="MATRIX_2:NM.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b1">M1</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> holds <br/><font color="Olive" title="b1">M1</font> is  <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">M1</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">K</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">M1</font> is   <a href="matrix_2.html#NM1" title="MATRIX_2:NM.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c3">M1</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> implies <font color="Maroon" title="c3">M1</font> is  <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:34"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
( <font color="Maroon" title="c3">M1</font> is   <a href="matrix_2.html#NM1" title="MATRIX_2:NM.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c3">M1</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</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">M1</font> is  <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> </span><br/>

<a NAME="E2:34"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E3:34"/>
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">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> <font color="Maroon" title="c3">M1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">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="c4">i</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c3">M1</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:34_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c3">M1</font>
 ; <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">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>)</span><br/>

<a NAME="E2:34_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E3:34_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E4:34_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c4">i</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> <font color="Maroon" title="c3">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:34_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j</font> or <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j</font>
 ; <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">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:34_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_1_1_2"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">j</font>
 ; <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">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>)</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:34_1_1_2_1"/>
( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> or <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">j</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1_2"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_1_1_2_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font>
 ; <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">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>)</span><br/><div class="add"><a NAME="E2:34_1_1_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <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="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c4">i</font>) <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="Maroon" title="c2">K</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E4:34_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="matrix_2.html#D3" title="MATRIX_2:def.3">MATRIX_2:def 3</a>, <a class="ref" href="matrix_2.html#D4" title="MATRIX_2:def.4">MATRIX_2:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:34_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_1_1_2_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">j</font>
 ; <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">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>)</span><br/><div class="add"><a NAME="E2:34_1_1_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <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="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c4">i</font>) <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="Maroon" title="c2">K</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E1:34_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E4:34_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="matrix_2.html#D3" title="MATRIX_2:def.3">MATRIX_2:def 3</a>, <a class="ref" href="matrix_2.html#D4" title="MATRIX_2:def.4">MATRIX_2:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:34_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E4:34"/><span class="kw">then </span>
<font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:34"/>
<font color="Maroon" title="c3">M1</font> is  <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_6.html#D5" target="_self" title="MATRIX_6:def.5">Def5</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/>


</div>
