<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#V12" title="VECTSP_1:attr.12">Fanoian</a>  <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">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="Olive" title="b1">n</font>,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b2">M1</font> is  <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a>  &amp; <font color="Olive" title="b2">M1</font> is  <a href="matrix_6.html#V3" title="MATRIX_6:attr.3">antisymmetric</a>  holds <br/><font color="Olive" title="b2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Olive" title="b1">n</font>,<font color="Olive" title="b1">n</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">M1</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">M1</font> is  <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a>  &amp; <font color="Olive" title="b1">M1</font> is  <a href="matrix_6.html#V3" title="MATRIX_6:attr.3">antisymmetric</a>  holds <br/><font color="Olive" title="b1">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</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="c2">n</font>,<font color="Maroon" title="c1">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>  &amp; <font color="Maroon" title="c3">M1</font> is  <a href="matrix_6.html#V3" title="MATRIX_6:attr.3">antisymmetric</a>  implies <font color="Maroon" title="c3">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>) )</span><br/>





<span class="kw">assume </span><a NAME="E1:39"/>
( <font color="Maroon" title="c3">M1</font> is  <a href="matrix_6.html#V2" title="MATRIX_6:attr.2">symmetric</a>  &amp; <font color="Maroon" title="c3">M1</font> is  <a href="matrix_6.html#V3" title="MATRIX_6:attr.3">antisymmetric</a>  )
 ; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span><br/>

<a NAME="E2:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A1</font></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> &amp; <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>  <a href="matrix_6.html#K1" title="MATRIX_6:func.1">-</a> <font color="Maroon" title="c3">M1</font> )
 
<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>, <a class="ref" href="matrix_6.html#D6" target="_self" title="MATRIX_6:def.6">Def6</a></span>;<br/>
<a NAME="E3:39"/>
 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/><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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</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>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_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 <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> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</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>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:39_1"/><span class="lab"><font color="Green" title="E19">A2</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"> <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> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</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>)</span><br/>

<a NAME="E2:39_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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><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>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="matrix_3.html#D2" title="MATRIX_3:def.2">MATRIX_3:def 2</a></span>;<br/>
<a NAME="E3:39_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><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>)</span> <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="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></span>;<br/>
<a NAME="E4:39_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><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>)</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="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>)</span> <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="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D8" title="VECTSP_1:def.8">VECTSP_1:def 8</a></span>;<br/>
<a NAME="E5:39_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><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>)</span></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"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <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="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D8" title="VECTSP_1:def.8">VECTSP_1:def 8</a></span>;<br/>
<a NAME="E6:39_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><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>)</span> <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="c1">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D7" title="VECTSP_1:def.7">VECTSP_1:def 7</a>, <a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a></span>;<br/>
<a NAME="E7:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A3</font></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="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T12" title="VECTSP_1:th.12">VECTSP_1:12</a></span>;<br/>
<a NAME="E8:39_1"/>
<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> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="matrix_1.html#T26" title="MATRIX_1:th.26">MATRIX_1:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:39_1"/>
<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> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="matrix_3.html#T1" title="MATRIX_3:th.1">MATRIX_3:1</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>
<span class="kw">hence </span><a NAME="E4:39"/>
<font color="Maroon" title="c3">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</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>; <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>
