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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="Red">::  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">p2</font>, <font color="Olive" title="b3">q2</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>  st <font color="Olive" title="b3">q2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">p2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  holds <br/> <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b2">p2</font>,<font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b3">q2</font>,<font color="Olive" title="b1">K</font></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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p2</font>, <font color="Olive" title="b2">q2</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>  st <font color="Olive" title="b2">q2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">p2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  holds <br/> <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b1">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b2">q2</font>,<font color="Maroon" title="c2">K</font></span><br/>



<a NAME="E1:59"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">q2</font> be    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">q2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">p2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">q2</font>,<font color="Maroon" title="c2">K</font> )</span><br/>



<span class="kw">assume </span><a NAME="E2:59"/>
<font color="Maroon" title="c4">q2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">p2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">q2</font>,<font color="Maroon" title="c2">K</font></span><br/>

<a NAME="E3:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A2</font></span>: 
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span>,<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span>,<font color="Maroon" title="c4">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="matrix_7.html#T29" title="MATRIX_7:th.29">MATRIX_7:29</a></span>;<br/>
<a NAME="E4:59"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span>,<font color="Maroon" title="c4">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">q2</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T26" target="_self" title="MATRIX11:th.26">Th26</a></span>;<br/>
<a NAME="E5:59"/>
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span>,<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T26" target="_self" title="MATRIX11:th.26">Th26</a></span>;<br/>
<a NAME="E6:59"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">q2</font>,<font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:59"/>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c4">q2</font>,<font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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