<?xml version="1.0"?>
<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">p2</font>, <font color="Olive" title="b2">q2</font>, <font color="Olive" title="b3">pq2</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><br/>  for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b6">K</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>  st <font color="Olive" title="b3">pq2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">p2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">q2</font> &amp; <font color="Olive" title="b2">q2</font> is  <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a>  &amp; <font color="Olive" title="b2">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">j</font> &amp;  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b6">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b6">K</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b1">p2</font>,<font color="Olive" title="b6">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b3">pq2</font>,<font color="Olive" title="b6">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; (  for <font color="Olive" title="b7">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b7">k</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> <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> &amp; <font color="Olive" title="b4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">k</font> &amp; <font color="Olive" title="b5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">k</font> holds <br/>( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b1">p2</font>,<font color="Olive" title="b6">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b7">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b3">pq2</font>,<font color="Olive" title="b6">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b7">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> iff <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b1">p2</font>,<font color="Olive" title="b6">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b5">j</font>,<font color="Olive" title="b7">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b3">pq2</font>,<font color="Olive" title="b6">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b5">j</font>,<font color="Olive" title="b7">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">n2</font> = <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c5">pq</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="firebrick">::  thesis: </font></span></a><span class="hide">  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><br/>  for <font color="Olive" title="b3">K</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>  st <font color="Maroon" title="c5">pq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c4">q</font> is  <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a>  &amp; <font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp;  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b3">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b3">K</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Olive" title="b3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; (  for <font color="Olive" title="b4">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">k</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> <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> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">k</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">k</font> holds <br/>( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b4">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Olive" title="b3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b4">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> iff <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b2">j</font>,<font color="Olive" title="b4">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Olive" title="b3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b2">j</font>,<font color="Olive" title="b4">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) ) )</span><br/><span class="kw">let </span><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>; <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>  st <font color="Maroon" title="c5">pq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c4">q</font> is  <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a>  &amp; <font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> &amp;  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b1">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b1">K</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Olive" title="b1">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; (  for <font color="Olive" title="b2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</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> <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> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">k</font> &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">k</font> holds <br/>( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Olive" title="b2">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Olive" title="b1">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Olive" title="b2">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> iff <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Olive" title="b2">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Olive" title="b1">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Olive" title="b2">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">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"> ( <font color="Maroon" title="c5">pq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c4">q</font> is  <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a>  &amp; <font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> &amp;  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font></span>)</span> implies ( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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> <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> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">k</font> holds <br/>( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> iff <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) ) ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c5">pq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">and </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c4">q</font> is  <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> 
 <span class="kw">and </span><br/><a NAME="E3:16"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">and </span><br/><a NAME="E4:16"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">and </span><br/><a NAME="E5:16"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</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"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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> <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> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">k</font> holds <br/>( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> iff <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) ) )</span><br/>

<a NAME="E6:16"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="matrix11.html#T8" target="_self" title="MATRIX11:th.8">Th8</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">P2</font> =  <a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c10">P1</font> =  <a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">p9</font> = <font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c12">q9</font> = <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c13">pq9</font> = <font color="Maroon" title="c5">pq</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<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></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D9" title="MATRIX_2:def.9">MATRIX_2:def 9</a></span>;<br/>
<a NAME="E8:16"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/>
<a NAME="E9:16"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="matrix11.html#T8" target="_self" title="MATRIX11:th.8">Th8</a></span>;<br/>
<a NAME="E10:16"/><span class="lab"><font color="Green" title="E21">A9</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">pq9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/>
<a NAME="E11:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<font color="Maroon" title="c5">pq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E12:16"/>
<font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="matrix11.html#T8" target="_self" title="MATRIX11:th.8">Th8</a></span>;<br/>
<a NAME="E13:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon" title="c5">pq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E14:16"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/>
<a NAME="E15:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A12</font></span>: 
<font color="Maroon" title="c11">p9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">p9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_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="E1:16_1_1"/>
( <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E24">A12</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="16_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_1_1_1"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:16_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="matrix11.html#D1" target="_self" title="MATRIX11:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_1_1_1"/>
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E13:16"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E1:16_1_1_1"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="matrix11.html#D1" target="_self" title="MATRIX11:def.1">Def1</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="16_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_1_1_2"/><span class="lab"><font color="Green" title="E25">A14</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:16_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="matrix11.html#D1" target="_self" title="MATRIX11:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_1_1_2"/>
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E13:16"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E1:16_1_1_2"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="ref" href="matrix11.html#D1" target="_self" title="MATRIX11:def.1">Def1</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>
<span class="kw">hence </span><a NAME="E17:16"/>
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <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="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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> <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> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">k</font> holds <br/>( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> iff <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c14">k</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"> ( <font color="Maroon" title="c14">k</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> <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> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">k</font> &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">k</font> implies ( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> iff <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E18:16"/><span class="lab"><font color="Green" title="E25">A15</font></span>: 
<font color="Maroon" title="c14">k</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> <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>
 <span class="kw">and </span><br/><a NAME="E19:16"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">k</font>
 <span class="kw">and </span><br/><a NAME="E20:16"/><span class="lab"><font color="Green" title="E27">A17</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">k</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"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> iff <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/>

<a NAME="E21:16"/><span class="lab"><font color="Green" title="E28">A18</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E19:16"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="txt" href="#E20:16"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="matrix11.html#T8" target="_self" title="MATRIX11:th.8">Th8</a></span>;<br/>
<a NAME="E22:16"/><span class="lab"><font color="Green" title="E29">A19</font></span>: 
<font color="Maroon" title="c5">pq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E23:16"/>
( <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k</font> or <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:16"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E24:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A20</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<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></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="ref" href="matrix11.html#T1" target="_self" title="MATRIX11:th.1">Th1</a></span>;<br/>
<a NAME="E25:16"/><span class="lab"><font color="Green" title="E31">A21</font></span>: 
( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> implies <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:16_2"/><span class="lab"><font color="Green" title="E32">A22</font></span>: 
( <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k</font> or <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E2:16_2"/><span class="lab"><font color="Green" title="E33">A23</font></span>: 
( <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k</font> or <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:16"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:16_2"/><span class="lab"><font color="Green" title="E34">A24</font></span>: 
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></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"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/>

<a NAME="E4:16_2"/>
( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">k</font>,<font color="Maroon" title="c6">i</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font> or <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">k</font>,<font color="Maroon" title="c6">i</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:16"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="ref" href="matrix11.html#T5" target="_self" title="MATRIX11:th.5">Th5</a></span>;<br/>
<a NAME="E5:16_2"/><span class="kw">then </span>
( ( <font color="Maroon" title="c5">pq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">pq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> &amp; <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> ) or ( <font color="Maroon" title="c5">pq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">pq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> &amp; <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E13:16"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E21:16"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="txt" href="#E22:16"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="txt" href="#E3:16_2"><span class="lab"><font color="Green" title="E34">A24</font></span></a>, <a class="txt" href="#E2:16_2"><span class="lab"><font color="Green" title="E33">A23</font></span></a>, <a class="txt" href="matrix11.html#E11"><span class="lab"><font color="Green" title="E11">Lm1</font></span></a></span>;<br/>
<a NAME="E6:16_2"/><span class="kw">then </span>
( ( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font> ) or ( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font></span>)</span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E1:16_2"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="ref" href="matrix11.html#D1" target="_self" title="MATRIX11:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:16_2"/>
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</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>
<a NAME="E26:16"/>
( <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k</font> or <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E27:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A25</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<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></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="ref" href="matrix11.html#T1" target="_self" title="MATRIX11:th.1">Th1</a></span>;<br/>
<a NAME="E28:16"/>
( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> implies <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:16_3"/><span class="lab"><font color="Green" title="E33">A26</font></span>: 
( <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k</font> or <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:16"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E2:16_3"/><span class="lab"><font color="Green" title="E34">A27</font></span>: 
( <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k</font> or <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:16_3"/><span class="lab"><font color="Green" title="E35">A28</font></span>: 
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></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"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/>

<a NAME="E4:16_3"/>
( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">k</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font> or <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c14">k</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:16"><span class="lab"><font color="Green" title="E32">A25</font></span></a>, <a class="ref" href="matrix11.html#T5" target="_self" title="MATRIX11:th.5">Th5</a></span>;<br/>
<a NAME="E5:16_3"/><span class="kw">then </span>
( ( <font color="Maroon" title="c5">pq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">pq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> &amp; <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> ) or ( <font color="Maroon" title="c5">pq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">pq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> &amp; <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E13:16"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E21:16"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="txt" href="#E22:16"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="txt" href="#E3:16_3"><span class="lab"><font color="Green" title="E35">A28</font></span></a>, <a class="txt" href="#E2:16_3"><span class="lab"><font color="Green" title="E34">A27</font></span></a>, <a class="txt" href="matrix11.html#E11"><span class="lab"><font color="Green" title="E11">Lm1</font></span></a></span>;<br/>
<a NAME="E6:16_3"/><span class="kw">then </span>
( ( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font> ) or ( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c8">K</font></span>)</span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E1:16_3"><span class="lab"><font color="Green" title="E33">A26</font></span></a>, <a class="ref" href="matrix11.html#D1" target="_self" title="MATRIX11:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:16_3"/>
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</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="E29:16"/>
( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> iff <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">pq</font>,<font color="Maroon" title="c8">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c14">k</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:16"><span class="lab"><font color="Green" title="E31">A21</font></span></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>
