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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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">IT</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> <font color="Maroon" title="c1">n</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/>( <font color="Olive" title="b1">IT</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  iff <font color="Olive" title="b1">IT</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">IT</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> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies ( <font color="Maroon" title="c2">IT</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  iff <font color="Maroon" title="c2">IT</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:33"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <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="c2">IT</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  iff <font color="Maroon" title="c2">IT</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  )</span><br/>

<span class="kw">hence </span><a NAME="E2:33"/>
( <font color="Maroon" title="c2">IT</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  implies <font color="Maroon" title="c2">IT</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="matrix_7.html#E33"><span class="lab"><font color="Green" title="E26">Lm2</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"> ( <font color="Maroon" title="c2">IT</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  implies <font color="Maroon" title="c2">IT</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:33_1"/>
<font color="Maroon" title="c2">IT</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</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="c2">IT</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> </span><br/><a NAME="E2:33_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">IT</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a></span>)</span> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="matrix_7.html#E33"><span class="lab"><font color="Green" title="E26">Lm2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:33_1"/>
<font color="Maroon" title="c2">IT</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T43" title="FUNCT_1:th.43">FUNCT_1:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:33"/>
( <font color="Maroon" title="c2">IT</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  implies <font color="Maroon" title="c2">IT</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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