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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:34_1_1"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:34_1_1"/>
 <a href="finseq_2.html#K1">idseq</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T12">FUNCT_2:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="matrix_2.html#E2:34_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E5:34_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="matrix_2.html#V1" target="_self">permutational</a>  <a href="hidden.html#M1">set</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:34_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>hence </b><a NAME="E2:34_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="matrix_2.html#E2:34_1_1"><i><font color="Green">E15</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="matrix_2.html#V1" target="_self">permutational</a>  <a href="hidden.html#M1">set</a> 
 <b>by </b><i><a class="ref" href="matrix_2.html#D9" target="_self">Def9</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="matrix_2.html#V1" target="_self">permutational</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E7:34_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:34_1_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>5</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="matrix_2.html#E2:34_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>

<b>assume </b><a NAME="E2:34_1_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:34_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T12">FUNCT_2:12</a></i>;<br/>
<b>hence </b><a NAME="E4:34_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="matrix_2.html#E2:34_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="matrix_2.html#E2:34_1_1_2"><i><font color="Green">E16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:34_1_1"/>
(  <a href="matrix_2.html#K11" target="_self">Permutations</a> <font color="Maroon">c<sub>1</sub></font> is <a href="matrix_2.html#V1" target="_self">permutational</a> &amp; not  <a href="matrix_2.html#K11" target="_self">Permutations</a> <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i><a class="ref" href="matrix_2.html#D11" target="_self">Def11</a></i>;<br/>


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