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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>] means <font color="Maroon">a<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:150_1_1"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:150_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:150_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="monoid_0.html#T79" target="_self">Th79</a></i>;<br/>
<b>hence </b><a NAME="E4:150_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>3</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:150_1_1"/><i><font color="Green">E71</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_1_1_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:150_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">c<sub>1</sub></font>) <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="monoid_0.html#D40" target="_self">Def40</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" target="_self">GFuncs</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>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E4:150_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V1">strict</a>  <a href="monoid_0.html#M5" target="_self">SubStr</a> of  <a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:150_1_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 <b>from </b><i><a class="ref" href="monoid_0.html#S2" target="_self">MONOID_0:sch 2</a>(<a class="txt" href="monoid_0.html#E1:150_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="monoid_0.html#E2:150_1_1"><i><font color="Green">E71</font></i></a>);<br/></i>
<a NAME="E5:150_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">c<sub>1</sub></font>) <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="monoid_0.html#D40" target="_self">Def40</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" target="_self">GFuncs</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/>
<a NAME="E7:150_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
 
<b>by </b><i><a class="txt" href="monoid_0.html#E4:150_1_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V1">strict</a> <a href="group_1.html#V2">unital</a>  <a href="monoid_0.html#M5" target="_self">SubStr</a> of  <a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="monoid_0.html#T80" target="_self">Th80</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E1:150_1_1_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">c<sub>1</sub></font>)
 <b>by </b><i><a class="ref" href="monoid_0.html#T25" target="_self">Th25</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="monoid_0.html#E4:150_1_1"><i><font color="Green">E72</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="monoid_0.html#E4:150_1_1"><i><font color="Green">E72</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K2">"</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="monoid_0.html#E5:150_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_2.html#T12">FUNCT_2:12</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="monoid_0.html#E4:150_1_1"><i><font color="Green">E72</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/><b>take </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font>;<br/><a NAME="E8:150_1_1_3"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="monoid_0.html#T79" target="_self">Th79</a>, <a class="ref" href="funct_2.html#T23">FUNCT_2:23</a>, <a class="ref" href="funct_2.html#T88">FUNCT_2:88</a></i>;<br/><b>hence </b><a NAME="E9:150_1_1_3"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E28"><i><font color="Green">Lemma14</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V1">strict</a> <a href="group_1.html#V2">unital</a> <a href="monoid_0.html#V13" target="_self">invertible</a>  <a href="monoid_0.html#M5" target="_self">SubStr</a> of  <a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="monoid_0.html#T12" target="_self">Th12</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E11:150_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> iff <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E4:150_1_1"><i><font color="Green">E72</font></i></a></i>;<br/>


</div>
