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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> );<br/>
<a NAME="E1:53_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E2:53_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> ;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E4:53_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:53_1_1"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E5:53_1_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<a NAME="E6:53_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:53_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:53_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:53_1_1_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E7:53_1_1_2"/>
<font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T140">GROUP_3:140</a></i>;<br/>
<a NAME="E8:53_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="txt" href="group_6.html#E1:53_1_1"><i><font color="Green">E45</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E9:53_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>consider </b><font color="Maroon">a1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:53_1_1_3"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">b1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:53_1_1_3"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> )
 <b>by </b><i/>;<br/><a NAME="E5:53_1_1_3"/><i><font color="Green">E111</font></i>: 
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> )
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">N</font>
<b>by </b><i>, , <a class="ref" href="group_3.html#T10">GROUP_3:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="group_2.html#K13">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="group_2.html#T128">GROUP_2:128</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font></span>)</span></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="group_3.html#T140">GROUP_3:140</a></i>
<br/>.= 
<font color="Maroon">a1</font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T116">GROUP_2:116</a></i>
<br/>.= 
<font color="Maroon">a1</font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="group_6.html#E1:53_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="group_2.html#T127">GROUP_2:127</a></i>
;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>

<a NAME="E10:53_1_1"/>
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:53_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font>
 ;<br/>


</div>
