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<div class="add">

<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">b</font> <a href="gcd_1.html#R3" target="_self">is_associated_to</a> <font color="Maroon">a</font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">b</font> <a href="gcd_1.html#R3" target="_self">is_associated_to</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/><a NAME="E2:32_1_1_1_1"/><b>then </b>
 ex <font color="Olive">B'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">B'</font> &amp; <font color="Olive">B'</font> <a href="gcd_1.html#R3" target="_self">is_associated_to</a> <font color="Maroon">a</font> )
 ;<br/><b>hence </b><a NAME="E3:32_1_1_1_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">b</font> <a href="gcd_1.html#R3" target="_self">is_associated_to</a> <font color="Maroon">a</font> </span>}</span>  implies <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:32_1_1_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">b</font> <a href="gcd_1.html#R3" target="_self">is_associated_to</a> <font color="Maroon">a</font> </span>}</span>  implies <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 ;<br/></div><b>end;</b></div>
<a NAME="E2:32_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">b</font> <a href="gcd_1.html#R3" target="_self">is_associated_to</a> <font color="Maroon">a</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2"><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">R</font>;<br/><a NAME="E1:32_1_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">b</font> <a href="gcd_1.html#R3" target="_self">is_associated_to</a> <font color="Maroon">a</font> </span>}</span>  implies <font color="Maroon">A</font> <a href="gcd_1.html#R3" target="_self">is_associated_to</a> <font color="Maroon">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1_1_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">b</font> <a href="gcd_1.html#R3" target="_self">is_associated_to</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/>

<a NAME="E2:32_1_1_2_1"/><b>then </b>
 ex <font color="Olive">A'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">A'</font> &amp; <font color="Olive">A'</font> <a href="gcd_1.html#R3" target="_self">is_associated_to</a> <font color="Maroon">a</font> )
 
;<br/>
<b>hence </b><a NAME="E3:32_1_1_2_1"/>
<font color="Maroon">A</font> <a href="gcd_1.html#R3" target="_self">is_associated_to</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E2:32_1_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">b</font> <a href="gcd_1.html#R3" target="_self">is_associated_to</a> <font color="Maroon">a</font> </span>}</span>  iff <font color="Maroon">A</font> <a href="gcd_1.html#R3" target="_self">is_associated_to</a> <font color="Maroon">a</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> st <br/>for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds <br/> ( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Olive">b</font> <a href="gcd_1.html#R3" target="_self">is_associated_to</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>


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