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<div class="add">

<b>set </b><font color="Maroon" title="c3">M</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">b</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">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" title="c4">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">b</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font> </span>}</span>  implies <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> )</span><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" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">b</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">b</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font> </span>}</span>  implies <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> )</span><br/><a NAME="E2:32_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">B'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B'</font> &amp; <font color="Olive" title="b1">B'</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font> )
 ;<br/><b>hence </b><a NAME="E3:32_1_1_1_1"/>
( <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">b</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font> </span>}</span>  implies <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E2:32_1_1_1"/>
( <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">b</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font> </span>}</span>  implies <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E2:32_1_1"/><b>then </b><i><font color="Green" title="E22">A1</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">b</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c4">A</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">b</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font> </span>}</span>  iff <font color="Maroon" title="c4">A</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font> )</span><br/><a NAME="E1:32_1_1_2"/>
( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">b</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font> </span>}</span>  implies <font color="Maroon" title="c4">A</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">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" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">b</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E2:32_1_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">A'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A'</font> &amp; <font color="Olive" title="b1">A'</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font> )
 
;<br/>
<b>hence </b><a NAME="E3:32_1_1_2_1"/>
<font color="Maroon" title="c4">A</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E2:32_1_1_2"/>
( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">b</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font> </span>}</span>  iff <font color="Maroon" title="c4">A</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> st <br/>for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Olive" title="b2">b</font> <a href="gcd_1.html#R3" title="GCD_1:pred.3">is_associated_to</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="#E2:32_1_1"><i><font color="Green" title="E22">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
