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<b>set </b><font color="Maroon">S</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font>;<br/>
<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">qa</font>, <font color="Olive">qb</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> st <br/>( <font color="Olive">qa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">qb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">i</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Olive">qa</font>,<font color="Olive">qb</font> );<br/>
<div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><b>now </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="E1:60_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">tfsm</font>
 ;<br/><b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> ;<br/><a NAME="E3:60_1_1_1"/>
<font color="Maroon">i</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">q</font>,<font color="Maroon">q</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:60_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">x</font>]
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 ;<br/><b>then consider </b><font color="Maroon">qa</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font>, <font color="Maroon">qb</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font><b> such that </b><br/><a NAME="E3:60_1_1_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">qa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">qb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">i</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">qa</font>,<font color="Maroon">qb</font> )
 ;<br/><a NAME="E4:60_1_1_2"/>
<font color="Maroon">i</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">qb</font>,<font color="Maroon">qa</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:60_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:60_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">x</font>]
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:60_1_1"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 ;<br/><b>then consider </b><font color="Maroon">qa</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font>, <font color="Maroon">qb</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font><b> such that </b><br/><a NAME="E3:60_1_1_3"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">qa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">qb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">i</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">qa</font>,<font color="Maroon">qb</font> )
 ;<br/><b>assume </b><a NAME="E4:60_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 ;<br/><b>then consider </b><font color="Maroon">qb'</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font>, <font color="Maroon">qc</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font><b> such that </b><br/><a NAME="E6:60_1_1_3"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">qb'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">qc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">i</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">qb'</font>,<font color="Maroon">qc</font> )
 ;<br/><a NAME="E7:60_1_1_3"/>
<font color="Maroon">i</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">qa</font>,<font color="Maroon">qc</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E8:60_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z</font>]
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">EqR</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font><b> such that </b><br/><a NAME="E5:60_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR</font> iff ( <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">tfsm</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="eqrel_1.html#S1">EQREL_1:sch 1</a>(, , <a class="txt" href="fsm_1.html#E5:60_1_1"><i><font color="Green">E43</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">EqR</font>
;<br/>

<b>let </b><font color="Maroon">qa</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font>, <font color="Maroon">qb</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:60_1_1_4"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">qa</font>,<font color="Maroon">qb</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR</font>
 ;<br/><b>then consider </b><font color="Maroon">qa'</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font>, <font color="Maroon">qb'</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font><b> such that </b><br/><a NAME="E3:60_1_1_4"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">qa'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qa</font> &amp; <font color="Maroon">qb'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qb</font> &amp; <font color="Maroon">i</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">qa'</font>,<font color="Maroon">qb'</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:60_1_1_4"><i><font color="Green">E44</font></i></a></i>;<br/><b>thus </b><a NAME="E4:60_1_1_4"/>
<font color="Maroon">i</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">qa</font>,<font color="Maroon">qb</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E7:60_1_1"/>
<font color="Maroon">i</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">qa</font>,<font color="Maroon">qb</font>
 ;<br/>

<b>hence </b><a NAME="E8:60_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">qa</font>,<font color="Maroon">qb</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR</font>
 <b>by </b><i/>;<br/>


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