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<b>set </b><font color="Maroon" title="c5">S</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">qa</font>, <font color="Olive" title="b2">qb</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> st <br/>( <font color="Olive" title="b1">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">qb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Maroon" title="c4">i</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Olive" title="b1">qa</font>,<font color="Olive" title="b2">qb</font> );<br/>
<a NAME="E1:58_1_1"/><i><font color="Green" title="E40">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c3">tfsm</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font>]
 
<b>by </b><i><a class="ref" href="fsm_1.html#T38" target="_self" title="FSM_1:th.38">Th38</a></i>;<br/>
<a NAME="E2:58_1_1"/><i><font color="Green" title="E41">A2</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>]
 
<b>by </b><i><a class="ref" href="fsm_1.html#T39" target="_self" title="FSM_1:th.39">Th39</a></i>;<br/>
<a NAME="E3:58_1_1"/><i><font color="Green" title="E42">A3</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>]
 
<b>by </b><i><a class="ref" href="fsm_1.html#T40" target="_self" title="FSM_1:th.40">Th40</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">EqR</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font><b> such that </b><br/><a NAME="E5:58_1_1"/><i><font color="Green" title="E43">A4</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">EqR</font> iff ( <font color="Olive" title="b1">x</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="c3">tfsm</font> &amp; <font color="Olive" title="b2">y</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="c3">tfsm</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>from </b><i><a class="ref" href="eqrel_1.html#S1" title="EQREL_1:sch.1">EQREL_1:sch 1</a>(<a class="txt" href="fsm_1.html#E1:58_1_1"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="fsm_1.html#E2:58_1_1"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="fsm_1.html#E3:58_1_1"><i><font color="Green" title="E42">A3</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c6">EqR</font>
;<br/>

<b>let </b><font color="Maroon" title="c7">qa</font>, <font color="Maroon" title="c8">qb</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:58_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">qa</font>,<font color="Maroon" title="c8">qb</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">EqR</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">qa'</font>, <font color="Maroon" title="c10">qb'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font><b> such that </b><br/><a NAME="E3:58_1_1_1"/><i><font color="Green" title="E44">A5</font></i>: 
( <font color="Maroon" title="c9">qa'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">qa</font> &amp; <font color="Maroon" title="c10">qb'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">qb</font> &amp; <font color="Maroon" title="c4">i</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c9">qa'</font>,<font color="Maroon" title="c10">qb'</font> )
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:58_1_1"><i><font color="Green" title="E43">A4</font></i></a></i>;<br/><b>thus </b><a NAME="E4:58_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c7">qa</font>,<font color="Maroon" title="c8">qb</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E3:58_1_1_1"><i><font color="Green" title="E44">A5</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E7:58_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="fsm_1.html#R6" title="FSM_1:pred.6">-equivalent</a> <font color="Maroon" title="c7">qa</font>,<font color="Maroon" title="c8">qb</font>
 ;<br/>

<b>hence </b><a NAME="E8:58_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">qa</font>,<font color="Maroon" title="c8">qb</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">EqR</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E5:58_1_1"><i><font color="Green" title="E43">A4</font></i></a></i>;<br/>


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