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<div class="add">

<b>consider </b><font color="Maroon" title="c1">Y</font> being    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E2:21_1"/><i><font color="Green" title="E13">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="c1">Y</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<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="relset_1.html#S1" title="RELSET_1:sch.1">RELSET_1:sch 1</a>();<br/></i>
<a NAME="E3:21_1"/><i><font color="Green" title="E14">A5</font></i>: 
<font color="Maroon" title="c1">Y</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a><br/>

<b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green" title="E15">A6</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<a NAME="E2:21_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font>]
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E1:21"><i><font color="Green" title="E10">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:21_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</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="c1">Y</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E2:21_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="eqrel_1.html#E1:21_1_1"><i><font color="Green" title="E15">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:21_1"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c1">Y</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a><br/><b>let </b><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:21_1_2"/><i><font color="Green" title="E16">A8</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">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="c1">Y</font> )
 ;<br/>

<a NAME="E2:21_1_2"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>]
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E2:21_1"><i><font color="Green" title="E13">A4</font></i></a></i>;<br/>
<a NAME="E3:21_1_2"/><b>then </b>
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font>] )
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E2:21"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="eqrel_1.html#E1:21_1_2"><i><font color="Green" title="E16">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:21_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</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="c1">Y</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E2:21_1"><i><font color="Green" title="E13">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:21_1"/><i><font color="Green" title="E16">A9</font></i>: 
<font color="Maroon" title="c1">Y</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a><br/><b>let </b><font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:21_1_3"/><i><font color="Green" title="E17">A10</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">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="c1">Y</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</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="c1">Y</font> )
 ;<br/>

<a NAME="E2:21_1_3"/><b>then </b><i><font color="Green" title="E18">A11</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>]
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E2:21_1"><i><font color="Green" title="E13">A4</font></i></a></i>;<br/>
<a NAME="E3:21_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>]
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E2:21_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="eqrel_1.html#E1:21_1_3"><i><font color="Green" title="E17">A10</font></i></a></i>;<br/>
<a NAME="E4:21_1_3"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font>]
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E3:21"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="eqrel_1.html#E2:21_1_3"><i><font color="Green" title="E18">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:21_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</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="c1">Y</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E2:21_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="eqrel_1.html#E1:21_1_3"><i><font color="Green" title="E17">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:21_1"/><i><font color="Green" title="E17">A12</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E3:21_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="orders_1.html#T98" title="ORDERS_1:th.98">ORDERS_1:98</a></i>;<br/>
<a NAME="E7:21_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E3:21_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="orders_1.html#T98" title="ORDERS_1:th.98">ORDERS_1:98</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">R1</font> = <font color="Maroon" title="c1">Y</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="eqrel_1.html#E4:21_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="eqrel_1.html#E5:21_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="eqrel_1.html#E6:21_1"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a>, <a class="ref" href="relat_2.html#D11" title="RELAT_2:def.11">RELAT_2:def 11</a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">R1</font>
;<br/>

<b>thus </b><a NAME="E9:21_1"/>
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="c2">R1</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>by </b><i><a class="txt" href="eqrel_1.html#E2:21_1"><i><font color="Green" title="E13">A4</font></i></a></i>;<br/>


</div>
