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<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>for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> holds <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> iff <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> );<br/>
<b>consider </b><font color="Maroon">R</font> being    <a href="relset_1.html#M2">Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:5_1_1"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>] )
 <b>from </b><i><a class="ref" href="relset_1.html#S2">RELSET_1:sch 2</a>();<br/></i>
<a NAME="E3:5_1_1"/>
<font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:5_1_1_1"/><i><font color="Green">E19</font></i>: 
<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">T</font>
 ;<br/>

<a NAME="E2:5_1_1_1"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> holds <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> )
 
;<br/>
<b>hence </b><a NAME="E3:5_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:5_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<a NAME="E5:5_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R3">is_symmetric_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2"><b>proof </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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a><br/>

<b>assume </b><a NAME="E1:5_1_1_2"/><i><font color="Green">E37</font></i>: 
( <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">T</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<a NAME="E2:5_1_1_2"/><b>then </b>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> holds <br/>( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:5_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:5_1_1"/>
<font color="Maroon">R</font> <a href="relat_2.html#R8">is_transitive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_3"><b>proof </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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a><br/>

<b>assume </b><a NAME="E1:5_1_1_3"/><i><font color="Green">E39</font></i>: 
( <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">T</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:5_1_1_3"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> holds <br/>( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> iff <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:5_1_1_3_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:5_1_1_3_1"/><b>then </b>
( ( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) &amp; ( <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) &amp; ( <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) &amp; ( <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:5_1_1_3_1"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> iff <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:5_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <b>by </b><i>, , <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a>, <a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon">R</font>
;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>

<b>thus </b><a NAME="E8:5_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> implies for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> holds <br/>( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> iff <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E9:5_1_1"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> holds <br/>( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> iff <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> )
 ;<br/>

<b>hence </b><a NAME="E10:5_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>


</div>
