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<div class="add">

<b>let </b><font color="Maroon" title="c2">R1</font>, <font color="Maroon" title="c3">R2</font> be  <a href="waybel_9.html#V1" title="WAYBEL_9:attr.1">strict</a>  <a href="waybel_9.html#L1" title="WAYBEL_9:struct.1">TopRelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:19_1_2"/><i><font color="Green" title="E9">A3</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">R1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">R1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> #)
 <b>and </b><br/><a NAME="E2:19_1_2"/><i><font color="Green" title="E10">A4</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R1</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> iff  ex <font color="Olive" title="b3">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b3">Y</font> ) )
 <b>and </b><br/><a NAME="E3:19_1_2"/><i><font color="Green" title="E11">A5</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">R2</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">R2</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> #)
 <b>and </b><br/><a NAME="E4:19_1_2"/><i><font color="Green" title="E12">A6</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R2</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> iff  ex <font color="Olive" title="b3">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b3">Y</font> ) )
 ;<br/>

<a NAME="E5:19_1_2"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</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_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a><br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:19_1_2_1_1"/><i><font color="Green" title="E13">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">R1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R1</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">x2</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c9">y2</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R2</font> <b>by </b><i><a class="txt" href="waybel25.html#E1:19_1_2"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="waybel25.html#E3:19_1_2"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="waybel25.html#E1:19_1_2_1_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:19_1_2_1_1"/>
<font color="Maroon" title="c6">x1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y1</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E1:19_1_2_1_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E6:19_1_2_1_1"/><i><font color="Green" title="E14">A8</font></i>: 
( <font color="Maroon" title="c10">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c10">Y</font> )
 <b>by </b><i><a class="txt" href="waybel25.html#E2:19_1_2"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/><a NAME="E7:19_1_2_1_1"/>
<font color="Maroon" title="c8">x2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">y2</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E4:19_1_2"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="waybel25.html#E6:19_1_2_1_1"><i><font color="Green" title="E14">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E8:19_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R2</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:19_1_2_1"/><i><font color="Green" title="E13">A9</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">x2</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y2</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R2</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c9">y1</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R1</font> <b>by </b><i><a class="txt" href="waybel25.html#E1:19_1_2"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="waybel25.html#E3:19_1_2"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="waybel25.html#E2:19_1_2_1"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:19_1_2_1"/>
<font color="Maroon" title="c6">x2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y2</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E2:19_1_2_1"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E7:19_1_2_1"/><i><font color="Green" title="E14">A10</font></i>: 
( <font color="Maroon" title="c10">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">y2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c10">Y</font> )
 <b>by </b><i><a class="txt" href="waybel25.html#E4:19_1_2"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/>
<a NAME="E8:19_1_2_1"/>
<font color="Maroon" title="c8">x1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">y1</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E2:19_1_2"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="waybel25.html#E7:19_1_2_1"><i><font color="Green" title="E14">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:19_1_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">R1</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:19_1_2"/>
<font color="Maroon" title="c2">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R2</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E1:19_1_2"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="waybel25.html#E3:19_1_2"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>


</div>
