<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c2">V</font> is <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a>
 ;<br/><b>let </b><font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">Y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span>;<br/><b>assume </b><a NAME="E2:22_1"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">Y</font>
 ;<br/><a NAME="E3:22_1"/>
<font color="Maroon" title="c3">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="waybel14.html#T18" target="_self" title="WAYBEL14:th.18">Th18</a></i>;<br/><a NAME="E4:22_1"/><b>then </b>
<font color="Maroon" title="c2">V</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">X</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E2:22_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E5:22_1"/><b>then </b>
( <font color="Maroon" title="c2">V</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">X</font> or <font color="Maroon" title="c2">V</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">Y</font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E1:22_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="waybel14.html#T14" target="_self" title="WAYBEL14:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E6:22_1"/>
( <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> or <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:22"/><i><font color="Green" title="E15">A3</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 <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span>  holds <br/>(  not <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">Y</font> or <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font> or <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">Y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span>;<br/><b>assume </b><a NAME="E1:22_2"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">X</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">Y</font>
 ;<br/><a NAME="E2:22_2"/>
<font color="Maroon" title="c3">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="waybel14.html#T18" target="_self" title="WAYBEL14:th.18">Th18</a></i>;<br/><a NAME="E3:22_2"/><b>then </b>
<font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E1:22_2"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E4:22_2"/><b>then </b>
( <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> or <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E2:22"><i><font color="Green" title="E15">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E5:22_2"/>
( <font color="Maroon" title="c2">V</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">X</font> or <font color="Maroon" title="c2">V</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">Y</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:22"/>
<font color="Maroon" title="c2">V</font> is <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a>
 <b>by </b><i><a class="ref" href="waybel14.html#T14" target="_self" title="WAYBEL14:th.14">Th14</a></i>;<br/>


</div>
