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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:2_1"/><i><font color="Green">E1</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/>

<a NAME="E2:2_1"/><i><font color="Green">E2</font></i>: 
 <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_1_1"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:2_1_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:2_1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E1:2_1_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="waybel_0.html#D15">WAYBEL_0:def 15</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:2_1_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E8:2_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E5:2_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E10:2_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E11:2_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>]
 <b>by </b><i><a class="txt" href="waybel31.html#E1:2_1"><i><font color="Green">E1</font></i></a>, <a class="txt" href="waybel31.html#E8:2_1_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E12:2_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="waybel31.html#E4:2_1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="waybel31.html#E7:2_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="waybel31.html#E10:2_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="waybel_0.html#D15">WAYBEL_0:def 15</a></i>;<br/>
<a NAME="E13:2_1_1"/>
 <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel31.html#E11:2_1_1"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:2_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel31.html#E12:2_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:2_1"/>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_1_2"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:2_1_2"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E6:2_1_2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E7:2_1_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="waybel31.html#E4:2_1_2"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="waybel31.html#E3:2_1_2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="waybel31.html#E6:2_1_2"><i><font color="Green">E5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E10:2_1_2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E11:2_1_2"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E3:2_1_2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="waybel31.html#E6:2_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="waybel_0.html#D15">WAYBEL_0:def 15</a></i>;<br/>
<a NAME="E12:2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel31.html#E1:2_1"><i><font color="Green">E1</font></i></a>, <a class="txt" href="waybel31.html#E7:2_1_2"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E13:2_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel31.html#E11:2_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E14:2_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel31.html#E10:2_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="waybel_0.html#D15">WAYBEL_0:def 15</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:2_1"/>
 <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel31.html#E2:2_1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
