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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="orders_2.html#V3">transitive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V1">with_suprema</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:17"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:17_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:17_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:17_1"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:17_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E7:17_1"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="yellow_4.html#E4:17_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E8:17_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_4.html#E4:17_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="yellow_4.html#E7:17_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E9:17_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_4.html#E3:17_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="yellow_4.html#E6:17_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/>
<b>hence </b><a NAME="E10:17_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_4.html#E7:17_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="yellow_4.html#E8:17_1"><i><font color="Green">E9</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>5</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="E2:17"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:17"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E5:17"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:17"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E8:17"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</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="yellow_4.html#E5:17"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E9:17"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_4.html#E5:17"><i><font color="Green">E6</font></i></a>, <a class="txt" href="yellow_4.html#E8:17"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E10:17"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_4.html#E4:17"><i><font color="Green">E5</font></i></a>, <a class="txt" href="yellow_4.html#E7:17"><i><font color="Green">E7</font></i></a>, <a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/>
<b>hence </b><a NAME="E11:17"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="yellow_4.html#E8:17"><i><font color="Green">E8</font></i></a>, <a class="txt" href="yellow_4.html#E9:17"><i><font color="Green">E9</font></i></a></i>;<br/>


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