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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span><b> such that </b><a NAME="E1:3"/><i><font color="Green">E1</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D5">YELLOW_0:def 5</a><br/>

<a NAME="E2:3"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:3"/><i><font color="Green">E2</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow10.html#E4:3"><i><font color="Green">E2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="yellow10.html#E4:3"><i><font color="Green">E2</font></i></a></i>;<br/>
<a NAME="E7:3"/><i><font color="Green">E3</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<a NAME="E8:3"/><i><font color="Green">E4</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="yellow10.html#E1:3"><i><font color="Green">E1</font></i></a>, <a class="txt" href="yellow10.html#E4:3"><i><font color="Green">E2</font></i></a>, <a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<b>thus </b><a NAME="E9:3"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="yellow_0.html#V2">upper-bounded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D5">YELLOW_0:def 5</a><br/>

<b>thus </b><a NAME="E1:3_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow10.html#E7:3"><i><font color="Green">E3</font></i></a>, <a class="txt" href="yellow10.html#E8:3"><i><font color="Green">E4</font></i></a>, <a class="ref" href="yellow_3.html#T29">YELLOW_3:29</a></i>;<br/>


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

<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D5">YELLOW_0:def 5</a><br/>

<b>thus </b><a NAME="E10:3"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="yellow10.html#E7:3"><i><font color="Green">E3</font></i></a>, <a class="txt" href="yellow10.html#E8:3"><i><font color="Green">E4</font></i></a>, <a class="ref" href="yellow_3.html#T29">YELLOW_3:29</a></i>;<br/>


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