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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:63"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#NR10" target="_self">is_a_projection_operator</a> 
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green">E41</font></i>: 
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:63"/><i><font color="Green">E42</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:63"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D14" target="_self">WAYBEL_1:def 14</a><br/>

<a NAME="E5:63"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="quantal1.html#V11">idempotent</a>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:63"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel_1.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E6:63"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:63"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel_1.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E7:63"/>
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E2:63"><i><font color="Green">E41</font></i></a>, <a class="ref" href="yellow_2.html#T10">YELLOW_2:10</a></i>;<br/>
<a NAME="E8:63"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>;<br/>
<a NAME="E9:63"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:63_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:63_1"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E3:63"><i><font color="Green">E42</font></i></a>, <a class="ref" href="yellow_0.html#T31">YELLOW_0:31</a></i>;<br/>
<a NAME="E3:63_1"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:63_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E4:63_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E6:63"><i><font color="Green">E45</font></i></a>, <a class="ref" href="waybel_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E5:63_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E4:63"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel_1.html#E5:63"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel_1.html#E1:63_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel_1.html#E46"><i><font color="Green">Lemma38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:63"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E3:63"><i><font color="Green">E42</font></i></a>, <a class="ref" href="yellow_0.html#T31">YELLOW_0:31</a></i>;<br/>
<b>hence </b><a NAME="E11:63"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_1.html#E8:63"><i><font color="Green">E46</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div>
