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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">L</font>;<br/>



<b>set </b><font color="Maroon">f</font> =  <a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span></span>)</span>;<br/><b>assume </b><a NAME="E1:95_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><b>reconsider </b><font color="Maroon">I</font> = <font color="Maroon">x</font>, <font color="Maroon">J</font> = <font color="Maroon">y</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/><b>consider </b><font color="Maroon">K1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:95_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font>
 <b>and </b><br/><a NAME="E5:95_1"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">K1</font>
 <b>by </b><i><a class="ref" href="waybel23.html#T3" target="_self">Th3</a></i>;<br/><b>consider </b><font color="Maroon">K2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:95_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">J</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K2</font>
 <b>and </b><br/><a NAME="E8:95_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">K2</font>
 <b>by </b><i><a class="ref" href="waybel23.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E9:95_1"/>
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">J</font>
 <b>by </b><i>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E10:95_1"/><b>then </b>
 <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">K1</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">K2</font>
 <b>by </b><i>, , <a class="ref" href="yellow_3.html#T6">YELLOW_3:6</a></i>;<br/><b>hence </b><a NAME="E11:95_1"/>
<span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:95"/>
 <a href="waybel23.html#K4" target="_self">idsMap</a> <font color="Maroon">S</font> is <a href="orders_3.html#V5">monotone</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/>


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