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<div class="add">

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="yellow_0.html#V2">upper-bounded</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>



<b>set </b><font color="Maroon">L</font> =  <a href="yellow_1.html#K7">MonMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">T</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K7">MonMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <b>by </b><i><a class="ref" href="waybel10.html#T10">WAYBEL10:10</a></i>;<br/>
<a NAME="E2:5"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">f</font> <a href="lattice3.html#R1">is_&lt;=_than</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="yellow_0.html#T6">YELLOW_0:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> ;<br/>
<a NAME="E4:5"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K7">MonMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>  st <font color="Olive">b</font> <a href="lattice3.html#R1">is_&lt;=_than</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">f</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K7">MonMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon">b</font> <a href="lattice3.html#R1">is_&lt;=_than</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> <b>by </b><i><a class="ref" href="waybel10.html#T10">WAYBEL10:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b''</font> = <font color="Maroon">b'</font>, <font color="Maroon">f''</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/>
<a NAME="E4:5_1"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>

<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="borsuk_1.html#D2">BORSUK_1:def 2</a></i>
<br/>.= 
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E2:5_1_1"/>
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:5_1"/><b>then </b>
<font color="Maroon">f'</font> <a href="yellow_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b'</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T10">YELLOW_2:10</a></i>;<br/>
<a NAME="E6:5_1"/><b>then </b>
<font color="Maroon">f''</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b''</font>
 
<b>by </b><i><a class="ref" href="waybel10.html#T12">WAYBEL10:12</a></i>;<br/>
<b>hence </b><a NAME="E7:5_1"/>
<font color="Maroon">f</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:5"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="yellow_1.html#K7">MonMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T31">YELLOW_0:31</a></i>;<br/>
<b>hence </b><a NAME="E6:5"/>
 <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K7">MonMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D12">YELLOW_0:def 12</a></i>;<br/>


</div>
