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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:94_1"/><i><font color="Green" title="E59">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="waybel_1.html#NR11" title="WAYBEL_1:NR.11">is_a_closure_operator</a> 
 ;<br/><b>thus </b><a NAME="E2:94_1"/>
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D18" title="YELLOW_0:def.18">YELLOW_0:def 18</a><br/>

<b>assume </b><a NAME="E1:94_1_1"/><i><font color="Green" title="E60">A2</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font>
 ;<br/>

<a NAME="E2:94_1_1"/><i><font color="Green" title="E61">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">X'</font> = <font color="Maroon" title="c3">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:94_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">X'</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">X'</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:94_1"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="waybel_1.html#E1:94_1_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="waybel_1.html#E2:94_1_1"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="waybel_1.html#T30" target="_self" title="WAYBEL_1:th.30">Th30</a></i>;<br/>
<b>hence </b><a NAME="E5:94_1_1"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:94_1_1"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>


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

<b>assume </b><a NAME="E2:94"/><i><font color="Green" title="E59">A4</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="waybel_1.html#NR12" title="WAYBEL_1:NR.12">is_a_kernel_operator</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D19" title="YELLOW_0:def.19">YELLOW_0:def 19</a><br/>

<b>assume </b><a NAME="E3:94"/><i><font color="Green" title="E60">A5</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font>
 ;<br/>

<a NAME="E4:94"/><i><font color="Green" title="E61">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">X'</font> = <font color="Maroon" title="c3">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:94"/>
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X'</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X'</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E2:94"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="waybel_1.html#E3:94"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="waybel_1.html#E4:94"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="waybel_1.html#T31" target="_self" title="WAYBEL_1:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E7:94"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_1.html#E4:94"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>


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