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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span> is  <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a>  &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">i</font> = <span class="p1">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/>
<div><span class="lab"><font color="Green" title="E59">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">I</font> be   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">I</font></span>)</span></span><br/><a NAME="E1:96_1"/>
<font color="Maroon" title="c3">I</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">I</font> = 
<span class="p1">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D3" title="YELLOW_2:def.3">YELLOW_2:def 3</a></span>
<br/>.= 
 <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D4" title="YELLOW_2:def.4">YELLOW_2:def 4</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:96"/>
( <span class="p1">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  &amp; <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is  <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T60" target="_self" title="WAYBEL_1:th.60">Th60</a>, <a class="ref" href="yellow_2.html#T14" title="YELLOW_2:th.14">YELLOW_2:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:96"/>
<span class="p1">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span> is  <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T41" target="_self" title="WAYBEL_1:th.41">Th41</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> </span><br/>

<span class="kw">take </span><font color="Maroon" title="c3">f</font> = <span class="p1">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_1.html#D8" target="_self" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> </span><br/>

<div><span class="lab"><font color="Green" title="E60">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">I</font> being  <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">I</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">I</font> be   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">I</font> implies <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">I</font> )</span><br/><span class="kw">assume </span><a NAME="E1:96_2"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">I</font></span><br/><span class="kw">hence </span><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">I</font> = 
<span class="p1">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">I</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_2"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">I</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D3" title="YELLOW_2:def.3">YELLOW_2:def 3</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:96"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T14" title="YELLOW_2:th.14">YELLOW_2:14</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E62">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="96_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> implies <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> ) &amp; ( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> implies <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> ) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c6">Ix</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:96_3"/><span class="lab"><font color="Green" title="E63">A6</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">Ix</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T12" title="YELLOW_2:th.12">YELLOW_2:12</a></span>;<br/><span class="kw">thus </span><a NAME="E3:96_3"/>
( <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> implies <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="waybel_1.html#D2" target="_self" title="WAYBEL_1:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> implies <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> )</span><br/><span class="kw">assume </span><a NAME="E4:96_3"/><span class="lab"><font color="Green" title="E64">A7</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font></span><br/><a NAME="E5:96_3"/>
( <font color="Maroon" title="c4">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">x9</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c8">y9</font> = <font color="Maroon" title="c5">y</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">Iy</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:96_3"/><span class="lab"><font color="Green" title="E65">A8</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">Iy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T12" title="YELLOW_2:th.12">YELLOW_2:12</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">Ix</font> = <font color="Maroon" title="c6">Ix</font>, <font color="Maroon" title="c11">Iy</font> = <font color="Maroon" title="c9">Iy</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">i1</font> =  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c10">Ix</font></span>)</span>, <font color="Maroon" title="c13">i2</font> =  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c11">Iy</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></span>;<br/><a NAME="E11:96_3"/><span class="lab"><font color="Green" title="E66">A9</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">Ix</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c10">Ix</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">Iy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c11">Iy</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E59">A1</font></span></a></span>;<br/><a NAME="E12:96_3"/><span class="lab"><font color="Green" title="E67">A10</font></span>: 
( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">x9</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c8">y9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96"><span class="lab"><font color="Green" title="E60">A2</font></span></a></span>;<br/><a NAME="E13:96_3"/>
(  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c10">Ix</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c10">Ix</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c11">Iy</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c11">Iy</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T34" title="WAYBEL_0:th.34">WAYBEL_0:34</a></span>;<br/><a NAME="E14:96_3"/><span class="kw">then </span>
 <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c10">Ix</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c11">Iy</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_3"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E2:96_3"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E8:96_3"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E11:96_3"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E12:96_3"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="ref" href="waybel_0.html#T21" title="WAYBEL_0:th.21">WAYBEL_0:21</a></span>;<br/><a NAME="E15:96_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">i1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E16:96_3"/>
<font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_3"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E8:96_3"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E11:96_3"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:96"/><span class="lab"><font color="Green" title="E63">A11</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:96_4"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</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 <font color="Maroon" title="c1">L</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:96_4"/>
<font color="Maroon" title="c4">x</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 <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<span class="lab"><font color="Green" title="E64">A12</font></span>: <span class="p1">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">x</font></span>)</span> = 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D3" title="YELLOW_2:def.3">YELLOW_2:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T34" title="WAYBEL_0:th.34">WAYBEL_0:34</a></span>
;<br/>
<a NAME="E5:96_4"/><span class="lab"><font color="Green" title="E65">A13</font></span>: 
 <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
 <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_4"><span class="lab"><font color="Green" title="E64">A12</font></span></a>, <a class="ref" href="yellow_2.html#D4" title="YELLOW_2:def.4">YELLOW_2:def 4</a></span>
;<br/>
<a NAME="E7:96_4"/><span class="kw">then </span>
 <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_4"><span class="lab"><font color="Green" title="E65">A13</font></span></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>;<br/>
<a NAME="E8:96_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A14</font></span>: 
 <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">x</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:96_4"><span class="lab"><font color="Green" title="E66">A14</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:96_4"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_4"><span class="lab"><font color="Green" title="E64">A12</font></span></a>, <a class="txt" href="#E8:96_4"><span class="lab"><font color="Green" title="E66">A14</font></span></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:96"/>
<font color="Maroon" title="c3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_1.html#D1" target="_self" title="WAYBEL_1:def.1">WAYBEL_1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:96_5"/><span class="lab"><font color="Green" title="E64">A15</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">Ix</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:96_5"/><span class="lab"><font color="Green" title="E65">A16</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">Ix</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T12" title="YELLOW_2:th.12">YELLOW_2:12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">Iy</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:96_5"/><span class="lab"><font color="Green" title="E66">A17</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">Iy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T12" title="YELLOW_2:th.12">YELLOW_2:12</a></span>;<br/>
<a NAME="E6:96_5"/>
( <font color="Maroon" title="c4">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">Ix</font> = <font color="Maroon" title="c6">Ix</font>, <font color="Maroon" title="c11">Iy</font> = <font color="Maroon" title="c7">Iy</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></span>;<br/>
<a NAME="E9:96_5"/><span class="lab"><font color="Green" title="E67">A18</font></span>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c10">Ix</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c10">Ix</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T34" title="WAYBEL_0:th.34">WAYBEL_0:34</a></span>;<br/>
<a NAME="E10:96_5"/><span class="lab"><font color="Green" title="E68">A19</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">Ix</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c10">Ix</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">Iy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c11">Iy</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E59">A1</font></span></a></span>;<br/>
<a NAME="E11:96_5"/>
( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c9">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96"><span class="lab"><font color="Green" title="E60">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:96_5"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_5"><span class="lab"><font color="Green" title="E64">A15</font></span></a>, <a class="txt" href="#E3:96_5"><span class="lab"><font color="Green" title="E65">A16</font></span></a>, <a class="txt" href="#E5:96_5"><span class="lab"><font color="Green" title="E66">A17</font></span></a>, <a class="txt" href="#E10:96_5"><span class="lab"><font color="Green" title="E68">A19</font></span></a>, <a class="txt" href="#E9:96_5"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="ref" href="waybel_0.html#T34" title="WAYBEL_0:th.34">WAYBEL_0:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:96"/>
<font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:96"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="txt" href="#E6:96"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
