<?xml version="1.0"?>
<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="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <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">S</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">S</font>  st  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b2">X</font>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b2">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 <font color="Olive" title="b1">S</font> holds <br/>(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b2">X</font>,<font color="Olive" title="b1">S</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b2">X</font>,<font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b2">X</font>,<font color="Maroon" title="c1">L</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</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"> for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>  st  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">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 <font color="Maroon" title="c2">S</font> holds <br/>(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c2">S</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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> &amp;  <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 <font color="Maroon" title="c2">S</font> implies (  <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="c2">S</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">S</font> <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="c3">X</font>,<font color="Maroon" title="c1">L</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:92"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E2:92"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
 <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 <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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="c2">S</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">S</font> <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="c3">X</font>,<font color="Maroon" title="c1">L</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">a</font> =  <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> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E56">A2</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E57">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:92_1"/>
 <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="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="yellow_0.html#D9" target="_self" title="YELLOW_0:def.9">Def9</a></span>;<br/><span class="kw">hence </span><a NAME="E2:92_1"/>
<font color="Maroon" title="c4">a</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T62" target="_self" title="YELLOW_0:th.62">Th62</a></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">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</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="c5">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c5">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">a</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">b9</font> = <font color="Maroon" title="c5">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T59" target="_self" title="YELLOW_0:th.59">Th59</a></span>;<br/><span class="kw">assume </span><a NAME="E4:92_1"/>
<font color="Maroon" title="c5">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</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="c5">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">a</font></span><br/><a NAME="E5:92_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">b9</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T63" target="_self" title="YELLOW_0:th.63">Th63</a></span>;<br/><a NAME="E6:92_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">b9</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="yellow_0.html#D9" target="_self" title="YELLOW_0:def.9">Def9</a></span>;<br/><span class="kw">hence </span><a NAME="E7:92_1"/>
<font color="Maroon" title="c5">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T61" target="_self" title="YELLOW_0:th.61">Th61</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">consider </span><font color="Maroon" title="c5">a9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E6:92"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c5">a9</font>
 <span class="kw">and </span><br/><a NAME="E7:92"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">a9</font>
 <span class="kw">and </span><br/><a NAME="E8:92"/>
for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">c</font> &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b2">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">c</font> ) holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="yellow_0.html#D7" target="_self" title="YELLOW_0:def.7">Def7</a></span>;<br/>
<a NAME="E9:92"/><span class="lab"><font color="Green" title="E60">A6</font></span>: 
<font color="Maroon" title="c5">a9</font> <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="c3">X</font>,<font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E6:92"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="#E7:92"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="ref" href="yellow_0.html#D9" target="_self" title="YELLOW_0:def.9">Def9</a></span>;<br/>
<span class="kw">thus </span><a NAME="E10:92"/>
 <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="c2">S</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">S</font> <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="c3">X</font>,<font color="Maroon" title="c1">L</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c4">a</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_0.html#D7" target="_self" title="YELLOW_0:def.7">YELLOW_0:def 7</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">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c4">a</font> &amp; ( for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">a</font> ) &amp; ( for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">c</font> &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b2">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">c</font> ) holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:92_2"/>
( <font color="Maroon" title="c4">a</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font> &amp; ( for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E57">A3</font></span></a></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">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">c</font> &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b2">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">c</font> ) holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</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">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">c</font> &amp; ( for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">c</font> ) implies <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">c9</font> = <font color="Maroon" title="c6">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T59" target="_self" title="YELLOW_0:th.59">Th59</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:92_2"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">c</font> ) or <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )</span><br/>

<a NAME="E4:92_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A7</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c7">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T63" target="_self" title="YELLOW_0:th.63">Th63</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:92_2"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">c</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="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font></span><br/>

<a NAME="E6:92_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A8</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E57">A3</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c8">b</font> implies <font color="Maroon" title="c8">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">c9</font> )</span><br/><span class="kw">assume </span><a NAME="E1:92_2_1"/>
<font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c8">b</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="c8">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">c9</font></span><br/><a NAME="E2:92_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A9</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:92"><span class="lab"><font color="Green" title="E59">A5</font></span></a></span>;<br/><a NAME="E3:92_2_1"/>
<font color="Maroon" title="c5">a9</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:92"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="txt" href="#E6:92_2"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="ref" href="yellow_0.html#T60" target="_self" title="YELLOW_0:th.60">Th60</a></span>;<br/><span class="kw">hence </span><a NAME="E4:92_2_1"/>
<font color="Maroon" title="c8">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_2_1"><span class="lab"><font color="Green" title="E63">A9</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</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="E8:92_2"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E4:92_2"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="ref" href="yellow_0.html#D9" target="_self" title="YELLOW_0:def.9">Def9</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="E11:92"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">S</font> <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="c3">X</font>,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="yellow_0.html#D9" target="_self" title="YELLOW_0:def.9">Def9</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>
