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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a><br/> for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">B</font> holds <br/> ( <font color="Maroon" title="c1">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b2">a</font> iff <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b3">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b3">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">B</font> : <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span>  <a href="lattice3.html#NR7" title="LATTICE3:NR.7">is_greater_than</a> <font color="Olive" title="b2">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">B</font> be   <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> holds <br/> ( <font color="Maroon" title="c1">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b1">a</font> iff <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> : <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span>  <a href="lattice3.html#NR7" title="LATTICE3:NR.7">is_greater_than</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  )</span><br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c3">a</font> iff <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span>  <a href="lattice3.html#NR7" title="LATTICE3:NR.7">is_greater_than</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  )</span><br/>





<b>set </b><font color="Maroon" title="c4">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:54"/>
( <font color="Maroon" title="c1">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c3">a</font> implies <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span>  <a href="lattice3.html#NR7" title="LATTICE3:NR.7">is_greater_than</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span>  <a href="lattice3.html#NR7" title="LATTICE3:NR.7">is_greater_than</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  implies <font color="Maroon" title="c1">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c3">a</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1"/><i><font color="Green" title="E30">A1</font></i>: 
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">B</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" target="_self" title="LATTICE3:def.17">LATTICE3:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span>  <a href="lattice3.html#NR7" title="LATTICE3:NR.7">is_greater_than</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span><br/>

<b>let </b><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">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" target="_self" title="LATTICE3:def.16">LATTICE3:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span>  implies <font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">b</font> )</span><br/>

<b>assume </b><a NAME="E2:54_1"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">b</font></span><br/>

<b>then consider </b><font color="Maroon" title="c6">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E4:54_1"/><i><font color="Green" title="E31">A2</font></i>: 
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>
<b>thus </b><a NAME="E5:54_1"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="lattice3.html#E1:54_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="lattice3.html#E4:54_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="lattices.html#T53" title="LATTICES:th.53">LATTICES:53</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:54"/><i><font color="Green" title="E30">A3</font></i>: 
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">B</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> : <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span>  holds <br/><font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" target="_self" title="LATTICE3:def.16">LATTICE3:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c3">a</font></span><br/>

<b>let </b><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">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" target="_self" title="LATTICE3:def.17">LATTICE3:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c5">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c3">a</font> )</span><br/>

<b>assume </b><a NAME="E3:54"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c3">a</font></span><br/>

<a NAME="E4:54"/><b>then </b>
<font color="Maroon" title="c5">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">B</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span> 
 
;<br/>
<a NAME="E5:54"/><b>then </b>
( <font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 
<b>by </b><i><a class="txt" href="lattice3.html#E2:54"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="lattices.html#T49" title="LATTICES:th.49">LATTICES:49</a></i>;<br/>
<b>hence </b><a NAME="E6:54"/>
<font color="Maroon" title="c5">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="ref" href="lattices.html#T53" title="LATTICES:th.53">LATTICES:53</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
