<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</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="c1">L</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>  iff  ex <font color="Olive" title="b1">C</font> being  <a href="conlat_1.html#NM1" title="CONLAT_1:NM.1">FormalContext</a> st  <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">L</font> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</a>  )</span><br/>

<div><span class="kw">hereby </span> <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">C</font> being  <a href="conlat_1.html#NM1" title="CONLAT_1:NM.1">FormalContext</a> st  <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">L</font> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</a>  implies <font color="Maroon" title="c1">L</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:25_1"/>
<font color="Maroon" title="c1">L</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> 
 ; <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">C</font> being  <a href="conlat_1.html#NM1" title="CONLAT_1:NM.1">FormalContext</a> st  <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">L</font> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</a> </span><br/><a NAME="E2:25_1"/><span class="kw">then </span>
 <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K6" title="CONLAT_2:func.6">Context</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_2.html#T15" target="_self" title="CONLAT_2:th.15">Th15</a></span>;<br/><span class="kw">hence </span><a NAME="E3:25_1"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="conlat_1.html#NM1" title="CONLAT_1:NM.1">FormalContext</a> st  <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">L</font> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</a> 
 ; <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">given </span><font color="Maroon" title="c2">C</font> being   <a href="conlat_1.html#NM1" title="CONLAT_1:NM.1">FormalContext</a><span class="kw"> such that </span><a NAME="E3:25"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">L</font> <a href="filter_1.html#R1" title="FILTER_1:pred.1">are_isomorphic</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="c1">L</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> </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="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="vectsp_8.html#D6" title="VECTSP_8:def.6">VECTSP_8:def 6</a> <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">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font> or <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">f</font> being    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c1">L</font>, <a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font><span class="kw"> such that </span><br/><a NAME="E5:25"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="lattice4.html#D5" title="LATTICE4:def.5">LATTICE4:def 5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">FX</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> ;<br/>
<a NAME="E6:25"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  <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 <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="c6">x</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="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  or <font color="Maroon" title="c6">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="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_2"/>
<font color="Maroon" title="c6">x</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="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </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="c6">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="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</span></span><br/>

<a NAME="E2:25_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:25_2"/>
<font color="Maroon" title="c6">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="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">FX</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">Fa</font> =  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c6">FX</font>,<span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">a</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="E9:25"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c6">FX</font>,<span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="lattice4.html#T9" title="LATTICE4:th.9">LATTICE4:9</a></span>;<br/>
<a NAME="E10:25"/><span class="lab"><font color="Green" title="E19">A4</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="Olive" title="b1">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">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="c8">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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="c9">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c9">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_3"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c9">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_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="c9">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">a</font></span><br/>

<a NAME="E2:25_3"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c6">FX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</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="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FX</font> or <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c10">q</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_3_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FX</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">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c10">q</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">y</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="E3:25_3_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font>
 <span class="kw">and </span><br/><a NAME="E4:25_3_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>
<a NAME="E5:25_3_1"/>
<font color="Maroon" title="c9">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_3"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E4:25_3_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:25_3_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c10">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:25_3_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="lattice4.html#T8" title="LATTICE4:th.8">LATTICE4:8</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="E3:25_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:25"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:25_3"/>
<font color="Maroon" title="c9">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="lattice4.html#T8" title="LATTICE4:th.8">LATTICE4:8</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="E11:25"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c6">FX</font>,<span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c6">FX</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></span>;<br/>
<a NAME="E12:25"/>
<font color="Maroon" title="c8">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</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="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or <font color="Maroon" title="c8">a</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c9">q</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_4"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c8">a</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c9">q</font></span><br/>

<a NAME="E2:25_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">FX</font>
 
;<br/>
<a NAME="E3:25_4"/><span class="kw">then </span>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c6">FX</font>,<span class="p1">(<span class="default"><a href="conlat_1.html#K11" title="CONLAT_1:func.11">ConceptLattice</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:25"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:25_4"/>
<font color="Maroon" title="c8">a</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c9">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E9:25"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="lattice4.html#T8" title="LATTICE4:th.8">LATTICE4:8</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="E13:25"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font> or <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:25"><span class="lab"><font color="Green" title="E19">A4</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"> verum</span><br/>


</div>
