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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">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> <a href="lattice8.html#R1" title="LATTICE8:pred.1">has_a_representation_of_type&lt;=</a> 2 implies <font color="Maroon" title="c1">L</font> is  <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:15"/>
<font color="Maroon" title="c1">L</font> <a href="lattice8.html#R1" title="LATTICE8:pred.1">has_a_representation_of_type&lt;=</a> 2
 ; <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="yellow11.html#V1" title="YELLOW11:attr.1">modular</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">A</font> being  non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c3">f</font> being   <a href="lattice5.html#NM1" title="LATTICE5:NM.1">Homomorphism</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c2">A</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:15"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E4:15"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
(  <a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c3">f</font> is  <a href="lattice8.html#V1" title="LATTICE8:attr.1">finitely_typed</a>  &amp;  ex <font color="Olive" title="b1">e</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b1">e</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="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">A</font> ) &amp;  <a href="lattice5.html#K3" title="LATTICE5:func.3">type_of</a> <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#D3" target="_self" title="LATTICE8:def.3">Def3</a></span>;<br/>
<a NAME="E5:15"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">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 <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/><span class="p1">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a>, <a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></span>;<br/>
<a NAME="E6:15"/>
(  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="lattice8.html#T7" target="_self" title="LATTICE8:th.7">Th7</a>, <a class="ref" href="waybel_1.html#T32" title="WAYBEL_1:th.32">WAYBEL_1:32</a></span>;<br/>
<a NAME="E7:15"/><span class="kw">then </span>
 <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <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="#E5:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></span>;<br/>
<a NAME="E8:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<font color="Maroon" title="c1">L</font>, <a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c3">f</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></span>;<br/>
<a NAME="E9:15"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
 <a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c3">f</font> is   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice8.html#T6" target="_self" title="LATTICE8:th.6">Th6</a></span>;<br/>
<a NAME="E10:15"/>
 <a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c3">f</font> is  <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="lattice8.html#T8" target="_self" title="LATTICE8:th.8">Th8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:15"/>
<font color="Maroon" title="c1">L</font> is  <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="lattice8.html#T5" target="_self" title="LATTICE8:th.5">Th5</a>, <a class="ref" href="waybel_1.html#T7" title="WAYBEL_1:th.7">WAYBEL_1:7</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/>


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