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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>  implies ( <font color="Maroon" title="c1">L</font> is  <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a>  iff  for <font color="Olive" title="b1">K</font> being   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="lattice5.html#NM2" title="LATTICE5:NM.2">Sublattice</a> of <font color="Maroon" title="c1">L</font>  holds  not  <a href="yellow11.html#K2" title="YELLOW11:func.2">M_3</a> ,<font color="Olive" title="b1">K</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c1">L</font> is  <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is  <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a>  iff  for <font color="Olive" title="b1">K</font> being   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="lattice5.html#NM2" title="LATTICE5:NM.2">Sublattice</a> of <font color="Maroon" title="c1">L</font>  holds  not  <a href="yellow11.html#K2" title="YELLOW11:func.2">M_3</a> ,<font color="Olive" title="b1">K</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:20"/>
( <font color="Maroon" title="c1">L</font> is  <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a>  implies  for <font color="Olive" title="b1">K</font> being   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="lattice5.html#NM2" title="LATTICE5:NM.2">Sublattice</a> of <font color="Maroon" title="c1">L</font>  holds  not  <a href="yellow11.html#K2" title="YELLOW11:func.2">M_3</a> ,<font color="Olive" title="b1">K</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">K</font> being   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="lattice5.html#NM2" title="LATTICE5:NM.2">Sublattice</a> of <font color="Maroon" title="c1">L</font>  holds  not  <a href="yellow11.html#K2" title="YELLOW11:func.2">M_3</a> ,<font color="Olive" title="b1">K</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  ) implies <font color="Maroon" title="c1">L</font> is  <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_1"/>
<font color="Maroon" title="c1">L</font> is  <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="lattice5.html#NM2" title="LATTICE5:NM.2">Sublattice</a> of <font color="Maroon" title="c1">L</font>  holds  not  <a href="yellow11.html#K2" title="YELLOW11:func.2">M_3</a> ,<font color="Olive" title="b1">K</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> </span><br/>

<a NAME="E2:20_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font>, <font color="Olive" title="b5">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>,<font color="Olive" title="b5">e</font> <a href="zfmisc_1.html#R3" title="ZFMISC_1:pred.3">are_mutually_distinct</a>  or  not <font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> or  not <font color="Olive" title="b3">c</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> or  not <font color="Olive" title="b4">d</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">d</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b3">c</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b3">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="yellow11.html#E17"><span class="lab"><font color="Green" title="E12">Lm3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:20_1"/>
 for <font color="Olive" title="b1">K</font> being   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="lattice5.html#NM2" title="LATTICE5:NM.2">Sublattice</a> of <font color="Maroon" title="c1">L</font>  holds  not  <a href="yellow11.html#K2" title="YELLOW11:func.2">M_3</a> ,<font color="Olive" title="b1">K</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="yellow11.html#T10" target="_self" onmouseover="rs('yellow11/T10')" onmouseout="rh()">Th10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:20"/>
 for <font color="Olive" title="b1">K</font> being   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="lattice5.html#NM2" title="LATTICE5:NM.2">Sublattice</a> of <font color="Maroon" title="c1">L</font>  holds  not  <a href="yellow11.html#K2" title="YELLOW11:func.2">M_3</a> ,<font color="Olive" title="b1">K</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">L</font> is  <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> </span><br/>

<a NAME="E4:20"/><span class="kw">then </span>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font>, <font color="Olive" title="b5">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>,<font color="Olive" title="b5">e</font> <a href="zfmisc_1.html#R3" title="ZFMISC_1:pred.3">are_mutually_distinct</a>  or  not <font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b1">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> or  not <font color="Olive" title="b3">c</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> or  not <font color="Olive" title="b4">d</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">d</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b3">c</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b2">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e</font> or  not <font color="Olive" title="b3">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow11.html#T10" target="_self" onmouseover="rs('yellow11/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:20"/>
<font color="Maroon" title="c1">L</font> is  <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="yellow11.html#E17"><span class="lab"><font color="Green" title="E12">Lm3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
