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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_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">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font><br/>  for <font color="Olive" title="b3">DI</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b2">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">DI</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> holds <br/><font color="Olive" title="b1">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Olive" title="b2">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Olive" title="b3">DI</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font>; <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">DI</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>  st <font color="Maroon" title="c3">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">DI</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> holds <br/><font color="Maroon" title="c2">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c3">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Olive" title="b1">DI</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">DI</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>; <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="c3">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">DI</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> implies <font color="Maroon" title="c2">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c3">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">DI</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> )</span><br/>







<span class="kw">assume </span><a NAME="E1:54"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c3">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">DI</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</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"> <font color="Maroon" title="c2">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c3">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">DI</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span><br/>

<span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">DI</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> = 
<span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">DI</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T34" target="_self" onmouseover="rs('filter_0/T34')" onmouseout="rh()">Th34</a></span>
<br/>.= 
<span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">DI</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T34" target="_self" onmouseover="rs('filter_0/T34')" onmouseout="rh()">Th34</a></span>
<br/>.= 
<span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">DI</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">i</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T24" target="_self" onmouseover="rs('filter_0/T24')" onmouseout="rh()">Th24</a></span>
<br/>.= 
<span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">DI</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">i</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T38" target="_self" onmouseover="rs('filter_0/T38')" onmouseout="rh()">Th38</a></span>
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">i1</font>, <font color="Maroon" title="c6">i2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font><span class="kw"> such that </span><br/><a NAME="E4:54"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">i2</font>
 <span class="kw">and </span><br/><a NAME="E5:54"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<font color="Maroon" title="c5">i1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">DI</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 <span class="kw">and </span><br/><a NAME="E6:54"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<font color="Maroon" title="c6">i2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">i</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a></span>;<br/>
<a NAME="E7:54"/>
<font color="Maroon" title="c2">i</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="filter_0.html#T15" target="_self" onmouseover="rs('filter_0/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<a NAME="E8:54"/><span class="kw">then </span>
<font color="Maroon" title="c5">i1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">i</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">i1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T9" onmouseover="rs('lattices/T9')" onmouseout="rh()">LATTICES:9</a></span>;<br/>
<a NAME="E9:54"/><span class="kw">then </span>
<font color="Maroon" title="c5">i1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c3">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="filter_0.html#D7" target="_self" onmouseover="rs('filter_0/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:54"/>
<font color="Maroon" title="c2">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c3">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">DI</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="filter_0.html#T9" target="_self" onmouseover="rs('filter_0/T9')" onmouseout="rh()">Th9</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/>


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