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<span class="kw">set </span><font color="Maroon" title="c2">OL</font> =  <a href="openlatt.html#K6" title="OPENLATT:func.6">Open_setLatt</a> <font color="Maroon" title="c1">T</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">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="openlatt.html#K6" title="OPENLATT:func.6">Open_setLatt</a> <font color="Maroon" title="c1">T</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_0.html#D6" onmouseover="rs('filter_0/D6')" onmouseout="rh()">FILTER_0:def 6</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">  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 <span class="p1">(<span class="default"><a href="openlatt.html#K6" title="OPENLATT:func.6">Open_setLatt</a> <font color="Maroon" title="c1">T</font></span>)</span> st <br/>( <font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">q</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 <span class="p1">(<span class="default"><a href="openlatt.html#K6" title="OPENLATT:func.6">Open_setLatt</a> <font color="Maroon" title="c1">T</font></span>)</span>  holds <br/>(  not <font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">q</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">reconsider </span><font color="Maroon" title="c5">p9</font> = <font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c6">q9</font> = <font color="Maroon" title="c4">q</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="openlatt.html#K1" title="OPENLATT:func.1">Topology_of</a> <font color="Maroon" title="c1">T</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">r9</font> =  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p9</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">q9</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="openlatt.html#K1" title="OPENLATT:func.1">Topology_of</a> <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c7">r9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K6" title="OPENLATT:func.6">Open_setLatt</a> <font color="Maroon" title="c1">T</font></span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">r</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">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c8">r</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">q</font> &amp; (  for <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 <span class="p1">(<span class="default"><a href="openlatt.html#K6" title="OPENLATT:func.6">Open_setLatt</a> <font color="Maroon" title="c1">T</font></span>)</span>  holds <br/>(  not <font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">q</font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c8">r</font> ) ) )</span><br/>

<a NAME="E4:12_1_1"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p9</font> <a href="openlatt.html#K3" title="OPENLATT:func.3">/\</a> <font color="Maroon" title="c7">r9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="openlatt.html#D3" target="_self" onmouseover="rs('openlatt/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E5:12_1_1"/>
<font color="Maroon" title="c5">p9</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c8">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">q9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="openlatt.html#T1" target="_self" onmouseover="rs('openlatt/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:12_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_1_1"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="openlatt.html#T7" target="_self" onmouseover="rs('openlatt/T7')" onmouseout="rh()">Th7</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">  for <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 <span class="p1">(<span class="default"><a href="openlatt.html#K6" title="OPENLATT:func.6">Open_setLatt</a> <font color="Maroon" title="c1">T</font></span>)</span>  holds <br/>(  not <font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">q</font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c8">r</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">r1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K6" title="OPENLATT:func.6">Open_setLatt</a> <font color="Maroon" title="c1">T</font></span>)</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"> (  not <font color="Maroon" title="c3">p</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c9">r1</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">q</font> or <font color="Maroon" title="c9">r1</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c8">r</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">r2</font> = <font color="Maroon" title="c9">r1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="openlatt.html#K1" title="OPENLATT:func.1">Topology_of</a> <font color="Maroon" title="c1">T</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">r19</font> = <font color="Maroon" title="c10">r2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E9:12_1_1"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c11">r19</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
;<br/>
<span class="kw">assume </span><a NAME="E10:12_1_1"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q</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="c9">r1</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c8">r</font></span><br/>

<a NAME="E11:12_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p9</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">r19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="openlatt.html#D3" target="_self" onmouseover="rs('openlatt/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E12:12_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p9</font> <a href="openlatt.html#K3" title="OPENLATT:func.3">/\</a> <font color="Maroon" title="c10">r2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">q9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:12_1_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="openlatt.html#T7" target="_self" onmouseover="rs('openlatt/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<a NAME="E13:12_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">r19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">r9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_1_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="openlatt.html#T2" target="_self" onmouseover="rs('openlatt/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:12_1_1"/>
<font color="Maroon" title="c9">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="openlatt.html#T7" target="_self" onmouseover="rs('openlatt/T7')" onmouseout="rh()">Th7</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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