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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</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">p</font>, <font color="Olive" title="b2">q</font> being   <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/>  for <font color="Olive" title="b3">p9</font>, <font color="Olive" title="b4">q9</font> being    <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>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p9</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q9</font> holds <br/>( <font color="Olive" title="b1">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q</font> iff <font color="Olive" title="b3">p9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">q9</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">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>; <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">p9</font>, <font color="Olive" title="b2">q9</font> being    <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>  st <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p9</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q9</font> holds <br/>( <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> iff <font color="Olive" title="b1">p9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">q9</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">p9</font>, <font color="Maroon" title="c5">q9</font> be    <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>; <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">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p9</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q9</font> implies ( <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> iff <font color="Maroon" title="c4">p9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">q9</font> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:11"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p9</font>
 <span class="kw">and </span><br/><a NAME="E2:11"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q9</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="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> iff <font color="Maroon" title="c4">p9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">q9</font> )</span><br/>

<div><span class="kw">hereby </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="c4">p9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">q9</font> implies <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:11_1"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">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="c4">p9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">q9</font></span><br/><font color="Maroon" title="c4">p9</font> <a href="openlatt.html#K2" title="OPENLATT:func.2">\/</a> <font color="Maroon" title="c5">q9</font> = 
<font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="openlatt.html#D2" target="_self" onmouseover="rs('openlatt/D2')" onmouseout="rh()">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c5">q9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E3:11_1"/>
<font color="Maroon" title="c4">p9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</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="E4:11"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c4">p9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">q9</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="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font></span><br/>

<font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">q</font> = 
<font color="Maroon" title="c4">p9</font> <a href="openlatt.html#K2" title="OPENLATT:func.2">\/</a> <font color="Maroon" title="c5">q9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="openlatt.html#D2" target="_self" onmouseover="rs('openlatt/D2')" onmouseout="rh()">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c3">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:11"/>
<font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">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"> verum</span><br/>


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