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<b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">p'</font>, <font color="Olive" title="b4">q'</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">p'</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q'</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">p'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">q'</font> )</span><br/><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p'</font>, <font color="Olive" title="b2">q'</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">p'</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q'</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">p'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">q'</font> )</span><br/><b>let </b><font color="Maroon" title="c4">p'</font>, <font color="Maroon" title="c5">q'</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">p'</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q'</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">p'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">q'</font> ) )</span><br/>







<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E8">A1</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p'</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">p'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">q'</font> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">p'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">q'</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"><b>assume </b><a NAME="E1:11_1"/><i><font color="Green" title="E9">A2</font></i>: 
<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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">p'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">q'</font></span><br/><font color="Maroon" title="c4">p'</font> <a href="openlatt.html#K2" title="OPENLATT:func.2">\/</a> <font color="Maroon" title="c5">q'</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>
<b>by </b><i><a class="txt" href="openlatt.html#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="openlatt.html#D2" target="_self" title="OPENLATT:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c5">q'</font>
<b>by </b><i><a class="txt" href="openlatt.html#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="openlatt.html#E1:11_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>
;<br/><b>hence </b><a NAME="E3:11_1"/>
<font color="Maroon" title="c4">p'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">q'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:11"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c4">p'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">q'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">p'</font> <a href="openlatt.html#K2" title="OPENLATT:func.2">\/</a> <font color="Maroon" title="c5">q'</font>
<b>by </b><i><a class="txt" href="openlatt.html#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="openlatt.html#D2" target="_self" title="OPENLATT:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c3">q</font>
<b>by </b><i><a class="txt" href="openlatt.html#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="openlatt.html#E3:11"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/>
<b>hence </b><a NAME="E5:11"/>
<font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font>
 <b>by </b><i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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