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<b>let </b><font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c2">T2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</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> being   <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c1">T1</font>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font> &amp; <font color="Olive" title="b1">P</font> is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c2">T2</font> holds <br/><font color="Maroon" title="c1">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T2</font></span><br/><b>let </b><font color="Maroon" title="c3">P</font> be    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c1">T1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font> &amp; <font color="Maroon" title="c3">P</font> is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c2">T2</font> implies <font color="Maroon" title="c1">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T2</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green" title="E20">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c3">P</font> is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c2">T2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T2</font></span><br/>

<b>consider </b><font color="Maroon" title="c4">B1</font> being    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T1</font><b> such that </b><br/><a NAME="E4:23"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c4">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="ref" href="cantor_1.html#D5" target="_self" title="CANTOR_1:def.5">Def5</a></i>;<br/>
<a NAME="E5:23"/><i><font color="Green" title="E23">A4</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c4">B1</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2" target="_self" title="CANTOR_1:def.2">Def2</a></i>;<br/>
<a NAME="E6:23"/>
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c4">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:23"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="cantor_1.html#T9" target="_self" title="CANTOR_1:th.9">Th9</a></i>;<br/>
<a NAME="E7:23"/><b>then </b><i><font color="Green" title="E24">A5</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:23"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">P'</font> = <font color="Maroon" title="c3">P</font> as    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c2">T2</font> <b>by </b><i><a class="txt" href="#E2:23"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">B2</font> being    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c2">T2</font><b> such that </b><br/><a NAME="E10:23"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c6">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c5">P'</font>
 <b>by </b><i><a class="ref" href="cantor_1.html#D5" target="_self" title="CANTOR_1:def.5">Def5</a></i>;<br/>
<a NAME="E11:23"/><i><font color="Green" title="E26">A7</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c6">B2</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2" target="_self" title="CANTOR_1:def.2">Def2</a></i>;<br/>
<a NAME="E12:23"/><i><font color="Green" title="E27">A8</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T7" target="_self" title="CANTOR_1:th.7">Th7</a></i>;<br/>
<a NAME="E13:23"/><i><font color="Green" title="E28">A9</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T7" target="_self" title="CANTOR_1:th.7">Th7</a></i>;<br/>
<a NAME="E14:23"/>
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c6">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c5">P'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E10:23"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="cantor_1.html#T9" target="_self" title="CANTOR_1:th.9">Th9</a></i>;<br/>
<a NAME="E15:23"/><b>then </b><i><font color="Green" title="E29">A10</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c5">P'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E11:23"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:23"/>
<font color="Maroon" title="c5">P'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D5" target="_self" title="CANTOR_1:def.5">Def5</a></i>;<br/>
<a NAME="E17:23"/><b>then </b>
 <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c5">P'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T16" target="_self" title="CANTOR_1:th.16">Th16</a></i>;<br/>
<a NAME="E18:23"/><b>then </b>
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c5">P'</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T9" target="_self" title="CANTOR_1:th.9">Th9</a></i>;<br/>
<a NAME="E19:23"/><b>then </b><i><font color="Green" title="E30">A11</font></i>: 
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c5">P'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</font>
 
<b>by </b><i><a class="txt" href="#E12:23"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="#E15:23"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E20:23"/>
<font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T1</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D5" target="_self" title="CANTOR_1:def.5">Def5</a></i>;<br/>
<a NAME="E21:23"/><b>then </b>
 <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T1</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T16" target="_self" title="CANTOR_1:th.16">Th16</a></i>;<br/>
<a NAME="E22:23"/><b>then </b>
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T9" target="_self" title="CANTOR_1:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E23:23"/>
<font color="Maroon" title="c1">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T2</font>
 <b>by </b><i><a class="txt" href="#E1:23"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E7:23"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="#E13:23"><i><font color="Green" title="E28">A9</font></i></a>, <a class="txt" href="#E19:23"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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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