<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">T</font> being <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> holds  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b1">T</font> <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 <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">T</font> be  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font> <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 <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span></span><br/>



<a NAME="E1:44"/><i><font color="Green" title="E20">A1</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">T</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">IB</font> = <font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">IS</font> = <font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> ;<br/>
<a NAME="E2:44"/>
 <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <span class="p2">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> 1</span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="lattice3.html#D2" title="LATTICE3:def.2">LATTICE3:def 2</a></i>;<br/>
<b>then </b><i><font color="Green" title="E21">A2</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#D2" title="YELLOW_1:def.2">YELLOW_1:def 2</a></i>
<br/>.= 
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="lattice3.html#D1" title="LATTICE3:def.1">LATTICE3:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="zfmisc_1.html#T30" title="ZFMISC_1:th.30">ZFMISC_1:30</a></i>
;<br/>
<i><font color="Green" title="E22">A3</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> = 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E5:44"/><i><font color="Green" title="E23">A4</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )</span><br/>

<b>assume </b><a NAME="E1:44_3"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/>

<a NAME="E2:44_3"/><b>then </b><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">R1</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E4:44_3"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c6">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>and </b><br/><a NAME="E5:44_3"/><i><font color="Green" title="E26">A7</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</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="c6">R1</font>
 <b>by </b><i><a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">R2</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E7:44_3"/><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c7">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>and </b><br/><a NAME="E8:44_3"/><i><font color="Green" title="E28">A9</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</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="c7">R2</font>
 <b>by </b><i><a class="txt" href="#E2:44_3"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">R1</font> = 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="txt" href="#E3:44"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="#E2:44_3"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="#E4:44_3"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> 
<b>by </b><i><a class="ref" href="waybel18.html#D9" target="_self" title="WAYBEL18:def.9">Def9</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">R2</font>
<b>by </b><i><a class="txt" href="#E2:44_3"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="#E7:44_3"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E10:44_3"/>
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="#E5:44_3"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="#E8:44_3"><i><font color="Green" title="E28">A9</font></i></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>
<a NAME="E6:44"/><b>then </b><i><font color="Green" title="E24">A10</font></i>: 
 <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:44"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<i><font color="Green" title="E25">A11</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> = 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:44"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="#E5:44"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel18.html#D3" target="_self" title="WAYBEL18:def.3">Def3</a></i>
;<br/>
<a NAME="E8:44"/><i><font color="Green" title="E26">A12</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">P</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">ii</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">ii</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : verum </span>}</span>  as    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of  <a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span> <b>by </b><i><a class="ref" href="waybel18.html#T14" target="_self" title="WAYBEL18:th.14">Th14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">T'</font> = <font color="Maroon" title="c2">T</font> as  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a> ;<br/>
<a NAME="E11:44"/>
<font color="Maroon" title="c6">T'</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="waybel_8.html#T17" title="WAYBEL_8:th.17">WAYBEL_8:17</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">R</font> being    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c6">T'</font><b> such that </b><br/><a NAME="E13:44"/><i><font color="Green" title="E27">A13</font></i>: 
<font color="Maroon" title="c7">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">yy</font></span>)</span> where <font color="Olive" title="b1">yy</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">T'</font> : <font color="Olive" title="b1">yy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c6">T'</font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel14.html#T42" title="WAYBEL14:th.42">WAYBEL14:42</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E15:44"/><i><font color="Green" title="E28">A14</font></i>: 
<font color="Maroon" title="c8">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>and </b><br/><a NAME="E16:44"/><i><font color="Green" title="E29">A15</font></i>: 
for <font color="Olive" title="b1">Y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="waybel18.html#T15" target="_self" title="WAYBEL18:th.15">Th15</a></i>;<br/>
<a NAME="E17:44"/><i><font color="Green" title="E30">A16</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">f</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 <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E15:44"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E18:44"/><i><font color="Green" title="E31">A17</font></i>: 
<font color="Maroon" title="c7">R</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font> or <font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c5">P</font> )</span><br/>

<b>assume </b><a NAME="E1:44_5"/><i><font color="Green" title="E32">A18</font></i>: 
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c5">P</font></span><br/>

<b>then consider </b><font color="Maroon" title="c10">Y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">T'</font><b> such that </b><br/><a NAME="E3:44_5"/><i><font color="Green" title="E33">A19</font></i>: 
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c10">Y</font>
 <b>and </b><br/><a NAME="E4:44_5"/><i><font color="Green" title="E34">A20</font></i>: 
<font color="Maroon" title="c10">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c6">T'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E13:44"><i><font color="Green" title="E27">A13</font></i></a></i>;<br/>
<a NAME="E5:44_5"/><i><font color="Green" title="E35">A21</font></i>: 
<font color="Maroon" title="c10">Y</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 
<b>by </b><i><a class="txt" href="#E4:44_5"><i><font color="Green" title="E34">A20</font></i></a>, <a class="ref" href="waybel_8.html#D1" title="WAYBEL_8:def.1">WAYBEL_8:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">X'</font> = <font color="Maroon" title="c9">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E7:44"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="#E1:44_5"><i><font color="Green" title="E32">A18</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">y</font> = <font color="Maroon" title="c10">Y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<a NAME="E8:44_5"/><i><font color="Green" title="E36">A22</font></i>: 
<font color="Maroon" title="c12">y</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E5:44_5"><i><font color="Green" title="E35">A21</font></i></a>, <a class="ref" href="waybel_8.html#T9" title="WAYBEL_8:th.9">WAYBEL_8:9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:44_5"/><i><font color="Green" title="E37">A23</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font>
 <b>and </b><br/><a NAME="E11:44_5"/><i><font color="Green" title="E38">A24</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="#E17:44"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">x</font> = <font color="Maroon" title="c13">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">I</font></span>)</span> <b>by </b><i><a class="txt" href="#E10:44_5"><i><font color="Green" title="E37">A23</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x'</font> = <font color="Maroon" title="c14">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="waybel_8.html#T28" title="WAYBEL_8:th.28">WAYBEL_8:28</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> $1,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/>
<a NAME="E14:44_5"/>
<font color="Maroon" title="c14">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 
<b>by </b><i><a class="txt" href="#E15:44"><i><font color="Green" title="E28">A14</font></i></a>, <a class="txt" href="#E8:44_5"><i><font color="Green" title="E36">A22</font></i></a>, <a class="txt" href="#E11:44_5"><i><font color="Green" title="E38">A24</font></i></a>, <a class="ref" href="waybel13.html#T28" title="WAYBEL13:th.28">WAYBEL13:28</a></i>;<br/>
<a NAME="E15:44_5"/><b>then </b><i><font color="Green" title="E39">A25</font></i>: 
<font color="Maroon" title="c14">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="waybel_8.html#T30" title="WAYBEL_8:th.30">WAYBEL_8:30</a></i>;<br/>
<i><font color="Green" title="E40">A26</font></i>:  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c10">Y</font> = 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">Y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="ref" href="waybel_0.html#D18" title="WAYBEL_0:def.18">WAYBEL_0:def 18</a></i>
<br/>.= 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></i>
<br/>.= 
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c12">y</font>
<b>by </b><i><a class="ref" href="waybel_0.html#D18" title="WAYBEL_0:def.18">WAYBEL_0:def 18</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c16">ZZ</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">jj</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">jj</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Olive" title="b1">jj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font> </span>}</span> ;<br/>
<a NAME="E17:44_5"/><i><font color="Green" title="E41">A27</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">jj</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">jj</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Olive" title="b1">jj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">jj</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">jj</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Olive" title="b1">jj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font> </span>}</span>  or <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> )</span><br/>

<b>assume </b><a NAME="E1:44_5_2"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">jj</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">jj</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Olive" title="b1">jj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font></span><br/>

<b>then consider </b><font color="Maroon" title="c18">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E3:44_5_2"/><i><font color="Green" title="E42">A28</font></i>: 
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c18">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:44_5_2"/>
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">x'</font>
 ;<br/>
<b>thus </b><a NAME="E5:44_5_2"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E3:44_5_2"><i><font color="Green" title="E42">A28</font></i></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>then reconsider </b><font color="Maroon" title="c17">ZZ</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">jj</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">jj</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Olive" title="b1">jj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font> </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E19:44_5"/><i><font color="Green" title="E42">A29</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">jj</font>) where <font color="Olive" title="b1">jj</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Olive" title="b1">jj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="#E15:44_5"><i><font color="Green" title="E39">A25</font></i></a>);<br/></i>
<a NAME="E20:44_5"/><i><font color="Green" title="E43">A30</font></i>: 
<font color="Maroon" title="c11">X'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c17">ZZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X'</font> or <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c17">ZZ</font> )</span><br/>

<b>assume </b><a NAME="E1:44_5_3"/><i><font color="Green" title="E44">A31</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c17">ZZ</font></span><br/>

<a NAME="E2:44_5_3"/><b>then </b><i><font color="Green" title="E45">A32</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E7:44"><i><font color="Green" title="E25">A11</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">z'</font> = <font color="Maroon" title="c18">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E7:44"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="#E1:44_5_3"><i><font color="Green" title="E44">A31</font></i></a></i>;<br/>
<a NAME="E4:44_5_3"/>
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:44_5_3"><i><font color="Green" title="E45">A32</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">gg</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:44_5_3"/><i><font color="Green" title="E46">A33</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">gg</font>
 <b>and </b><br/><a NAME="E7:44_5_3"/><i><font color="Green" title="E47">A34</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E8:44_5_3"/><i><font color="Green" title="E48">A35</font></i>: 
for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c20">gg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E9:44_5_3"/>
<font color="Maroon" title="c12">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">z'</font>
 
<b>by </b><i><a class="txt" href="#E3:44_5"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="#E16:44_5"><i><font color="Green" title="E40">A26</font></i></a>, <a class="txt" href="#E1:44_5_3"><i><font color="Green" title="E44">A31</font></i></a>, <a class="ref" href="waybel_0.html#T18" title="WAYBEL_0:th.18">WAYBEL_0:18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">h1</font>, <font color="Maroon" title="c22">h2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E11:44_5_3"/><i><font color="Green" title="E49">A36</font></i>: 
<font color="Maroon" title="c21">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font>
 <b>and </b><br/><a NAME="E12:44_5_3"/><i><font color="Green" title="E50">A37</font></i>: 
<font color="Maroon" title="c22">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">z'</font>
 <b>and </b><br/><a NAME="E13:44_5_3"/><i><font color="Green" title="E51">A38</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b3">xi</font>, <font color="Olive" title="b4">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">R</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">yi</font> )
 <b>by </b><i><a class="txt" href="#E8:44"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<i><font color="Green" title="E52">A39</font></i>: <font color="Maroon" title="c21">h1</font> = 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x'</font>
<b>by </b><i><a class="txt" href="#E11:44_5"><i><font color="Green" title="E38">A24</font></i></a>, <a class="txt" href="#E11:44_5_3"><i><font color="Green" title="E49">A36</font></i></a></i>
<br/>.= 
 <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="txt" href="#E16:44"><i><font color="Green" title="E29">A15</font></i></a></i>
;<br/>
<a NAME="E15:44_5_3"/>
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">ZZ</font> holds <br/><font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c23">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">ZZ</font> implies <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Z</font> )</span><br/>

<b>assume </b><a NAME="E1:44_5_3_2"/>
<font color="Maroon" title="c23">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">ZZ</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Z</font></span><br/>

<b>then consider </b><font color="Maroon" title="c24">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E3:44_5_3_2"/><i><font color="Green" title="E53">A40</font></i>: 
<font color="Maroon" title="c23">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:44_5_3_2"/><i><font color="Green" title="E54">A41</font></i>: 
<font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font>
 ;<br/>
<i><font color="Green" title="E55">A42</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c25">RS</font> being    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> , <font color="Maroon" title="c26">xj</font>, <font color="Maroon" title="c27">yj</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c25">RS</font><b> such that </b><br/><a NAME="E7:44_5_3_2"/><i><font color="Green" title="E56">A43</font></i>: 
<font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c24">j</font>
 <b>and </b><br/><a NAME="E8:44_5_3_2"/><i><font color="Green" title="E57">A44</font></i>: 
<font color="Maroon" title="c26">xj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font>
 <b>and </b><br/><a NAME="E9:44_5_3_2"/><i><font color="Green" title="E58">A45</font></i>: 
<font color="Maroon" title="c27">yj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font>
 <b>and </b><br/><a NAME="E10:44_5_3_2"/><i><font color="Green" title="E59">A46</font></i>: 
<font color="Maroon" title="c26">xj</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c27">yj</font>
 <b>by </b><i><a class="txt" href="#E13:44_5_3"><i><font color="Green" title="E51">A38</font></i></a></i>;<br/>
<a NAME="E11:44_5_3_2"/><i><font color="Green" title="E60">A47</font></i>: 
<font color="Maroon" title="c26">xj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="#E14:44_5_3"><i><font color="Green" title="E52">A39</font></i></a>, <a class="txt" href="#E4:44_5_3_2"><i><font color="Green" title="E54">A41</font></i></a>, <a class="txt" href="#E8:44_5_3_2"><i><font color="Green" title="E57">A44</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">a</font> = <font color="Maroon" title="c26">xj</font>, <font color="Maroon" title="c29">b</font> = <font color="Maroon" title="c27">yj</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span> <b>by </b><i><a class="txt" href="#E7:44_5_3_2"><i><font color="Green" title="E56">A43</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E13:44_5_3_2"/><i><font color="Green" title="E61">A48</font></i>: 
<font color="Maroon" title="c28">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c29">b</font>
 
<b>by </b><i><a class="txt" href="#E7:44_5_3_2"><i><font color="Green" title="E56">A43</font></i></a>, <a class="txt" href="#E10:44_5_3_2"><i><font color="Green" title="E59">A46</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E14:44_5_3_2"/><i><font color="Green" title="E62">A49</font></i>: 
<font color="Maroon" title="c29">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span>
 
;<br/>
<a NAME="E15:44_5_3_2"/><i><font color="Green" title="E63">A50</font></i>: 
<font color="Maroon" title="c27">yj</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5_3_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_5_3_2_2"/>
<font color="Maroon" title="c27">yj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:44_5_3_2_2"/><b>then </b><i><font color="Green" title="E64">A51</font></i>: 
1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E11:44_5_3_2"><i><font color="Green" title="E60">A47</font></i></a>, <a class="txt" href="#E13:44_5_3_2"><i><font color="Green" title="E61">A48</font></i></a>, <a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/>
<b>thus </b><a NAME="E3:44_5_3_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:44_5_3_2_2"><i><font color="Green" title="E64">A51</font></i></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>
<a NAME="E16:44_5_3_2"/><i><font color="Green" title="E64">A52</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c22">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:44_5_3"><i><font color="Green" title="E46">A33</font></i></a>, <a class="txt" href="#E7:44_5_3"><i><font color="Green" title="E47">A34</font></i></a>, <a class="txt" href="#E12:44_5_3"><i><font color="Green" title="E50">A37</font></i></a>, <a class="txt" href="#E5:44_5_3_2"><i><font color="Green" title="E55">A42</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E17:44_5_3_2"/>
for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds <br/><font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5_3_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c30">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> implies <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> )</span><br/>

<b>assume </b><a NAME="E1:44_5_3_2_3"/>
<font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font></span><br/>

<a NAME="E2:44_5_3_2_3"/><b>then </b><i><font color="Green" title="E65">A53</font></i>: 
<font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:44_5_3_2"><i><font color="Green" title="E55">A42</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_5_3_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_5_3_2_3_1"/>
( <font color="Maroon" title="c30">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j</font> or <font color="Maroon" title="c30">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_5_3_2_3_1_1"><b>suppose </b></a><a NAME="E1:44_5_3_2_3_1_1"/><i><font color="Green" title="E66">A54</font></i>: 
<font color="Maroon" title="c30">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font></span><br/><div class="add"><a NAME="E2:44_5_3_2_3_1_1"/><b>then </b><i><font color="Green" title="E67">A55</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:44_5_3_2_3"><i><font color="Green" title="E65">A53</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:44_5_3_2_3_1_1"/>
<font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="#E3:44"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="#E9:44_5_3_2"><i><font color="Green" title="E58">A45</font></i></a>, <a class="txt" href="#E14:44_5_3_2"><i><font color="Green" title="E62">A49</font></i></a>, <a class="txt" href="#E15:44_5_3_2"><i><font color="Green" title="E63">A50</font></i></a>, <a class="txt" href="#E1:44_5_3_2_3_1_1"><i><font color="Green" title="E66">A54</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:44_5_3_2_3_1_1"/>
<font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font>
 <b>by </b><i><a class="txt" href="#E2:44_5_3_2_3_1_1"><i><font color="Green" title="E67">A55</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_5_3_2_3_1_2"><b>suppose </b></a><a NAME="E1:44_5_3_2_3_1_2"/>
<font color="Maroon" title="c30">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font></span><br/><div class="add"><a NAME="E2:44_5_3_2_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:44_5_3_2_3_1_2"/>
<font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font>
 <b>by </b><i><a class="txt" href="#E6:44"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="#E6:44_5_3"><i><font color="Green" title="E46">A33</font></i></a>, <a class="txt" href="#E8:44_5_3"><i><font color="Green" title="E48">A35</font></i></a>, <a class="txt" href="#E12:44_5_3"><i><font color="Green" title="E50">A37</font></i></a>, <a class="txt" href="#E2:44_5_3_2_3"><i><font color="Green" title="E65">A53</font></i></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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E18:44_5_3_2"/>
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Z</font>
 <b>by </b><i><a class="txt" href="#E12:44_5_3"><i><font color="Green" title="E50">A37</font></i></a>, <a class="txt" href="#E3:44_5_3_2"><i><font color="Green" title="E53">A40</font></i></a>, <a class="txt" href="#E16:44_5_3_2"><i><font color="Green" title="E64">A52</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</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>hence </b><a NAME="E16:44_5_3"/>
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c17">ZZ</font>
 <b>by </b><i><a class="txt" href="#E1:44_5_3"><i><font color="Green" title="E44">A31</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</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>
<a NAME="E21:44_5"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c17">ZZ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">X'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c17">ZZ</font> or <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X'</font> )</span><br/>

<b>assume </b><a NAME="E1:44_5_4"/><i><font color="Green" title="E44">A56</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c17">ZZ</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X'</font></span><br/>

<a NAME="E2:44_5_4"/><b>then </b><i><font color="Green" title="E45">A57</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E7:44"><i><font color="Green" title="E25">A11</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">z'</font> = <font color="Maroon" title="c18">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E7:44"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="#E1:44_5_4"><i><font color="Green" title="E44">A56</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c20">h1</font> =  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font>;<br/>
<i><font color="Green" title="E46">A58</font></i>:  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font> = 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x'</font>
<b>by </b><i><a class="txt" href="#E16:44"><i><font color="Green" title="E29">A15</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c12">y</font>
<b>by </b><i><a class="txt" href="#E11:44_5"><i><font color="Green" title="E38">A24</font></i></a></i>
;<br/>
<a NAME="E5:44_5_4"/>
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:44_5_4"><i><font color="Green" title="E45">A57</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">h2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:44_5_4"/><i><font color="Green" title="E47">A59</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font>
 <b>and </b><br/><a NAME="E8:44_5_4"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E9:44_5_4"/><i><font color="Green" title="E48">A60</font></i>: 
for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E10:44_5_4"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b3">xi</font>, <font color="Olive" title="b4">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">R</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">yi</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies  ex <font color="Olive" title="b1">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b2">xi</font>, <font color="Olive" title="b3">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">R</font> st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b2">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b3">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b2">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">yi</font> ) )</span><br/>

<b>assume </b><a NAME="E1:44_5_4_2"/><i><font color="Green" title="E49">A61</font></i>: 
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b2">xi</font>, <font color="Olive" title="b3">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">R</font> st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b2">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b3">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b2">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">yi</font> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c23">i'</font> = <font color="Maroon" title="c22">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/>
<b>take </b><font color="Maroon" title="c24">RS</font> =  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">xi</font>, <font color="Olive" title="b2">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c24">RS</font> st <br/>( <font color="Maroon" title="c24">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b2">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">yi</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c25">RB</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E4:44_5_4_2"/><i><font color="Green" title="E50">A62</font></i>: 
<font color="Maroon" title="c25">RB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <b>and </b><br/><a NAME="E5:44_5_4_2"/><i><font color="Green" title="E51">A63</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</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="c25">RB</font>
 <b>by </b><i><a class="txt" href="#E1:44_5_4_2"><i><font color="Green" title="E49">A61</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
<a NAME="E6:44_5_4_2"/><i><font color="Green" title="E52">A64</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="#E3:44"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="#E1:44_5_4_2"><i><font color="Green" title="E49">A61</font></i></a>, <a class="txt" href="#E4:44_5_4_2"><i><font color="Green" title="E50">A62</font></i></a>, <a class="txt" href="#E5:44_5_4_2"><i><font color="Green" title="E51">A63</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E7:44_5_4_2"/><i><font color="Green" title="E53">A65</font></i>: 
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:44_5_4_2"><i><font color="Green" title="E49">A61</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:44_5_4_2"/><b>then </b><i><font color="Green" title="E54">A66</font></i>: 
<font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 
<b>by </b><i><a class="txt" href="#E9:44_5_4"><i><font color="Green" title="E48">A60</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_5_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_5_4_2_1"/>
( <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font> or  not <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_5_4_2_1_1"><b>suppose </b></a><a NAME="E1:44_5_4_2_1_1"/><i><font color="Green" title="E55">A67</font></i>: 
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">xi</font>, <font color="Olive" title="b2">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c24">RS</font> st <br/>( <font color="Maroon" title="c24">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b2">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">yi</font> )</span><br/><div class="add"><a NAME="E2:44_5_4_2_1_1"/><b>then </b>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c23">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">ZZ</font>
 ;<br/><a NAME="E3:44_5_4_2_1_1"/><b>then </b>
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c23">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:44_5_4"><i><font color="Green" title="E44">A56</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/><b>then consider </b><font color="Maroon" title="c26">h2'</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:44_5_4_2_1_1"/><i><font color="Green" title="E56">A68</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">h2'</font>
 <b>and </b><br/><a NAME="E6:44_5_4_2_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c26">h2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c23">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E7:44_5_4_2_1_1"/><i><font color="Green" title="E57">A69</font></i>: 
for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c23">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds <br/><font color="Maroon" title="c26">h2'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c23">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/><a NAME="E8:44_5_4_2_1_1"/>
<font color="Maroon" title="c23">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c23">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:44"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="#E7:44_5_4_2"><i><font color="Green" title="E53">A65</font></i></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/><a NAME="E9:44_5_4_2_1_1"/><b>then </b><i><font color="Green" title="E58">A70</font></i>: 
<font color="Maroon" title="c26">h2'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c23">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i'</font>
 <b>by </b><i><a class="txt" href="#E7:44_5_4_2_1_1"><i><font color="Green" title="E57">A69</font></i></a></i>;<br/><a NAME="E10:44_5_4_2_1_1"/><i><font color="Green" title="E59">A71</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c23">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E4:44"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="#E7:44_5_4_2"><i><font color="Green" title="E53">A65</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">xi</font> = 1, <font color="Maroon" title="c28">yi</font> = 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c24">RS</font> <b>by </b><i><a class="txt" href="#E3:44"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>take </b>
<font color="Maroon" title="c27">xi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c24">RS</font> st <br/>( <font color="Maroon" title="c24">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b1">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">yi</font> )</span><br/><b>take </b>
<font color="Maroon" title="c28">yi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c24">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c28">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">yi</font> )</span><br/><b>thus </b><a NAME="E12:44_5_4_2_1_1"/>
<font color="Maroon" title="c24">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_5_4_2"><i><font color="Green" title="E49">A61</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c27">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c28">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">yi</font> )</span><br/><b>thus </b><a NAME="E13:44_5_4_2_1_1"/>
<font color="Maroon" title="c27">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_5_4_2"><i><font color="Green" title="E49">A61</font></i></a>, <a class="txt" href="#E1:44_5_4_2_1_1"><i><font color="Green" title="E55">A67</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c28">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">yi</font> )</span><br/><b>thus </b><a NAME="E14:44_5_4_2_1_1"/>
<font color="Maroon" title="c28">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <b>by </b><i><a class="txt" href="#E7:44_5_4"><i><font color="Green" title="E47">A59</font></i></a>, <a class="txt" href="#E5:44_5_4_2_1_1"><i><font color="Green" title="E56">A68</font></i></a>, <a class="txt" href="#E9:44_5_4_2_1_1"><i><font color="Green" title="E58">A70</font></i></a>, <a class="txt" href="#E10:44_5_4_2_1_1"><i><font color="Green" title="E59">A71</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">yi</font></span><br/><b>thus </b><a NAME="E15:44_5_4_2_1_1"/>
<font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c28">yi</font>
 ; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_5_4_2_1_2"><b>suppose </b></a><a NAME="E1:44_5_4_2_1_2"/><i><font color="Green" title="E55">A72</font></i>: 
not <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">xi</font>, <font color="Olive" title="b2">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c24">RS</font> st <br/>( <font color="Maroon" title="c24">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b2">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">yi</font> )</span><br/><div class="add"><b>thus </b><a NAME="E2:44_5_4_2_1_2"/>
 ex <font color="Olive" title="b1">xi</font>, <font color="Olive" title="b2">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c24">RS</font> st <br/>( <font color="Maroon" title="c24">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b2">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">yi</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5_4_2_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_5_4_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_5_4_2_1_2_1_1"/>
( <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="#E6:44_5_4_2"><i><font color="Green" title="E52">A64</font></i></a>, <a class="txt" href="#E8:44_5_4_2"><i><font color="Green" title="E54">A66</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_5_4_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:44_5_4_2_1_2_1_1_1"/><i><font color="Green" title="E56">A73</font></i>: 
<font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">xi</font>, <font color="Olive" title="b2">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c24">RS</font> st <br/>( <font color="Maroon" title="c24">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b2">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">yi</font> )</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c26">xi</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <font color="Maroon" title="c27">yi</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c24">RS</font> <b>by </b><i><a class="txt" href="#E3:44"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>take </b>
<font color="Maroon" title="c26">xi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c24">RS</font> st <br/>( <font color="Maroon" title="c24">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c26">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b1">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c26">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">yi</font> )</span><br/><b>take </b>
<font color="Maroon" title="c27">yi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c24">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c26">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c27">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c26">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c27">yi</font> )</span><br/><b>thus </b><a NAME="E3:44_5_4_2_1_2_1_1_1"/>
<font color="Maroon" title="c24">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_5_4_2"><i><font color="Green" title="E49">A61</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c26">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c27">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c26">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c27">yi</font> )</span><br/><b>thus </b><a NAME="E4:44_5_4_2_1_2_1_1_1"/>
<font color="Maroon" title="c26">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_5_4_2"><i><font color="Green" title="E49">A61</font></i></a>, <a class="txt" href="#E1:44_5_4_2_1_2"><i><font color="Green" title="E55">A72</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c27">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c26">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c27">yi</font> )</span><br/><b>thus </b><a NAME="E5:44_5_4_2_1_2_1_1_1"/>
<font color="Maroon" title="c27">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_5_4_2_1_2_1_1_1"><i><font color="Green" title="E56">A73</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c26">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c27">yi</font></span><br/><b>thus </b><a NAME="E6:44_5_4_2_1_2_1_1_1"/>
<font color="Maroon" title="c26">xi</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c27">yi</font>
 ; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_5_4_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:44_5_4_2_1_2_1_1_2"/><i><font color="Green" title="E56">A74</font></i>: 
<font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">xi</font>, <font color="Olive" title="b2">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c24">RS</font> st <br/>( <font color="Maroon" title="c24">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b2">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">yi</font> )</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c26">xi</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <font color="Maroon" title="c27">yi</font> = 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c24">RS</font> <b>by </b><i><a class="txt" href="#E3:44"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>take </b>
<font color="Maroon" title="c26">xi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c24">RS</font> st <br/>( <font color="Maroon" title="c24">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c26">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Olive" title="b1">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c26">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">yi</font> )</span><br/><b>take </b>
<font color="Maroon" title="c27">yi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c24">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c26">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c27">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c26">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c27">yi</font> )</span><br/><b>thus </b><a NAME="E3:44_5_4_2_1_2_1_1_2"/>
<font color="Maroon" title="c24">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_5_4_2"><i><font color="Green" title="E49">A61</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c26">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c27">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c26">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c27">yi</font> )</span><br/><b>thus </b><a NAME="E4:44_5_4_2_1_2_1_1_2"/>
<font color="Maroon" title="c26">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_5_4_2"><i><font color="Green" title="E49">A61</font></i></a>, <a class="txt" href="#E1:44_5_4_2_1_2"><i><font color="Green" title="E55">A72</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c27">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c26">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c27">yi</font> )</span><br/><b>thus </b><a NAME="E5:44_5_4_2_1_2_1_1_2"/>
<font color="Maroon" title="c27">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_5_4_2_1_2_1_1_2"><i><font color="Green" title="E56">A74</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c26">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c27">yi</font></span><br/><a NAME="E6:44_5_4_2_1_2_1_1_2"/>
<font color="Maroon" title="c26">xi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">yi</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><b>hence </b><a NAME="E7:44_5_4_2_1_2_1_1_2"/>
<font color="Maroon" title="c26">xi</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c27">yi</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</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></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:44_5_4"/><b>then </b>
<font color="Maroon" title="c12">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">z'</font>
 
<b>by </b><i><a class="txt" href="#E8:44"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="#E4:44_5_4"><i><font color="Green" title="E46">A58</font></i></a>, <a class="txt" href="#E7:44_5_4"><i><font color="Green" title="E47">A59</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E12:44_5_4"/>
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X'</font>
 <b>by </b><i><a class="txt" href="#E3:44_5"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="#E16:44_5"><i><font color="Green" title="E40">A26</font></i></a>, <a class="ref" href="waybel_0.html#T18" title="WAYBEL_0:th.18">WAYBEL_0:18</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>
<a NAME="E22:44_5"/><b>then </b>
<font color="Maroon" title="c11">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c17">ZZ</font>
 
<b>by </b><i><a class="txt" href="#E20:44_5"><i><font color="Green" title="E43">A30</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E23:44_5"/>
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="txt" href="#E17:44_5"><i><font color="Green" title="E41">A27</font></i></a>, <a class="txt" href="#E19:44_5"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="cantor_1.html#D4" title="CANTOR_1:def.4">CANTOR_1:def 4</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>
<a NAME="E19:44"/>
 <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> <font color="Maroon" title="c7">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c5">P</font> or <font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font> )</span><br/>

<b>assume </b><a NAME="E1:44_6"/><i><font color="Green" title="E32">A75</font></i>: 
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c5">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c10">X'</font> = <font color="Maroon" title="c9">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c11">ZZ</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:44_6"/><i><font color="Green" title="E33">A76</font></i>: 
<font color="Maroon" title="c11">ZZ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font>
 <b>and </b><br/><a NAME="E5:44_6"/><i><font color="Green" title="E34">A77</font></i>: 
<font color="Maroon" title="c11">ZZ</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>and </b><br/><a NAME="E6:44_6"/><i><font color="Green" title="E35">A78</font></i>: 
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c11">ZZ</font>
 <b>by </b><i><a class="txt" href="#E1:44_6"><i><font color="Green" title="E32">A75</font></i></a>, <a class="ref" href="cantor_1.html#D4" title="CANTOR_1:def.4">CANTOR_1:def 4</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> $1,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c12">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:44_6"/><i><font color="Green" title="E36">A79</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>and </b><br/><a NAME="E9:44_6"/><i><font color="Green" title="E37">A80</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">e</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E10:44_6"/>
<font color="Maroon" title="c5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> or <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font> )</span><br/>

<b>assume </b><a NAME="E1:44_6_1"/>
<font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font></span><br/>

<b>then consider </b><font color="Maroon" title="c14">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E3:44_6_1"/><i><font color="Green" title="E38">A81</font></i>: 
<font color="Maroon" title="c13">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c14">e</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ;<br/>
<a NAME="E4:44_6_1"/>
<font color="Maroon" title="c13">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 
<b>by </b><i><a class="txt" href="#E9:44_6"><i><font color="Green" title="E37">A80</font></i></a>, <a class="txt" href="#E3:44_6_1"><i><font color="Green" title="E38">A81</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:44_6_1"/>
<font color="Maroon" title="c13">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font>
 <b>by </b><i><a class="txt" href="#E8:44_6"><i><font color="Green" title="E36">A79</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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>
<a NAME="E11:44_6"/><b>then </b>
<font color="Maroon" title="c11">ZZ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F</font>
 
<b>by </b><i><a class="txt" href="#E4:44_6"><i><font color="Green" title="E33">A76</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">x'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:44_6"/><i><font color="Green" title="E38">A82</font></i>: 
<font color="Maroon" title="c13">x'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">F</font>
 <b>and </b><br/><a NAME="E14:44_6"/><i><font color="Green" title="E39">A83</font></i>: 
<font color="Maroon" title="c13">x'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>and </b><br/><a NAME="E15:44_6"/><i><font color="Green" title="E40">A84</font></i>: 
<font color="Maroon" title="c12">F</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">ZZ</font>
 <b>by </b><i><a class="txt" href="#E5:44_6"><i><font color="Green" title="E34">A77</font></i></a>, <a class="ref" href="orders_1.html#T195" title="ORDERS_1:th.195">ORDERS_1:195</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">x'</font> = <font color="Maroon" title="c13">x'</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="#E8:44_6"><i><font color="Green" title="E36">A79</font></i></a>, <a class="txt" href="#E13:44_6"><i><font color="Green" title="E38">A82</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">PP</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">i</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x'</font> </span>}</span> ;<br/>
<a NAME="E17:44_6"/><i><font color="Green" title="E41">A85</font></i>: 
<font color="Maroon" title="c11">ZZ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">i</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c16">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">ZZ</font> or <font color="Maroon" title="c16">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">i</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x'</font> </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:44_6_2"/>
<font color="Maroon" title="c16">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">ZZ</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">i</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x'</font> </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c17">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:44_6_2"/><i><font color="Green" title="E42">A86</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">F</font>
 <b>and </b><br/><a NAME="E4:44_6_2"/><i><font color="Green" title="E43">A87</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x'</font>
 <b>and </b><br/><a NAME="E5:44_6_2"/><i><font color="Green" title="E44">A88</font></i>: 
<font color="Maroon" title="c16">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E15:44_6"><i><font color="Green" title="E40">A84</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">e</font> = <font color="Maroon" title="c17">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="#E8:44_6"><i><font color="Green" title="E36">A79</font></i></a>, <a class="txt" href="#E3:44_6_2"><i><font color="Green" title="E42">A86</font></i></a></i>;<br/>
<a NAME="E7:44_6_2"/>
<font color="Maroon" title="c16">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c18">e</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E9:44_6"><i><font color="Green" title="E37">A80</font></i></a>, <a class="txt" href="#E5:44_6_2"><i><font color="Green" title="E44">A88</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:44_6_2"/>
<font color="Maroon" title="c16">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">i</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x'</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E4:44_6_2"><i><font color="Green" title="E43">A87</font></i></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>
<a NAME="E18:44_6"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">i</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x'</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">ZZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c16">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">i</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x'</font> </span>}</span>  or <font color="Maroon" title="c16">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">ZZ</font> )</span><br/>

<b>assume </b><a NAME="E1:44_6_3"/>
<font color="Maroon" title="c16">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">i</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x'</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">ZZ</font></span><br/>

<b>then consider </b><font color="Maroon" title="c17">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E3:44_6_3"/><i><font color="Green" title="E42">A89</font></i>: 
<font color="Maroon" title="c16">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c17">e</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:44_6_3"/><i><font color="Green" title="E43">A90</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x'</font>
 ;<br/>
<a NAME="E5:44_6_3"/>
<font color="Maroon" title="c16">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 
<b>by </b><i><a class="txt" href="#E9:44_6"><i><font color="Green" title="E37">A80</font></i></a>, <a class="txt" href="#E3:44_6_3"><i><font color="Green" title="E42">A89</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:44_6_3"/>
<font color="Maroon" title="c16">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">ZZ</font>
 <b>by </b><i><a class="txt" href="#E8:44_6"><i><font color="Green" title="E36">A79</font></i></a>, <a class="txt" href="#E15:44_6"><i><font color="Green" title="E40">A84</font></i></a>, <a class="txt" href="#E4:44_6_3"><i><font color="Green" title="E43">A90</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</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>
<a NAME="E19:44_6"/><b>then </b><i><font color="Green" title="E42">A91</font></i>: 
<font color="Maroon" title="c11">ZZ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">i</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x'</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E17:44_6"><i><font color="Green" title="E41">A85</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">x</font> = <font color="Maroon" title="c14">x'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">I</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_8.html#T28" title="WAYBEL_8:th.28">WAYBEL_8:28</a></i>;<br/>
<a NAME="E21:44_6"/><i><font color="Green" title="E43">A92</font></i>: 
<font color="Maroon" title="c16">x</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 
<b>by </b><i><a class="txt" href="#E14:44_6"><i><font color="Green" title="E39">A83</font></i></a>, <a class="ref" href="waybel_8.html#T30" title="WAYBEL_8:th.30">WAYBEL_8:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">y</font> = <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> ;<br/>
<a NAME="E23:44_6"/><i><font color="Green" title="E44">A93</font></i>: 
<font color="Maroon" title="c17">y</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 
<b>by </b><i><a class="txt" href="#E15:44"><i><font color="Green" title="E28">A14</font></i></a>, <a class="txt" href="#E21:44_6"><i><font color="Green" title="E43">A92</font></i></a>, <a class="ref" href="waybel13.html#T28" title="WAYBEL13:th.28">WAYBEL13:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">Y</font> = <font color="Maroon" title="c17">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">T'</font> <b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<i><font color="Green" title="E45">A94</font></i>:  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c18">Y</font> = 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">Y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="ref" href="waybel_0.html#D18" title="WAYBEL_0:def.18">WAYBEL_0:def 18</a></i>
<br/>.= 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></i>
<br/>.= 
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c17">y</font>
<b>by </b><i><a class="ref" href="waybel_0.html#D18" title="WAYBEL_0:def.18">WAYBEL_0:def 18</a></i>
;<br/>
<a NAME="E26:44_6"/><i><font color="Green" title="E46">A95</font></i>: 
<font color="Maroon" title="c10">X'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c17">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_6_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X'</font> or <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c17">y</font> )</span><br/>

<b>assume </b><a NAME="E1:44_6_5"/><i><font color="Green" title="E47">A96</font></i>: 
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c17">y</font></span><br/>

<a NAME="E2:44_6_5"/><b>then </b><i><font color="Green" title="E48">A97</font></i>: 
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E7:44"><i><font color="Green" title="E25">A11</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">z'</font> = <font color="Maroon" title="c19">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E7:44"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="#E1:44_6_5"><i><font color="Green" title="E47">A96</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c21">h1</font> =  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font>;<br/>
<a NAME="E4:44_6_5"/><i><font color="Green" title="E49">A98</font></i>: 
 <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font>
 
<b>by </b><i><a class="txt" href="#E16:44"><i><font color="Green" title="E29">A15</font></i></a></i>;<br/>
<a NAME="E5:44_6_5"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:44_6_5"><i><font color="Green" title="E48">A97</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">h2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:44_6_5"/><i><font color="Green" title="E50">A99</font></i>: 
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font>
 <b>and </b><br/><a NAME="E8:44_6_5"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c22">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E9:44_6_5"/><i><font color="Green" title="E51">A100</font></i>: 
for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E10:44_6_5"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b3">xi</font>, <font color="Olive" title="b4">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">R</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">yi</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_6_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies  ex <font color="Olive" title="b1">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b2">xi</font>, <font color="Olive" title="b3">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">R</font> st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b2">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b3">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b2">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">yi</font> ) )</span><br/>

<b>assume </b><a NAME="E1:44_6_5_1"/><i><font color="Green" title="E52">A101</font></i>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b2">xi</font>, <font color="Olive" title="b3">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">R</font> st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b2">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b3">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b2">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">yi</font> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c24">i'</font> = <font color="Maroon" title="c23">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/>
<b>take </b><font color="Maroon" title="c25">RS</font> =  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">xi</font>, <font color="Olive" title="b2">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c25">RS</font> st <br/>( <font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b2">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">yi</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c26">RB</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E4:44_6_5_1"/><i><font color="Green" title="E53">A102</font></i>: 
<font color="Maroon" title="c26">RB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 <b>and </b><br/><a NAME="E5:44_6_5_1"/><i><font color="Green" title="E54">A103</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</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="c26">RB</font>
 <b>by </b><i><a class="txt" href="#E1:44_6_5_1"><i><font color="Green" title="E52">A101</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
<a NAME="E6:44_6_5_1"/><i><font color="Green" title="E55">A104</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="#E3:44"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="#E1:44_6_5_1"><i><font color="Green" title="E52">A101</font></i></a>, <a class="txt" href="#E4:44_6_5_1"><i><font color="Green" title="E53">A102</font></i></a>, <a class="txt" href="#E5:44_6_5_1"><i><font color="Green" title="E54">A103</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E7:44_6_5_1"/><i><font color="Green" title="E56">A105</font></i>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:44_6_5_1"><i><font color="Green" title="E52">A101</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:44_6_5_1"/><b>then </b><i><font color="Green" title="E57">A106</font></i>: 
<font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 
<b>by </b><i><a class="txt" href="#E9:44_6_5"><i><font color="Green" title="E51">A100</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_6_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_6_5_1_1"/>
( <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font> or  not <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_6_5_1_1_1"><b>suppose </b></a><a NAME="E1:44_6_5_1_1_1"/><i><font color="Green" title="E58">A107</font></i>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">xi</font>, <font color="Olive" title="b2">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c25">RS</font> st <br/>( <font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b2">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">yi</font> )</span><br/><div class="add"><a NAME="E2:44_6_5_1_1_1"/><b>then </b>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">ZZ</font>
 <b>by </b><i><a class="txt" href="#E19:44_6"><i><font color="Green" title="E42">A91</font></i></a></i>;<br/><a NAME="E3:44_6_5_1_1_1"/><b>then </b>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:44_6"><i><font color="Green" title="E35">A78</font></i></a>, <a class="txt" href="#E1:44_6_5"><i><font color="Green" title="E47">A96</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/><b>then consider </b><font color="Maroon" title="c27">h2'</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:44_6_5_1_1_1"/><i><font color="Green" title="E59">A108</font></i>: 
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">h2'</font>
 <b>and </b><br/><a NAME="E6:44_6_5_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c27">h2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E7:44_6_5_1_1_1"/><i><font color="Green" title="E60">A109</font></i>: 
for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds <br/><font color="Maroon" title="c27">h2'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/><a NAME="E8:44_6_5_1_1_1"/>
<font color="Maroon" title="c24">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:44"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="#E7:44_6_5_1"><i><font color="Green" title="E56">A105</font></i></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/><a NAME="E9:44_6_5_1_1_1"/><b>then </b><i><font color="Green" title="E61">A110</font></i>: 
<font color="Maroon" title="c27">h2'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i'</font>
 <b>by </b><i><a class="txt" href="#E7:44_6_5_1_1_1"><i><font color="Green" title="E60">A109</font></i></a></i>;<br/><a NAME="E10:44_6_5_1_1_1"/><i><font color="Green" title="E62">A111</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c24">i'</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E4:44"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="#E7:44_6_5_1"><i><font color="Green" title="E56">A105</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">xi</font> = 1, <font color="Maroon" title="c29">yi</font> = 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c25">RS</font> <b>by </b><i><a class="txt" href="#E3:44"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>take </b>
<font color="Maroon" title="c28">xi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c25">RS</font> st <br/>( <font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c28">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b1">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c28">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">yi</font> )</span><br/><b>take </b>
<font color="Maroon" title="c29">yi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c28">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c29">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c28">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c29">yi</font> )</span><br/><b>thus </b><a NAME="E12:44_6_5_1_1_1"/>
<font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_6_5_1"><i><font color="Green" title="E52">A101</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c28">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c29">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c28">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c29">yi</font> )</span><br/><b>thus </b><a NAME="E13:44_6_5_1_1_1"/>
<font color="Maroon" title="c28">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_6_5_1_1_1"><i><font color="Green" title="E58">A107</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c29">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c28">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c29">yi</font> )</span><br/><b>thus </b><a NAME="E14:44_6_5_1_1_1"/>
<font color="Maroon" title="c29">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 <b>by </b><i><a class="txt" href="#E7:44_6_5"><i><font color="Green" title="E50">A99</font></i></a>, <a class="txt" href="#E5:44_6_5_1_1_1"><i><font color="Green" title="E59">A108</font></i></a>, <a class="txt" href="#E9:44_6_5_1_1_1"><i><font color="Green" title="E61">A110</font></i></a>, <a class="txt" href="#E10:44_6_5_1_1_1"><i><font color="Green" title="E62">A111</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c28">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c29">yi</font></span><br/><b>thus </b><a NAME="E15:44_6_5_1_1_1"/>
<font color="Maroon" title="c28">xi</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c29">yi</font>
 ; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_6_5_1_1_2"><b>suppose </b></a><a NAME="E1:44_6_5_1_1_2"/><i><font color="Green" title="E58">A112</font></i>: 
not <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">xi</font>, <font color="Olive" title="b2">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c25">RS</font> st <br/>( <font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b2">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">yi</font> )</span><br/><div class="add"><b>thus </b><a NAME="E2:44_6_5_1_1_2"/>
 ex <font color="Olive" title="b1">xi</font>, <font color="Olive" title="b2">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c25">RS</font> st <br/>( <font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b2">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">yi</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_6_5_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_6_5_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_6_5_1_1_2_1_1"/>
( <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="#E6:44_6_5_1"><i><font color="Green" title="E55">A104</font></i></a>, <a class="txt" href="#E8:44_6_5_1"><i><font color="Green" title="E57">A106</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_6_5_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:44_6_5_1_1_2_1_1_1"/><i><font color="Green" title="E59">A113</font></i>: 
<font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">xi</font>, <font color="Olive" title="b2">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c25">RS</font> st <br/>( <font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b2">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">yi</font> )</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c27">xi</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <font color="Maroon" title="c28">yi</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c25">RS</font> <b>by </b><i><a class="txt" href="#E3:44"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>take </b>
<font color="Maroon" title="c27">xi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c25">RS</font> st <br/>( <font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b1">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">yi</font> )</span><br/><b>take </b>
<font color="Maroon" title="c28">yi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c28">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">yi</font> )</span><br/><b>thus </b><a NAME="E3:44_6_5_1_1_2_1_1_1"/>
<font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_6_5_1"><i><font color="Green" title="E52">A101</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c27">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c28">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">yi</font> )</span><br/><b>thus </b><a NAME="E4:44_6_5_1_1_2_1_1_1"/>
<font color="Maroon" title="c27">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_6_5_1"><i><font color="Green" title="E52">A101</font></i></a>, <a class="txt" href="#E1:44_6_5_1_1_2"><i><font color="Green" title="E58">A112</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c28">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">yi</font> )</span><br/><b>thus </b><a NAME="E5:44_6_5_1_1_2_1_1_1"/>
<font color="Maroon" title="c28">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_6_5_1_1_2_1_1_1"><i><font color="Green" title="E59">A113</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">yi</font></span><br/><b>thus </b><a NAME="E6:44_6_5_1_1_2_1_1_1"/>
<font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c28">yi</font>
 ; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_6_5_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:44_6_5_1_1_2_1_1_2"/><i><font color="Green" title="E59">A114</font></i>: 
<font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">xi</font>, <font color="Olive" title="b2">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c25">RS</font> st <br/>( <font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b2">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b1">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">yi</font> )</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c27">xi</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <font color="Maroon" title="c28">yi</font> = 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c25">RS</font> <b>by </b><i><a class="txt" href="#E3:44"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>take </b>
<font color="Maroon" title="c27">xi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c25">RS</font> st <br/>( <font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Olive" title="b1">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">yi</font> )</span><br/><b>take </b>
<font color="Maroon" title="c28">yi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c28">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">yi</font> )</span><br/><b>thus </b><a NAME="E3:44_6_5_1_1_2_1_1_2"/>
<font color="Maroon" title="c25">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_6_5_1"><i><font color="Green" title="E52">A101</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c27">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c28">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">yi</font> )</span><br/><b>thus </b><a NAME="E4:44_6_5_1_1_2_1_1_2"/>
<font color="Maroon" title="c27">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_6_5_1"><i><font color="Green" title="E52">A101</font></i></a>, <a class="txt" href="#E1:44_6_5_1_1_2"><i><font color="Green" title="E58">A112</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c28">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">yi</font> )</span><br/><b>thus </b><a NAME="E5:44_6_5_1_1_2_1_1_2"/>
<font color="Maroon" title="c28">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 <b>by </b><i><a class="txt" href="#E1:44_6_5_1_1_2_1_1_2"><i><font color="Green" title="E59">A114</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">yi</font></span><br/><a NAME="E6:44_6_5_1_1_2_1_1_2"/>
<font color="Maroon" title="c27">xi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">yi</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><b>hence </b><a NAME="E7:44_6_5_1_1_2_1_1_2"/>
<font color="Maroon" title="c27">xi</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c28">yi</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</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></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:44_6_5"/><b>then </b>
<font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c20">z'</font>
 
<b>by </b><i><a class="txt" href="#E8:44"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="#E4:44_6_5"><i><font color="Green" title="E49">A98</font></i></a>, <a class="txt" href="#E7:44_6_5"><i><font color="Green" title="E50">A99</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E12:44_6_5"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c17">y</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18" title="WAYBEL_0:th.18">WAYBEL_0:18</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>
<a NAME="E27:44_6"/>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c17">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">X'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_6_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c17">y</font> or <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X'</font> )</span><br/>

<b>assume </b><a NAME="E1:44_6_6"/><i><font color="Green" title="E47">A115</font></i>: 
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c17">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X'</font></span><br/>

<a NAME="E2:44_6_6"/><b>then </b><i><font color="Green" title="E48">A116</font></i>: 
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon" title="c20">z'</font> = <font color="Maroon" title="c19">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E1:44_6_6"><i><font color="Green" title="E47">A115</font></i></a></i>;<br/>
<a NAME="E4:44_6_6"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:44_6_6"><i><font color="Green" title="E48">A116</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">gg</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:44_6_6"/><i><font color="Green" title="E49">A117</font></i>: 
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">gg</font>
 <b>and </b><br/><a NAME="E7:44_6_6"/><i><font color="Green" title="E50">A118</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E8:44_6_6"/><i><font color="Green" title="E51">A119</font></i>: 
for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c21">gg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E9:44_6_6"/>
<font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c20">z'</font>
 
<b>by </b><i><a class="txt" href="#E1:44_6_6"><i><font color="Green" title="E47">A115</font></i></a>, <a class="ref" href="waybel_0.html#T18" title="WAYBEL_0:th.18">WAYBEL_0:18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">h1</font>, <font color="Maroon" title="c23">h2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E11:44_6_6"/><i><font color="Green" title="E52">A120</font></i>: 
<font color="Maroon" title="c22">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font>
 <b>and </b><br/><a NAME="E12:44_6_6"/><i><font color="Green" title="E53">A121</font></i>: 
<font color="Maroon" title="c23">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">z'</font>
 <b>and </b><br/><a NAME="E13:44_6_6"/><i><font color="Green" title="E54">A122</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b3">xi</font>, <font color="Olive" title="b4">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">R</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">yi</font> )
 <b>by </b><i><a class="txt" href="#E8:44"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E14:44_6_6"/><i><font color="Green" title="E55">A123</font></i>: 
<font color="Maroon" title="c22">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c16">x</font>,<font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="txt" href="#E16:44"><i><font color="Green" title="E29">A15</font></i></a>, <a class="txt" href="#E11:44_6_6"><i><font color="Green" title="E52">A120</font></i></a></i>;<br/>
<a NAME="E15:44_6_6"/>
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">ZZ</font> holds <br/><font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_6_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c24">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">ZZ</font> implies <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Z</font> )</span><br/>

<b>assume </b><a NAME="E1:44_6_6_1"/>
<font color="Maroon" title="c24">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">ZZ</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Z</font></span><br/>

<b>then consider </b><font color="Maroon" title="c25">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E3:44_6_6_1"/><i><font color="Green" title="E56">A124</font></i>: 
<font color="Maroon" title="c24">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c25">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:44_6_6_1"/><i><font color="Green" title="E57">A125</font></i>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x</font>
 <b>by </b><i><a class="txt" href="#E19:44_6"><i><font color="Green" title="E42">A91</font></i></a></i>;<br/>
<i><font color="Green" title="E58">A126</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c25">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c26">RS</font> being    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> , <font color="Maroon" title="c27">xj</font>, <font color="Maroon" title="c28">yj</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c26">RS</font><b> such that </b><br/><a NAME="E7:44_6_6_1"/><i><font color="Green" title="E59">A127</font></i>: 
<font color="Maroon" title="c26">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c25">j</font>
 <b>and </b><br/><a NAME="E8:44_6_6_1"/><i><font color="Green" title="E60">A128</font></i>: 
<font color="Maroon" title="c27">xj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>
 <b>and </b><br/><a NAME="E9:44_6_6_1"/><i><font color="Green" title="E61">A129</font></i>: 
<font color="Maroon" title="c28">yj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>
 <b>and </b><br/><a NAME="E10:44_6_6_1"/><i><font color="Green" title="E62">A130</font></i>: 
<font color="Maroon" title="c27">xj</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c28">yj</font>
 <b>by </b><i><a class="txt" href="#E13:44_6_6"><i><font color="Green" title="E54">A122</font></i></a></i>;<br/>
<a NAME="E11:44_6_6_1"/><i><font color="Green" title="E63">A131</font></i>: 
<font color="Maroon" title="c27">xj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="#E14:44_6_6"><i><font color="Green" title="E55">A123</font></i></a>, <a class="txt" href="#E4:44_6_6_1"><i><font color="Green" title="E57">A125</font></i></a>, <a class="txt" href="#E8:44_6_6_1"><i><font color="Green" title="E60">A128</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E12:44_6_6_1"/><i><font color="Green" title="E64">A132</font></i>: 
<font color="Maroon" title="c26">RS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1
 
<b>by </b><i><a class="txt" href="#E7:44_6_6_1"><i><font color="Green" title="E59">A127</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c29">b</font> = <font color="Maroon" title="c28">yj</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span> <b>by </b><i><a class="txt" href="#E7:44_6_6_1"><i><font color="Green" title="E59">A127</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E14:44_6_6_1"/><i><font color="Green" title="E65">A133</font></i>: 
<font color="Maroon" title="c29">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span>
 
;<br/>
<a NAME="E15:44_6_6_1"/><i><font color="Green" title="E66">A134</font></i>: 
<font color="Maroon" title="c28">yj</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_6_6_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_6_6_1_2"/>
<font color="Maroon" title="c28">yj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:44_6_6_1_2"/><b>then </b><i><font color="Green" title="E67">A135</font></i>: 
1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E10:44_6_6_1"><i><font color="Green" title="E62">A130</font></i></a>, <a class="txt" href="#E11:44_6_6_1"><i><font color="Green" title="E63">A131</font></i></a>, <a class="txt" href="#E12:44_6_6_1"><i><font color="Green" title="E64">A132</font></i></a>, <a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/>
<b>thus </b><a NAME="E3:44_6_6_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:44_6_6_1_2"><i><font color="Green" title="E67">A135</font></i></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>
<a NAME="E16:44_6_6_1"/><i><font color="Green" title="E67">A136</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c23">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c25">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:44_6_6"><i><font color="Green" title="E49">A117</font></i></a>, <a class="txt" href="#E7:44_6_6"><i><font color="Green" title="E50">A118</font></i></a>, <a class="txt" href="#E12:44_6_6"><i><font color="Green" title="E53">A121</font></i></a>, <a class="txt" href="#E5:44_6_6_1"><i><font color="Green" title="E58">A126</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E17:44_6_6_1"/>
for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c25">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds <br/><font color="Maroon" title="c23">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c25">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_6_6_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c30">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c25">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> implies <font color="Maroon" title="c23">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c25">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> )</span><br/>

<b>assume </b><a NAME="E1:44_6_6_1_3"/>
<font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c25">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c23">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c25">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font></span><br/>

<a NAME="E2:44_6_6_1_3"/><b>then </b><i><font color="Green" title="E68">A137</font></i>: 
<font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:44_6_6_1"><i><font color="Green" title="E58">A126</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_6_6_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_6_6_1_3_1"/>
( <font color="Maroon" title="c30">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j</font> or <font color="Maroon" title="c30">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_6_6_1_3_1_1"><b>suppose </b></a><a NAME="E1:44_6_6_1_3_1_1"/><i><font color="Green" title="E69">A138</font></i>: 
<font color="Maroon" title="c30">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c23">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c25">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font></span><br/><div class="add"><a NAME="E2:44_6_6_1_3_1_1"/><b>then </b><i><font color="Green" title="E70">A139</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c25">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:44_6_6_1_3"><i><font color="Green" title="E68">A137</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:44_6_6_1_3_1_1"/>
<font color="Maroon" title="c23">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="#E3:44"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="#E9:44_6_6_1"><i><font color="Green" title="E61">A129</font></i></a>, <a class="txt" href="#E14:44_6_6_1"><i><font color="Green" title="E65">A133</font></i></a>, <a class="txt" href="#E15:44_6_6_1"><i><font color="Green" title="E66">A134</font></i></a>, <a class="txt" href="#E1:44_6_6_1_3_1_1"><i><font color="Green" title="E69">A138</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:44_6_6_1_3_1_1"/>
<font color="Maroon" title="c23">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c25">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font>
 <b>by </b><i><a class="txt" href="#E2:44_6_6_1_3_1_1"><i><font color="Green" title="E70">A139</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_6_6_1_3_1_2"><b>suppose </b></a><a NAME="E1:44_6_6_1_3_1_2"/>
<font color="Maroon" title="c30">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c23">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c25">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font></span><br/><div class="add"><a NAME="E2:44_6_6_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c25">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:44_6_6_1_3_1_2"/>
<font color="Maroon" title="c23">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c25">j</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font>
 <b>by </b><i><a class="txt" href="#E6:44"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="#E6:44_6_6"><i><font color="Green" title="E49">A117</font></i></a>, <a class="txt" href="#E8:44_6_6"><i><font color="Green" title="E51">A119</font></i></a>, <a class="txt" href="#E12:44_6_6"><i><font color="Green" title="E53">A121</font></i></a>, <a class="txt" href="#E2:44_6_6_1_3"><i><font color="Green" title="E68">A137</font></i></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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E18:44_6_6_1"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Z</font>
 <b>by </b><i><a class="txt" href="#E12:44_6_6"><i><font color="Green" title="E53">A121</font></i></a>, <a class="txt" href="#E3:44_6_6_1"><i><font color="Green" title="E56">A124</font></i></a>, <a class="txt" href="#E16:44_6_6_1"><i><font color="Green" title="E67">A136</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</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>hence </b><a NAME="E16:44_6_6"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X'</font>
 <b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E7:44"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="#E6:44_6"><i><font color="Green" title="E35">A78</font></i></a>, <a class="txt" href="#E1:44_6_6"><i><font color="Green" title="E47">A115</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</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>
<a NAME="E28:44_6"/><b>then </b><i><font color="Green" title="E47">A140</font></i>: 
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c18">Y</font>
 
<b>by </b><i><a class="txt" href="#E25:44_6"><i><font color="Green" title="E45">A94</font></i></a>, <a class="txt" href="#E26:44_6"><i><font color="Green" title="E46">A95</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="E29:44_6"/>
<font color="Maroon" title="c18">Y</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E23:44_6"><i><font color="Green" title="E44">A93</font></i></a>, <a class="ref" href="waybel_8.html#T9" title="WAYBEL_8:th.9">WAYBEL_8:9</a></i>;<br/>
<a NAME="E30:44_6"/><b>then </b>
<font color="Maroon" title="c18">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c6">T'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_8.html#D1" title="WAYBEL_8:def.1">WAYBEL_8:def 1</a></i>;<br/>
<b>hence </b><a NAME="E31:44_6"/>
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font>
 <b>by </b><i><a class="txt" href="#E13:44"><i><font color="Green" title="E27">A13</font></i></a>, <a class="txt" href="#E28:44_6"><i><font color="Green" title="E47">A140</font></i></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>
<a NAME="E20:44"/><b>then </b><i><font color="Green" title="E32">A141</font></i>: 
<font color="Maroon" title="c7">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="txt" href="#E18:44"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">P'</font> = <font color="Maroon" title="c5">P</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">T</font> <b>by </b><i><a class="txt" href="#E7:44"><i><font color="Green" title="E25">A11</font></i></a></i>;<br/>
<a NAME="E22:44"/>
<font color="Maroon" title="c9">P'</font> is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="txt" href="#E7:44"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="#E20:44"><i><font color="Green" title="E32">A141</font></i></a>, <a class="ref" href="yellow_9.html#T23" title="YELLOW_9:th.23">YELLOW_9:23</a></i>;<br/>
<b>hence </b><a NAME="E23:44"/>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font> <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 <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:44"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="yellow_9.html#T26" title="YELLOW_9:th.26">YELLOW_9:26</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>
