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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">S</font> being  <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">T</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">S</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> holds <br/> ex <font color="Olive" title="b3">R</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a> <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b3">R</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> &amp; ( <font color="Olive" title="b1">F</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> &amp; <font color="Olive" title="b2">S</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Olive" title="b3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( for <font color="Olive" title="b4">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b5">Si</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b3">R</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b5">Si</font></span>)</span> &amp; <font color="Olive" title="b5">Si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b6">j</font></span>)</span> where <font color="Olive" title="b6">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b6">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b4">i</font> </span>}</span>  ) ) )</span><br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">T</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b1">S</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> holds <br/> ex <font color="Olive" title="b2">R</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a> <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">R</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> &amp; ( <font color="Maroon" title="c2">F</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> &amp; <font color="Olive" title="b1">S</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( for <font color="Olive" title="b3">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b4">Si</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b4">Si</font></span>)</span> &amp; <font color="Olive" title="b4">Si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b5">j</font></span>)</span> where <font color="Olive" title="b5">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">i</font> </span>}</span>  ) ) )</span><br/><b>let </b><font color="Maroon" title="c3">S</font> be   <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c3">S</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> implies  ex <font color="Olive" title="b1">R</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a> <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">R</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> &amp; ( <font color="Maroon" title="c2">F</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( for <font color="Olive" title="b2">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b3">Si</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">R</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b3">Si</font></span>)</span> &amp; <font color="Olive" title="b3">Si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b4">j</font></span>)</span> where <font color="Olive" title="b4">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">i</font> </span>}</span>  ) ) ) )</span><br/>





<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E31">A1</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c3">S</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</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">R</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a> <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">R</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> &amp; ( <font color="Maroon" title="c2">F</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( for <font color="Olive" title="b2">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b3">Si</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">R</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b3">Si</font></span>)</span> &amp; <font color="Olive" title="b3">Si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b4">j</font></span>)</span> where <font color="Olive" title="b4">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">i</font> </span>}</span>  ) ) )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> ex <font color="Olive" title="b2">SS</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">SS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b2">SS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b3">j</font></span>)</span> where <font color="Olive" title="b3">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">i</font> </span>}</span>  &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b2">SS</font></span>)</span> );<br/>
<a NAME="E2:38"/><i><font color="Green" title="E32">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:38_1"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<b>reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E1:38_1"><i><font color="Green" title="E33">A3</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">SS</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">x'</font> </span>}</span> ;<br/>
<a NAME="E3:38_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">x'</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">y</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="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">x'</font> </span>}</span>  or <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )</span><br/>

<b>assume </b><a NAME="E1:38_1_1"/><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">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="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><br/>

<a NAME="E2:38_1_1"/>
 ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">x'</font> )
 
<b>by </b><i><a class="txt" href="#E1:38_1_1"><i><font color="Green" title="E34">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">SS</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">x'</font> </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>take </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">SS</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">SS</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>, <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">SS</font></span>)</span>] )</span><br/>

<a NAME="E5:38_1"/>
<font color="Maroon" title="c7">SS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">y</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="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">SS</font> or <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )</span><br/>

<b>assume </b><a NAME="E1:38_1_2"/><i><font color="Green" title="E34">A5</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">SS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/>

<b>consider </b><font color="Maroon" title="c9">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:38_1_2"/><i><font color="Green" title="E35">A6</font></i>: 
( <font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">x'</font> )
 <b>by </b><i><a class="txt" href="#E1:38_1_2"><i><font color="Green" title="E34">A5</font></i></a></i>;<br/>
<a NAME="E4:38_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:38_1_2"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="#E3:38_1_2"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:38_1_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E1:38"><i><font color="Green" title="E31">A1</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>hence </b><a NAME="E6:38_1"/>
(  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">SS</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>, <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">SS</font></span>)</span>] )
 ; <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>consider </b><font color="Maroon" title="c4">R</font> being   <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:38"/><i><font color="Green" title="E33">A7</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E2:38"><i><font color="Green" title="E32">A2</font></i></a>);<br/></i>
<div><i><font color="Green" title="E34">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n</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">n</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="c4">R</font> implies  not <font color="Maroon" title="c4">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/><b>assume </b><a NAME="E1:38_2"/><i><font color="Green" title="E35">A9</font></i>: 
<font color="Maroon" title="c5">n</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="c4">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c4">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/><b>reconsider </b><font color="Maroon" title="c6">n'</font> = <font color="Maroon" title="c5">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E1:38_2"><i><font color="Green" title="E35">A9</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c7">SS</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:38_2"/>
<font color="Maroon" title="c7">SS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 <b>and </b><br/><a NAME="E5:38_2"/><i><font color="Green" title="E36">A10</font></i>: 
( <font color="Maroon" title="c7">SS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">n'</font> </span>}</span>  &amp; <font color="Maroon" title="c4">R</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c6">n'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">SS</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:38"><i><font color="Green" title="E33">A7</font></i></a></i>;<br/><a NAME="E6:38_2"/><i><font color="Green" title="E37">B11</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E7:38_2"/>
<font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c6">n'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">SS</font>
 <b>by </b><i><a class="txt" href="#E5:38_2"><i><font color="Green" title="E36">A10</font></i></a></i>;<br/><a NAME="E8:38_2"/><b>then </b>
<font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c6">n'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">SS</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E9:38_2"/><b>then </b>
( <font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c6">n'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c6">n'</font></span>)</span> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c6">n'</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">SS</font></span>)</span> )
 <b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a>, <a class="ref" href="pre_topc.html#T49" title="PRE_TOPC:th.49">PRE_TOPC:49</a></i>;<br/><b>hence </b><a NAME="E10:38_2"/>
not <font color="Maroon" title="c4">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="#E5:38_2"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="#E6:38_2"><i><font color="Green" title="E37">B11</font></i></a>, <a class="txt" href="#E1:38"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</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="hs2(this)" href="javascript:()" title="38_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">R</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/><b>reconsider </b><font color="Maroon" title="c6">n'</font> = <font color="Maroon" title="c5">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E2:38_3"/>
 ex <font color="Olive" title="b1">SS</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">SS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b1">SS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span> where <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">n'</font> </span>}</span>  &amp; <font color="Maroon" title="c4">R</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c6">n'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">SS</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:38"><i><font color="Green" title="E33">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E3:38_3"/>
<font color="Maroon" title="c4">R</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <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="c5">R</font> = <font color="Maroon" title="c4">R</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a> <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="#E5:38"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="compl_sp.html#D6" target="_self" title="COMPL_SP:def.6">Def6</a>, <a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">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="c5">R</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> &amp; ( <font color="Maroon" title="c2">F</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c5">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( for <font color="Olive" title="b1">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b2">Si</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c5">R</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b2">Si</font></span>)</span> &amp; <font color="Olive" title="b2">Si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b3">j</font></span>)</span> where <font color="Olive" title="b3">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">i</font> </span>}</span>  ) ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">R</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/><b>consider </b><font color="Maroon" title="c7">Sn</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E2:38_4"/>
<font color="Maroon" title="c7">Sn</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 <b>and </b><br/><a NAME="E3:38_4"/><i><font color="Green" title="E35">A12</font></i>: 
<font color="Maroon" title="c7">Sn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">n</font> </span>}</span> 
 <b>and </b><br/><a NAME="E4:38_4"/><i><font color="Green" title="E36">A13</font></i>: 
<font color="Maroon" title="c5">R</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">Sn</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:38"><i><font color="Green" title="E33">A7</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">Sn1</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E6:38_4"/>
<font color="Maroon" title="c8">Sn1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 <b>and </b><br/><a NAME="E7:38_4"/><i><font color="Green" title="E37">A14</font></i>: 
<font color="Maroon" title="c8">Sn1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 </span>}</span> 
 <b>and </b><br/><a NAME="E8:38_4"/><i><font color="Green" title="E38">A15</font></i>: 
<font color="Maroon" title="c5">R</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">Sn1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:38"><i><font color="Green" title="E33">A7</font></i></a></i>;<br/><a NAME="E9:38_4"/>
<font color="Maroon" title="c8">Sn1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Sn</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">y</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Sn1</font> or <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Sn</font> )</span><br/>

<b>assume </b><a NAME="E1:38_4_1"/><i><font color="Green" title="E39">A16</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Sn1</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Sn</font></span><br/>

<b>consider </b><font color="Maroon" title="c10">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:38_4_1"/><i><font color="Green" title="E40">A17</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="#E7:38_4"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="#E1:38_4_1"><i><font color="Green" title="E39">A16</font></i></a></i>;<br/>
<a NAME="E4:38_4_1"/>
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="txt" href="#E3:38_4_1"><i><font color="Green" title="E40">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E5:38_4_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Sn</font>
 <b>by </b><i><a class="txt" href="#E3:38_4"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="#E3:38_4_1"><i><font color="Green" title="E40">A17</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="E10:38_4"/><b>then </b>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">Sn1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">Sn</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></i>;<br/><b>hence </b><a NAME="E11:38_4"/>
<font color="Maroon" title="c5">R</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="#E4:38_4"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="#E8:38_4"><i><font color="Green" title="E38">A15</font></i></a>, <a class="ref" href="pre_topc.html#T49" title="PRE_TOPC:th.49">PRE_TOPC:49</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="E9:38"/>
<font color="Maroon" title="c5">R</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>
 <b>by </b><i><a class="ref" href="kurato_2.html#D5" title="KURATO_2:def.5">KURATO_2:def 5</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="c2">F</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c5">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( for <font color="Olive" title="b1">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b2">Si</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c5">R</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b2">Si</font></span>)</span> &amp; <font color="Olive" title="b2">Si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b3">j</font></span>)</span> where <font color="Olive" title="b3">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">i</font> </span>}</span>  ) ) )</span><br/>

<b>thus </b><a NAME="E10:38"/>
( <font color="Maroon" title="c2">F</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c5">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b2">Si</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c5">R</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b2">Si</font></span>)</span> &amp; <font color="Olive" title="b2">Si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b3">j</font></span>)</span> where <font color="Olive" title="b3">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">i</font> </span>}</span>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_5"/><i><font color="Green" title="E35">A18</font></i>: 
( <font color="Maroon" title="c2">F</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c5">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<b>assume </b><a NAME="E2:38_5"/>
 <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c5">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:38_5"/><i><font color="Green" title="E36">A19</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="#E4:38_5"><i><font color="Green" title="E36">A19</font></i></a></i>;<br/>
<a NAME="E6:38_5"/><i><font color="Green" title="E37">A20</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">rngS</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">S</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<a NAME="E8:38_5"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">S</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a>
 
<b>by </b><i><a class="txt" href="#E1:38"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="#E1:38_5"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="pcomps_1.html#T12" title="PCOMPS_1:th.12">PCOMPS_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E10:38_5"/><i><font color="Green" title="E38">A21</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">W</font> &amp; <font color="Maroon" title="c9">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>and </b><br/><a NAME="E11:38_5"/><i><font color="Green" title="E39">A22</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">V</font> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rngS</font> &amp; <font color="Olive" title="b1">V</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">W</font> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D2" title="PCOMPS_1:def.2">PCOMPS_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">Sp</font> = <font color="Maroon" title="c3">S</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c8">rngS</font> <b>by </b><i><a class="txt" href="#E6:38_5"><i><font color="Green" title="E37">A20</font></i></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">S'</font> = <font color="Maroon" title="c10">Sp</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">S'</font> = <font color="Maroon" title="c11">S'</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c8">rngS</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E1:38_5"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="funct_2.html#T31" title="FUNCT_2:th.31">FUNCT_2:31</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">rngS</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = <font color="Maroon" title="c12">S'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/>
<b>set </b><font color="Maroon" title="c13">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">V</font> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rngS</font> &amp; <font color="Olive" title="b1">V</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">W</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c14">Y</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">rngS</font> : <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">V</font> where <font color="Olive" title="b2">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rngS</font> &amp; <font color="Olive" title="b2">V</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">W</font> ) </span>}</span>  </span>}</span> ;<br/>
<a NAME="E15:38_5"/><i><font color="Green" title="E40">A23</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">rngS</font> : <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">V</font> where <font color="Olive" title="b2">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rngS</font> &amp; <font color="Olive" title="b2">V</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">W</font> ) </span>}</span>  </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="#E11:38_5"><i><font color="Green" title="E39">A22</font></i></a>);<br/></i>
<a NAME="E16:38_5"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">rngS</font> : <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">V</font> where <font color="Olive" title="b2">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rngS</font> &amp; <font color="Olive" title="b2">V</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">W</font> ) </span>}</span>  </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">y</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="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">rngS</font> : <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">V</font> where <font color="Olive" title="b2">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rngS</font> &amp; <font color="Olive" title="b2">V</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">W</font> ) </span>}</span>  </span>}</span>  or <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<b>assume </b><a NAME="E1:38_5_1"/><i><font color="Green" title="E41">A24</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">rngS</font> : <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">V</font> where <font color="Olive" title="b2">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rngS</font> &amp; <font color="Olive" title="b2">V</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">W</font> ) </span>}</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="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<a NAME="E2:38_5_1"/>
 ex <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">rngS</font> st <br/>( <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">z</font>) &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">V</font> where <font color="Olive" title="b2">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rngS</font> &amp; <font color="Olive" title="b2">V</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">W</font> ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="#E1:38_5_1"><i><font color="Green" title="E41">A24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:38_5_1"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <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="c15">Y</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">rngS</font> : <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">V</font> where <font color="Olive" title="b2">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rngS</font> &amp; <font color="Olive" title="b2">V</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">W</font> ) </span>}</span>  </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c16">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E19:38_5"/><i><font color="Green" title="E41">A25</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Y</font> holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font>
 <b>by </b><i><a class="txt" href="#E15:38_5"><i><font color="Green" title="E40">A23</font></i></a>, <a class="ref" href="stirl2_1.html#T66" title="STIRL2_1:th.66">STIRL2_1:66</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">n1</font> = <font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<b>consider </b><font color="Maroon" title="c18">Sn</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E21:38_5"/><i><font color="Green" title="E42">A26</font></i>: 
<font color="Maroon" title="c18">Sn</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 <b>and </b><br/><a NAME="E22:38_5"/><i><font color="Green" title="E43">A27</font></i>: 
<font color="Maroon" title="c18">Sn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 </span>}</span> 
 <b>and </b><br/><a NAME="E23:38_5"/><i><font color="Green" title="E44">A28</font></i>: 
<font color="Maroon" title="c5">R</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c18">Sn</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:38"><i><font color="Green" title="E33">A7</font></i></a></i>;<br/>
<a NAME="E24:38_5"/>
<font color="Maroon" title="c18">Sn</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a>
 
<b>by </b><i><a class="txt" href="#E1:38_5"><i><font color="Green" title="E35">A18</font></i></a>, <a class="txt" href="#E21:38_5"><i><font color="Green" title="E42">A26</font></i></a>, <a class="ref" href="pcomps_1.html#T12" title="PCOMPS_1:th.12">PCOMPS_1:12</a></i>;<br/>
<a NAME="E25:38_5"/><b>then </b>
(  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c18">Sn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c18">Sn</font></span>)</span> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="#E4:38_5"><i><font color="Green" title="E36">A19</font></i></a>, <a class="ref" href="kurato_2.html#T6" title="KURATO_2:th.6">KURATO_2:6</a>, <a class="ref" href="pcomps_1.html#T23" title="PCOMPS_1:th.23">PCOMPS_1:23</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">CLF</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E27:38_5"/><i><font color="Green" title="E45">A29</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">CLF</font> &amp; <font color="Maroon" title="c19">CLF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c18">Sn</font> )
 <b>by </b><i><a class="txt" href="#E23:38_5"><i><font color="Green" title="E44">A28</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">CLF</font> = <font color="Maroon" title="c19">CLF</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="#E27:38_5"><i><font color="Green" title="E45">A29</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">U</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E30:38_5"/><i><font color="Green" title="E46">A30</font></i>: 
( <font color="Maroon" title="c20">CLF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c21">U</font> &amp; <font color="Maroon" title="c21">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Sn</font> )
 <b>by </b><i><a class="txt" href="#E27:38_5"><i><font color="Green" title="E45">A29</font></i></a>, <a class="ref" href="pcomps_1.html#D3" title="PCOMPS_1:def.3">PCOMPS_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E32:38_5"/><i><font color="Green" title="E47">A31</font></i>: 
( <font color="Maroon" title="c21">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c22">j</font> &amp; <font color="Maroon" title="c22">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="#E22:38_5"><i><font color="Green" title="E43">A27</font></i></a>, <a class="txt" href="#E30:38_5"><i><font color="Green" title="E46">A30</font></i></a></i>;<br/>
<a NAME="E33:38_5"/>
( <font color="Maroon" title="c10">Sp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rngS</font> &amp; <font color="Maroon" title="c10">Sp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">j</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">W</font> )
 
<b>by </b><i><a class="txt" href="#E10:38_5"><i><font color="Green" title="E38">A21</font></i></a>, <a class="txt" href="#E27:38_5"><i><font color="Green" title="E45">A29</font></i></a>, <a class="txt" href="#E30:38_5"><i><font color="Green" title="E46">A30</font></i></a>, <a class="txt" href="#E32:38_5"><i><font color="Green" title="E47">A31</font></i></a>, <a class="ref" href="tops_1.html#T39" title="TOPS_1:th.39">TOPS_1:39</a></i>;<br/>
<a NAME="E34:38_5"/><b>then </b><i><font color="Green" title="E48">X</font></i>: 
<font color="Maroon" title="c10">Sp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">V</font> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rngS</font> &amp; <font color="Olive" title="b1">V</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">W</font> ) </span>}</span> 
 
;<br/>
<a NAME="E35:38_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c22">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j</font>
 
<b>by </b><i><a class="txt" href="#E1:38_5"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></i>;<br/>
<a NAME="E36:38_5"/><b>then </b>
<font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Y</font>
 
<b>by </b><i><a class="txt" href="#E34:38_5"><i><font color="Green" title="E48">X</font></i></a></i>;<br/>
<a NAME="E37:38_5"/><b>then </b>
<font color="Maroon" title="c22">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font>
 
<b>by </b><i><a class="txt" href="#E19:38_5"><i><font color="Green" title="E41">A25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E38:38_5"/>
contradiction
 <b>by </b><i><a class="txt" href="#E32:38_5"><i><font color="Green" title="E47">A31</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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>let </b><font color="Maroon" title="c6">i</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">Si</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c5">R</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">Si</font></span>)</span> &amp; <font color="Olive" title="b1">Si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span> where <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> </span>}</span>  )</span><br/>

<a NAME="E11:38"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E12:38"/><b>then </b>
 ex <font color="Olive" title="b1">SS</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">SS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b1">SS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span> where <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> </span>}</span>  &amp; <font color="Maroon" title="c5">R</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">SS</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E4:38"><i><font color="Green" title="E33">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:38"/>
 ex <font color="Olive" title="b1">Si</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c5">R</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">Si</font></span>)</span> &amp; <font color="Olive" title="b1">Si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span> where <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> </span>}</span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
