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<b>let </b><font color="Maroon" title="c1">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="fintopo4.html#K4" title="FINTOPO4:func.4">FTSC1</a> <font color="Maroon" title="c1">n</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> )</span><br/>

<b>assume </b><a NAME="E1:24"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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">  <a href="fintopo4.html#K4" title="FINTOPO4:func.4">FTSC1</a> <font color="Maroon" title="c1">n</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a></span><br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green" title="E12">A1</font></i>: 
 <a href="fintopo4.html#K4" title="FINTOPO4:func.4">FTSC1</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="fintopo4.html#K3" title="FINTOPO4:func.3">Nbdc1</a> <font color="Maroon" title="c1">n</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="fintopo4.html#D6" target="_self" title="FINTOPO4:def.6">Def6</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K4" title="FINTOPO4:func.4">FTSC1</a> <font color="Maroon" title="c1">n</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="fin_topo.html#D4" title="FIN_TOPO:def.4">FIN_TOPO:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c2">x</font></span><br/>

<a NAME="E3:24"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E2:24"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">i</font> = <font color="Maroon" title="c2">x</font> as    <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>  ;<br/>
<b>set </b><font color="Maroon" title="c4">f</font> =  <a href="fintopo4.html#K3" title="FINTOPO4:func.3">Nbdc1</a> <font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E5:24"/><i><font color="Green" title="E13">A2</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="fintopo4.html#E2:24"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:24"/><i><font color="Green" title="E14">A3</font></i>: 
( ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> implies  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <span class="p1">(<span class="default"><a href="fintopo4.html#K3" title="FINTOPO4:func.3">Nbdc1</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> ) &amp; ( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> implies  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <span class="p1">(<span class="default"><a href="fintopo4.html#K3" title="FINTOPO4:func.3">Nbdc1</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> ) &amp; ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> implies  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <span class="p1">(<span class="default"><a href="fintopo4.html#K3" title="FINTOPO4:func.3">Nbdc1</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> ) &amp; ( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> implies  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <span class="p1">(<span class="default"><a href="fintopo4.html#K3" title="FINTOPO4:func.3">Nbdc1</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c3">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"><font color="Maroon" title="c3">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 
<b>by </b><i><a class="txt" href="fintopo4.html#E2:24"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="fintopo4.html#D5" target="_self" title="FINTOPO4:def.5">Def5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1"/>
( ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> ) or ( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> ) or ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> ) or ( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> ) )
 </span><b>by </b><i><a class="txt" href="fintopo4.html#E5:24"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>suppose </b></a><a NAME="E1:24_1_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c2">x</font></span><br/><div class="add"><b>hence </b><a NAME="E2:24_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="fintopo4.html#E2:24"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="fintopo4.html#E6:24"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1: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="24_1_2"><b>suppose </b></a><a NAME="E1:24_1_2"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c2">x</font></span><br/><div class="add"><b>hence </b><a NAME="E2:24_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="fintopo4.html#E2:24"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="fintopo4.html#E6:24"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1: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="24_1_3"><b>suppose </b></a><a NAME="E1:24_1_3"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c2">x</font></span><br/><div class="add"><b>hence </b><a NAME="E2:24_1_3"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="fintopo4.html#E2:24"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="fintopo4.html#E6:24"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1: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="24_1_4"><b>suppose </b></a><a NAME="E1:24_1_4"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c2">x</font></span><br/><div class="add"><b>hence </b><a NAME="E2:24_1_4"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="fintopo4.html#E2:24"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="fintopo4.html#E6:24"><i><font color="Green" title="E14">A3</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><b>end;</b></div>

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