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<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">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> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ] <b>means </b> <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c1">m</font>,$1,$1 <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> ;<br/>
<a NAME="E1:8"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="calcul_2.html#T2" target="_self" title="CALCUL_2:th.2">Th2</a></i>;<br/>
<a NAME="E2:8"/><i><font color="Green" title="E8">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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> ;<br/>

<b>assume </b><a NAME="E1:8_1"/><i><font color="Green" title="E9">A3</font></i>: 
 <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c3">n</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">i</font> = <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">n</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/>
<a NAME="E3:8_1"/><i><font color="Green" title="E10">A4</font></i>: 
 <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c1">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="calcul_2.html#T5" target="_self" title="CALCUL_2:th.5">Th5</a></i>;<br/>
<i><font color="Green" title="E11">A5</font></i>: <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">n</font>
<b>by </b><i><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></i>
<br/>.= 
<font color="Maroon" title="c3">n</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></i>
;<br/>
<a NAME="E5:8_1"/><i><font color="Green" title="E12">A6</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T48" title="CARD_1:th.48">CARD_1:48</a></i>;<br/>
<div><i><font color="Green" title="E13">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_2"/>
<font color="Maroon" title="c3">n</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:8_1_2"/><i><font color="Green" title="E14">A8</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:8_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="calcul_2.html#E3:8_1_2"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:8_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="calcul_2.html#E3:8_1_2"><i><font color="Green" title="E14">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_3"/>
 <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c3">n</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:8_1_3"/><i><font color="Green" title="E14">A9</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:8_1_3"/><i><font color="Green" title="E15">A10</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> )
 <b>by </b><i><a class="txt" href="calcul_2.html#E3:8_1_3"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:8_1_3"/>
not <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E6:8_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="calcul_2.html#E3:8_1_3"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="calcul_2.html#E4:8_1_3"><i><font color="Green" title="E15">A10</font></i></a>, <a class="ref" href="calcul_2.html#T1" target="_self" title="CALCUL_2:th.1">Th1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:8_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="calcul_2.html#E1:8_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="calcul_2.html#E3:8_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="calcul_2.html#E4:8_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="calcul_2.html#E5:8_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="calcul_2.html#E6:8_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="card_1.html#T58" title="CARD_1:th.58">CARD_1:58</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:8"/>
for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="calcul_2.html#E1:8"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="calcul_2.html#E2:8"><i><font color="Green" title="E8">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:8"/>
 <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 ;<br/>


</div>
