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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</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> ex <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 <br/>( <font color="Olive" title="b1">n</font>,$1 <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> $1 );<br/>
<b>assume </b><a NAME="E1:49"/>
( <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">m</font> )
 ;<br/>

<a NAME="E2:49"/><b>then </b><i><font color="Green" title="E34">A1</font></i>: 
 ex <font color="Olive" title="b1">m</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">m</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c3">m</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> <b> such that </b><br/><a NAME="E4:49"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">m</font>]
 <b>and </b><br/><a NAME="E5:49"/><i><font color="Green" title="E36">A3</font></i>: 
for <font color="Olive" title="b1">k</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">k</font>] holds <br/><font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="card_1.html#E2:49"><i><font color="Green" title="E34">A1</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c4">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> <b> such that </b><br/><a NAME="E7:49"/><i><font color="Green" title="E37">A4</font></i>: 
( <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">m</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">m</font> )
 <b>by </b><i><a class="txt" href="card_1.html#E4:49"><i><font color="Green" title="E35">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:49"/><i><font color="Green" title="E38">A5</font></i>: 
( <font color="Maroon" title="c5">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> )
 <b>by </b><i><a class="txt" href="card_1.html#E7:49"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E10:49"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="card_1.html#E7:49"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="card_1.html#E9:49"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">m1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E12:49"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">m1</font> = <font color="Maroon" title="c6">m1</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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E14:49"/><i><font color="Green" title="E40">A7</font></i>: 
not <font color="Maroon" title="c7">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">m1</font>
 
;<br/>
<i><font color="Green" title="E41">A8</font></i>: <font color="Maroon" title="c7">m1</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="c7">m1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">m1</font>
<b>by </b><i><a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="txt" href="card_1.html#E12:49"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="card_1.html#T52" target="_self" title="CARD_1:th.52">Th52</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E42">A9</font></i>: <font color="Maroon" title="c3">m</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">m1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
<font color="Maroon" title="c7">m1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">m1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40" title="XBOOLE_1:th.40">XBOOLE_1:40</a></i>
<br/>.= 
<font color="Maroon" title="c7">m1</font>
<b>by </b><i><a class="txt" href="card_1.html#E14:49"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></i>
;<br/>
<a NAME="E17:49"/>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="card_1.html#E7:49"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="card_1.html#E9:49"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">n1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E19:49"/><i><font color="Green" title="E43">A10</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">n1</font> = <font color="Maroon" title="c8">n1</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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E21:49"/><i><font color="Green" title="E44">A11</font></i>: 
not <font color="Maroon" title="c9">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">n1</font>
 
;<br/>
<i><font color="Green" title="E45">A12</font></i>: <font color="Maroon" title="c9">n1</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="c9">n1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c9">n1</font>
<b>by </b><i><a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">n</font>
<b>by </b><i><a class="txt" href="card_1.html#E19:49"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="card_1.html#T52" target="_self" title="CARD_1:th.52">Th52</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E46">A13</font></i>: <font color="Maroon" title="c4">n</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">n1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = 
<font color="Maroon" title="c9">n1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">n1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40" title="XBOOLE_1:th.40">XBOOLE_1:40</a></i>
<br/>.= 
<font color="Maroon" title="c9">n1</font>
<b>by </b><i><a class="txt" href="card_1.html#E21:49"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></i>
;<br/>
<a NAME="E24:49"/>
( <font color="Maroon" title="c9">n1</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="c9">n1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c7">m1</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="c7">m1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E25:49"/><b>then </b>
( <font color="Maroon" title="c9">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c7">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">m</font> )
 
<b>by </b><i><a class="txt" href="card_1.html#E15:49"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="card_1.html#E22:49"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E26:49"/><b>then </b>
<font color="Maroon" title="c9">n1</font>,<font color="Maroon" title="c7">m1</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="card_1.html#E7:49"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="card_1.html#E16:49"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="card_1.html#E23:49"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="card_1.html#T61" target="_self" title="CARD_1:th.61">Th61</a></i>;<br/>
<a NAME="E27:49"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m1</font>
 
<b>by </b><i><a class="txt" href="card_1.html#E5:49"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="card_1.html#E7:49"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="card_1.html#E12:49"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="card_1.html#E19:49"><i><font color="Green" title="E43">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E28:49"/>
contradiction
 <b>by </b><i><a class="txt" href="card_1.html#E12:49"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>


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