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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">N</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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>   st <font color="Olive" title="b1">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Olive" title="b1">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c2">N</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> implies  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c2">N</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:14"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c2">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E2:14"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c2">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E3:14"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c2">N</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">N</font> implies <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:14_1"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c4">k9</font> = <font color="Maroon" title="c3">k</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a></span>;<br/><a NAME="E3:14_1"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c2">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></span>;<br/><a NAME="E4:14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:14_1"/>
<font color="Maroon" title="c4">k9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="stirl2_1.html#T10" target="_self" title="STIRL2_1:th.10">Th10</a></span>;<br/><a NAME="E6:14_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:14_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:14"/><span class="kw">then </span>
<font color="Maroon" title="c2">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:14"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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