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<span class="kw">let </span><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>  st <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a>  &amp; <font color="Olive" title="b1">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c2">n</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> holds <br/><font color="Olive" title="b1">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c2">n</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> implies <font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
( <font color="Maroon" title="c3">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a>  )
 <span class="kw">and </span><br/><a NAME="E2:43"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c2">n</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font></span><br/>

<span class="kw">assume </span><a NAME="E3:43"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_1_1"/>
( <font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">k</font> or <font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</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">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_1_1_1"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:43_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</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">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" onmouseover="rs('nat_1/T44')" onmouseout="rh()">NAT_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:43_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_1_1_2"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</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">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:43_1_1_2"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c1">k</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#T16" target="_self" onmouseover="rs('stirl2_1/T16')" onmouseout="rh()">Th16</a></span>;<br/><a NAME="E3:43_1_1_2"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/><a NAME="E4:43_1_1_2"/>
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E5:43_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="nat_1.html#T44" onmouseover="rs('nat_1/T44')" onmouseout="rh()">NAT_1:44</a></span>;<br/><a NAME="E6:43_1_1_2"/><span class="kw">then </span>
(  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p2"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c1">k</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> &amp; <font color="Maroon" title="c1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <font color="Maroon" title="c2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E1:43_1_1_2"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E3:43_1_1_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E7:43_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E2:43_1_1_2"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="stirl2_1.html#T5" target="_self" onmouseover="rs('stirl2_1/T5')" onmouseout="rh()">Th5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:43"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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