<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">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> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c1">k</font></span>)</span><br/>  for <font color="Olive" title="b2">g</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="c3">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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">n</font> holds <br/> for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">i</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="Olive" title="b2">g</font> &amp; <font color="Olive" title="b4">j</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="Olive" title="b2">g</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> holds <br/> <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</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="Olive" title="b3">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="Olive" title="b2">g</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="Olive" title="b4">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c1">k</font></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">  for <font color="Olive" title="b1">g</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="c3">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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>  st <font color="Maroon" title="c4">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">n</font> holds <br/> for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">i</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="Olive" title="b1">g</font> &amp; <font color="Olive" title="b3">j</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="Olive" title="b1">g</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> holds <br/> <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</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="Olive" title="b2">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="Olive" title="b1">g</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="Olive" title="b3">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">g</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="c3">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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></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="c4">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">n</font> implies  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c5">g</font> &amp; <font color="Olive" title="b2">j</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="c5">g</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> holds <br/> <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="Olive" title="b1">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="c5">g</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="Olive" title="b2">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
( <font color="Maroon" title="c4">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="stirl2_1.html#V1" title="STIRL2_1:attr.1">"increasing</a>  )
 <span class="kw">and </span><br/><a NAME="E2:48"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">n</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">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c5">g</font> &amp; <font color="Olive" title="b2">j</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="c5">g</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> holds <br/> <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="Olive" title="b1">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="c5">g</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="Olive" title="b2">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span><br/>

<a NAME="E3:48"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_1"/>
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  iff <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E29">A1</font></span></a></span>;<br/>
<a NAME="E2:48_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">i</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"> ( <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> implies <font color="Maroon" title="c4">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c6">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:48_1"/>
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</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"> <font color="Maroon" title="c4">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span><br/>

<a NAME="E4:48_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c3">n</font>
 
<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="E5:48_1"/>
<font color="Maroon" title="c4">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</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>
<a NAME="E4:48"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
 for <font color="Olive" title="b1">l</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">l</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="c5">g</font> &amp;  not <font color="Olive" title="b1">l</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="c4">f</font> holds <br/><font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c3">n</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">l</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"> ( <font color="Maroon" title="c6">l</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="c5">g</font> &amp;  not <font color="Maroon" title="c6">l</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="c4">f</font> implies <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c3">n</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48_2"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon" title="c6">l</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="c5">g</font>
 <span class="kw">and </span><br/><a NAME="E2:48_2"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
 not <font color="Maroon" title="c6">l</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="c4">f</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"> <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:48_2"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font>
 <span class="kw">and </span><br/><a NAME="E5:48_2"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_2"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:48_2"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
<font color="Maroon" title="c6">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c3">n</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/>

<a NAME="E7:48_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</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="#E1:48_2"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_2"><span class="lab"><font color="Green" title="E35">A8</font></span></a></span>;<br/>
<a NAME="E9:48_2"/>
<font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</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="#E4:48_2"><span class="lab"><font color="Green" title="E35">A8</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="E10:48_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 
<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="E11:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_2"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E5:48_2"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<a NAME="E12:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A12</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c3">n</font>
 
<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/>
<a NAME="E13:48_2"/>
<font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E11:48_2"><span class="lab"><font color="Green" title="E38">A11</font></span></a></span>;<br/>
<a NAME="E14:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A13</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E15:48_2"/>
<font color="Maroon" title="c4">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E11:48_2"><span class="lab"><font color="Green" title="E38">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:48_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_2"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E5:48_2"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E14:48_2"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="txt" href="#E12:48_2"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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>
<a NAME="E5:48"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
 for <font color="Olive" title="b1">l</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">l</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="c5">g</font> &amp;  not <font color="Olive" title="b1">l</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="c4">f</font> holds <br/> <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="Olive" title="b1">l</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_3"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</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/>
<span class="kw">let </span><font color="Maroon" title="c6">l</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"> ( <font color="Maroon" title="c6">l</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="c5">g</font> &amp;  not <font color="Maroon" title="c6">l</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="c4">f</font> implies  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">l</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:48_3"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<font color="Maroon" title="c6">l</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="c5">g</font>
 <span class="kw">and </span><br/><a NAME="E3:48_3"/><span class="lab"><font color="Green" title="E36">A17</font></span>: 
 not <font color="Maroon" title="c6">l</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="c4">f</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">  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">l</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font></span><br/>

<a NAME="E4:48_3"/><span class="lab"><font color="Green" title="E37">A18</font></span>: 
<font color="Maroon" title="c6">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">l</font></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" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:48_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</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="#E2:48_3"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:48_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A19</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_3"><span class="lab"><font color="Green" title="E34">A15</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="E7:48_3"/>
<font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E2:48_3"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E3:48_3"><span class="lab"><font color="Green" title="E36">A17</font></span></a></span>;<br/>
<a NAME="E8:48_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">l</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_3"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="txt" href="#E4:48_3"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E9:48_3"/><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="c5">g</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="c6">l</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">l</font></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="nat_1.html#D1" onmouseover="rs('nat_1/D1')" onmouseout="rh()">NAT_1:def 1</a></span>;<br/>
<a NAME="E10:48_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A20</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p3"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">l</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">l</font></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="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E11:48_3"/><span class="lab"><font color="Green" title="E40">A21</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="c5">g</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="c6">l</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">n</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/>

<a NAME="E12:48_3"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">l</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="c3">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="E13:48_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A22</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="c5">g</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="c6">l</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_3"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<a NAME="E14:48_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E29">A1</font></span></a></span>;<br/>
<a NAME="E15:48_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A23</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E16:48_3"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">l</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_3"><span class="lab"><font color="Green" title="E41">A22</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="E17:48_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A24</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p3"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">l</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span>)</span> <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="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48_3"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E18:48_3"/>
<font color="Maroon" title="c4">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p3"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">l</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p3"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c6">l</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E13:48_3"><span class="lab"><font color="Green" title="E41">A22</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:48_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_3"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E10:48_3"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="txt" href="#E17:48_3"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:48"/><span class="lab"><font color="Green" title="E34">A25</font></span>: 
 for <font color="Olive" title="b1">k1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">k1</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="c4">f</font> holds <br/> <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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"><font color="Olive" title="b1">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Olive" title="b1">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_4"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</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#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E2:48_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A26</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c3">n</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" onmouseover="rs('nat_1/T39')" onmouseout="rh()">NAT_1:39</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">k1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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"> ( <font color="Maroon" title="c6">k1</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="c4">f</font> implies  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:48_4"/><span class="lab"><font color="Green" title="E36">A27</font></span>: 
<font color="Maroon" title="c6">k1</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="c4">f</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">  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:48_4"/><span class="lab"><font color="Green" title="E37">A28</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E6:48_4"/><span class="lab"><font color="Green" title="E38">A29</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_4"><span class="lab"><font color="Green" title="E36">A27</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E7:48_4"/><span class="lab"><font color="Green" title="E39">A30</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_4"><span class="lab"><font color="Green" title="E37">A28</font></span></a></span>;<br/>
<a NAME="E8:48_4"/>
 not  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c1">k</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_4"><span class="lab"><font color="Green" title="E36">A27</font></span></a></span>;<br/>
<a NAME="E9:48_4"/><span class="kw">then </span>
 not <font color="Maroon" title="c1">k</font> is  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a> 
 
;<br/>
<a NAME="E10:48_4"/><span class="kw">then </span>
 not <font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font> is  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a> 
 
;<br/>
<a NAME="E11:48_4"/><span class="kw">then </span>
 not  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">m</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
;<br/>
<a NAME="E12:48_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A31</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E13:48_4"/>
<font color="Maroon" title="c3">n</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#T8" target="_self" onmouseover="rs('stirl2_1/T8')" onmouseout="rh()">Th8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_4"><span class="lab"><font color="Green" title="E39">A30</font></span></a></span>;<br/>
<a NAME="E15:48_4"/>
<font color="Maroon" title="c6">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E16:48_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A32</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_4"><span class="lab"><font color="Green" title="E37">A28</font></span></a>, <a class="txt" href="#E6:48_4"><span class="lab"><font color="Green" title="E38">A29</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E17:48_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A33</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="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D1" onmouseover="rs('nat_1/D1')" onmouseout="rh()">NAT_1:def 1</a></span>;<br/>
<a NAME="E18:48_4"/><span class="lab"><font color="Green" title="E43">A34</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_4"><span class="lab"><font color="Green" title="E37">A28</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="E19:48_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A35</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="c4">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"><font color="Maroon" title="c6">k1</font></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> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:48_4"><span class="lab"><font color="Green" title="E42">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E20:48_4"/><span class="lab"><font color="Green" title="E45">A36</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_4"><span class="lab"><font color="Green" title="E36">A27</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E46">A37</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_4_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">  for <font color="Olive" title="b1">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">g</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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/> <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n1</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">n1</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"> ( <font color="Maroon" title="c9">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">g</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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n1</font> )</span><br/><span class="kw">assume </span><a NAME="E1:48_4_1"/><span class="lab"><font color="Green" title="E47">A38</font></span>: 
<font color="Maroon" title="c9">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">g</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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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">  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n1</font></span><br/><a NAME="E2:48_4_1"/>
<font color="Maroon" title="c9">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48_4"><span class="lab"><font color="Green" title="E40">A31</font></span></a>, <a class="txt" href="#E1:48_4_1"><span class="lab"><font color="Green" title="E47">A38</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/><a NAME="E3:48_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</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/><a NAME="E4:48_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A39</font></span>: 
<font color="Maroon" title="c9">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <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/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_4_1_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">  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n1</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_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="E1:48_4_1_1_1"/>
( <font color="Maroon" title="c9">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> or <font color="Maroon" title="c9">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_4_1"><span class="lab"><font color="Green" title="E48">A39</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="48_4_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_4_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A40</font></span>: 
<font color="Maroon" title="c9">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</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">  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n1</font></span><br/><div class="add"><a NAME="E2:48_4_1_1_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c9">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_4_1"><span class="lab"><font color="Green" title="E47">A38</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/><a NAME="E3:48_4_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A41</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c9">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E1:48_4_1_1_1_1"><span class="lab"><font color="Green" title="E49">A40</font></span></a></span>;<br/><a NAME="E4:48_4_1_1_1_1"/>
<font color="Maroon" title="c9">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_4"><span class="lab"><font color="Green" title="E45">A36</font></span></a>, <a class="txt" href="#E1:48_4_1_1_1_1"><span class="lab"><font color="Green" title="E49">A40</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="E5:48_4_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_4_1_1_1_1"><span class="lab"><font color="Green" title="E50">A41</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E6:48_4_1_1_1_1"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D1" onmouseover="rs('nat_1/D1')" onmouseout="rh()">NAT_1:def 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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_4_1_1_1_2"/>
<font color="Maroon" title="c9">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</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">  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n1</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:48_4_1_1_1_2"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:48_4"><span class="lab"><font color="Green" title="E42">A33</font></span></a>, <a class="txt" href="#E18:48_4"><span class="lab"><font color="Green" title="E43">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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="E6:48_4_1"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n1</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"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E22:48_4"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_4"><span class="lab"><font color="Green" title="E41">A32</font></span></a>, <a class="ref" href="nat_1.html#D1" onmouseover="rs('nat_1/D1')" onmouseout="rh()">NAT_1:def 1</a></span>;<br/>
<a NAME="E23:48_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E24:48_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A42</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="stirl2_1.html#K1" title="STIRL2_1:func.1">"</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E19:48_4"><span class="lab"><font color="Green" title="E44">A35</font></span></a></span>;<br/>
<a NAME="E25:48_4"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:48_4"><span class="lab"><font color="Green" title="E44">A35</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="E26:48_4"/><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="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">g</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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48_4"><span class="lab"><font color="Green" title="E40">A31</font></span></a>, <a class="txt" href="#E2:48_4"><span class="lab"><font color="Green" title="E35">A26</font></span></a>, <a class="txt" href="#E24:48_4"><span class="lab"><font color="Green" title="E47">A42</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:48_4"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c6">k1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48_4"><span class="lab"><font color="Green" title="E46">A37</font></span></a>, <a class="ref" href="nat_1.html#D1" onmouseover="rs('nat_1/D1')" onmouseout="rh()">NAT_1:def 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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</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"> ( <font color="Maroon" title="c6">i</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="c5">g</font> &amp; <font color="Maroon" title="c7">j</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="c5">g</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> implies  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="c5">g</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="c7">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:48"/><span class="lab"><font color="Green" title="E35">A43</font></span>: 
<font color="Maroon" title="c6">i</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="c5">g</font>
 <span class="kw">and </span><br/><a NAME="E8:48"/><span class="lab"><font color="Green" title="E36">A44</font></span>: 
<font color="Maroon" title="c7">j</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="c5">g</font>
 <span class="kw">and </span><br/><a NAME="E9:48"/><span class="lab"><font color="Green" title="E37">A45</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</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">  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="c5">g</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="c7">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span><br/>

<a NAME="E10:48"/><span class="lab"><font color="Green" title="E38">A46</font></span>: 
 for <font color="Olive" title="b1">l</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">l</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="c5">g</font> &amp;  not <font color="Olive" title="b1">l</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="c4">f</font> holds <br/><font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">k</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">l</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"> ( <font color="Maroon" title="c8">l</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="c5">g</font> &amp;  not <font color="Maroon" title="c8">l</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="c4">f</font> implies <font color="Maroon" title="c8">l</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48_5"/>
<font color="Maroon" title="c8">l</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="c5">g</font>
 <span class="kw">and </span><br/><a NAME="E2:48_5"/><span class="lab"><font color="Green" title="E39">A47</font></span>: 
 not <font color="Maroon" title="c8">l</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="c4">f</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"> <font color="Maroon" title="c8">l</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">k</font></span><br/>

<span class="kw">assume </span><a NAME="E3:48_5"/>
<font color="Maroon" title="c8">l</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/>

<a NAME="E4:48_5"/><span class="kw">then </span>
<font color="Maroon" title="c8">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <font color="Maroon" title="c1">k</font>
 
<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="E5:48_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:48_5"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</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>
<a NAME="E11:48"/><span class="lab"><font color="Green" title="E39">A48</font></span>: 
( <font color="Maroon" title="c6">i</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="c7">j</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="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_6"><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">  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="c5">g</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="c7">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_6_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="E1:48_6_1"/>
( ( <font color="Maroon" title="c6">i</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="c4">f</font> &amp; <font color="Maroon" title="c7">j</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="c4">f</font> ) or ( <font color="Maroon" title="c6">i</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="c4">f</font> &amp;  not <font color="Maroon" title="c7">j</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="c4">f</font> ) or (  not <font color="Maroon" title="c6">i</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="c4">f</font> &amp; <font color="Maroon" title="c7">j</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="c4">f</font> ) or (  not <font color="Maroon" title="c6">i</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="c4">f</font> &amp;  not <font color="Maroon" title="c7">j</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="c4">f</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_6_1_1"/><span class="lab"><font color="Green" title="E40">A49</font></span>: 
( <font color="Maroon" title="c6">i</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="c4">f</font> &amp; <font color="Maroon" title="c7">j</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="c4">f</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">  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="c5">g</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="c7">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:48_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A50</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="c5">g</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="c7">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c7">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E34">A25</font></span></a>, <a class="txt" href="#E11:48"><span class="lab"><font color="Green" title="E39">A48</font></span></a></span>;<br/><a NAME="E3:48_6_1_1"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c6">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E34">A25</font></span></a>, <a class="txt" href="#E1:48_6_1_1"><span class="lab"><font color="Green" title="E40">A49</font></span></a>, <a class="txt" href="#E11:48"><span class="lab"><font color="Green" title="E39">A48</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_6_1_1"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="c5">g</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="c7">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E37">A45</font></span></a>, <a class="txt" href="#E1:48_6_1_1"><span class="lab"><font color="Green" title="E40">A49</font></span></a>, <a class="txt" href="#E2:48_6_1_1"><span class="lab"><font color="Green" title="E41">A50</font></span></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="48_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_6_1_2"/><span class="lab"><font color="Green" title="E40">A51</font></span>: 
( <font color="Maroon" title="c6">i</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="c4">f</font> &amp;  not <font color="Maroon" title="c7">j</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="c4">f</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">  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="c5">g</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="c7">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:48_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A52</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E3:48_6_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="c4">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="c6">i</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> <font color="Maroon" title="c3">n</font> <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="E4:48_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A53</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="c5">g</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="c6">i</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> <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E34">A25</font></span></a>, <a class="txt" href="#E1:48_6_1_2"><span class="lab"><font color="Green" title="E40">A51</font></span></a>, <a class="txt" href="#E11:48"><span class="lab"><font color="Green" title="E39">A48</font></span></a></span>;<br/><a NAME="E5:48_6_1_2"/>
<font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_6_1_2"><span class="lab"><font color="Green" title="E41">A52</font></span></a>, <a class="ref" href="nat_1.html#T20" onmouseover="rs('nat_1/T20')" onmouseout="rh()">NAT_1:20</a></span>;<br/><a NAME="E6:48_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A54</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="E7:48_6_1_2"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c7">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="txt" href="#E1:48_6_1_2"><span class="lab"><font color="Green" title="E40">A51</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:48_6_1_2"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="c5">g</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="c7">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_6_1_2"><span class="lab"><font color="Green" title="E42">A53</font></span></a>, <a class="txt" href="#E6:48_6_1_2"><span class="lab"><font color="Green" title="E43">A54</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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="48_6_1_3"><span class="kw">suppose </span></a><a NAME="E1:48_6_1_3"/><span class="lab"><font color="Green" title="E40">A55</font></span>: 
(  not <font color="Maroon" title="c6">i</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="c4">f</font> &amp; <font color="Maroon" title="c7">j</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="c4">f</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">  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="c5">g</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="c7">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:48_6_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A56</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font>
 <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/><a NAME="E3:48_6_1_3"/>
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E35">A43</font></span></a>, <a class="txt" href="#E10:48"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E1:48_6_1_3"><span class="lab"><font color="Green" title="E40">A55</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_6_1_3"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="c5">g</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="c7">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E37">A45</font></span></a>, <a class="txt" href="#E2:48_6_1_3"><span class="lab"><font color="Green" title="E41">A56</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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="48_6_1_4"><span class="kw">suppose </span></a><a NAME="E1:48_6_1_4"/><span class="lab"><font color="Green" title="E40">A57</font></span>: 
(  not <font color="Maroon" title="c6">i</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="c4">f</font> &amp;  not <font color="Maroon" title="c7">j</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="c4">f</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">  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="c5">g</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="c7">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:48_6_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="stirl2_1.html#K2" title="STIRL2_1:func.2">.</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E35">A43</font></span></a>, <a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E32">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_6_1_4"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="c5">g</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="c7">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E37">A45</font></span></a>, <a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E1:48_6_1_4"><span class="lab"><font color="Green" title="E40">A57</font></span></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="E13:48"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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="c6">i</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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="c5">g</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="c7">j</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></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"> verum</span><br/>


</div>
