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<b>thus </b><a NAME="E1:18_1_2"/>
for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>  st ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/> <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Olive" title="b3">i</font>,<span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> ) &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/> <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Olive" title="b2">f2</font>,<font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Olive" title="b3">i</font>,<span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> ) holds <br/><font color="Olive" title="b1">f1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">f2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>;<br/>

<b>assume </b><a NAME="E1:18_1_2_1"/><i><font color="Green" title="E8">A11</font></i>: 
( ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/> <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c2">f1</font>,<font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Olive" title="b1">i</font>,<span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/> <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">f2</font>,<font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Olive" title="b1">i</font>,<span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> ) )
 ;<br/>

<a NAME="E2:18_1_2_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/> <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c2">f1</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">f2</font>,<font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:18_1_2_1_1"/><i><font color="Green" title="E9">A12</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">i2</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:18_1_2_1_1"/>
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c5">i2</font>,<span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E1:18_1_2_1"><i><font color="Green" title="E8">A11</font></i></a>, <a class="txt" href="fintopo4.html#E1:18_1_2_1_1"><i><font color="Green" title="E9">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:18_1_2_1_1"/>
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="fintopo4.html#E1:18_1_2_1"><i><font color="Green" title="E8">A11</font></i></a>, <a class="txt" href="fintopo4.html#E1:18_1_2_1_1"><i><font color="Green" title="E9">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:18_1_2_1"/>
<font color="Maroon" title="c2">f1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f2</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T54" title="RELSET_1:th.54">RELSET_1:54</a></i>;<br/>


</div><b>end;</b></div>


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