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<span class="kw">set </span><font color="Maroon" title="c2">n</font> =  <a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<span class="p3">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,$1) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> );<br/>
<span class="kw">set </span><font color="Maroon" title="c3">X</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> 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>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] </span> } </span> ;<br/>
<a NAME="E1:5_1_1"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> 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>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] </span> } </span>  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">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" onmouseover="rs('domain_1/S7')" onmouseout="rh()">DOMAIN_1:sch 7</a>();<br/></span>
<a NAME="E2:5_1_1"/>
 <a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font> <a href="jordan1h.html#R1" title="JORDAN1H:pred.1">is_sufficiently_large_for</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan11.html#D1" target="_self" onmouseover="rs('jordan11/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:5_1_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">j</font>]
 ;<br/>
<a NAME="E5:5_1_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c4">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/>
<a NAME="E6:5_1_1"/>
<font color="Maroon" title="c4">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> 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>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E5:5_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">X</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> 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>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c5">X</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="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c5">X</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<span class="p3">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c5">X</font></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> &amp; (  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<span class="p3">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b1">j</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c5">X</font> ) )</span><br/>

<a NAME="E8:5_1_1"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c5">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D7" onmouseover="rs('xxreal_2/D7')" onmouseout="rh()">XXREAL_2:def 7</a></span>;<br/>
<a NAME="E9:5_1_1"/><span class="kw">then </span>
 ex <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#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] )
 
;<br/>
<span class="kw">hence </span><a NAME="E10:5_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c5">X</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">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<span class="p3">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b1">j</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c5">X</font></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>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<span class="p2">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<span class="p3">(<span class="default"><a href="jordan11.html#K1" title="JORDAN11:func.1">ApproxIndex</a> <font color="Maroon" title="c1">C</font></span>)</span>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c6">i</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c5">X</font> )</span><br/>

<a NAME="E11:5_1_1"/><span class="lab"><font color="Green" title="E8">A4</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> 
 
<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/>
<span class="kw">assume </span><a NAME="E12:5_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i</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">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c5">X</font></span><br/>

<a NAME="E13:5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:5_1_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:5_1_1"/>
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D7" onmouseover="rs('xxreal_2/D7')" onmouseout="rh()">XXREAL_2:def 7</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>
