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<span class="kw">let </span><font color="Maroon" title="c2">it1</font>, <font color="Maroon" title="c3">it2</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="c2">it1</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="c2">it1</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> &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> <font color="Maroon" title="c2">it1</font> ) &amp; <font color="Maroon" title="c3">it2</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="c3">it2</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> &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> <font color="Maroon" title="c3">it2</font> ) implies <font color="Maroon" title="c2">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">it2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:5_1_2"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
<font color="Maroon" title="c2">it1</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>
 <span class="kw">and </span><br/><a NAME="E2:5_1_2"/><span class="lab"><font color="Green" title="E6">A6</font></span>: 
 <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="c2">it1</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>
 <span class="kw">and </span><br/><a NAME="E3:5_1_2"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
 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> <font color="Maroon" title="c2">it1</font>
 <span class="kw">and </span><br/><a NAME="E4:5_1_2"/><span class="lab"><font color="Green" title="E8">A8</font></span>: 
<font color="Maroon" title="c3">it2</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>
 <span class="kw">and </span><br/><a NAME="E5:5_1_2"/><span class="lab"><font color="Green" title="E9">A9</font></span>: 
 <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="c3">it2</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>
 <span class="kw">and </span><br/><a NAME="E6:5_1_2"/><span class="lab"><font color="Green" title="E10">A10</font></span>: 
 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> <font color="Maroon" title="c3">it2</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="c2">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">it2</font></span><br/>

<a NAME="E7:5_1_2"/><span class="lab"><font color="Green" title="E11">A11</font></span>: 
<font color="Maroon" title="c3">it2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">it1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1_2"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="txt" href="#E2:5_1_2"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E6:5_1_2"><span class="lab"><font color="Green" title="E10">A10</font></span></a></span>;<br/>
<a NAME="E8:5_1_2"/>
<font color="Maroon" title="c2">it1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">it2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_1_2"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="txt" href="#E4:5_1_2"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="txt" href="#E5:5_1_2"><span class="lab"><font color="Green" title="E9">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:5_1_2"/>
<font color="Maroon" title="c2">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">it2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:5_1_2"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</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/>


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