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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>set </b><font color="Maroon" title="c2">IT</font> =  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font>;<br/>
<b>let </b><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font></span>)</span>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font></span>)</span>
 <b>and </b><br/><a NAME="E3:16"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">m</font>
 ;<br/>

<a NAME="E4:16"/><i><font color="Green" title="E14">A4</font></i>: 
 <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a>  <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="jordan1h.html#T11" target="_self" title="JORDAN1H:th.11">Th11</a>, <a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a></i>;<br/>
<a NAME="E5:16"/><b>then </b>
 <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="triang_1.html#T8" title="TRIANG_1:th.8">TRIANG_1:8</a></i>;<br/>
<a NAME="E6:16"/><b>then </b><i><font color="Green" title="E15">A5</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="jordan1h.html#E2:16"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="jordan1h.html#E3:16"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E7:16"/>
( <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c3">n</font> &amp; <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">m</font> )
 
<b>by </b><i><a class="txt" href="jordan1h.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="jordan1h.html#E2:16"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E8:16"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> 
 
<b>by </b><i><a class="txt" href="jordan1h.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="jordan1h.html#E2:16"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="jordan1h.html#E3:16"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="jordan1h.html#E4:16"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E9:16"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 
<b>by </b><i><a class="ref" href="jordan1h.html#T9" target="_self" title="JORDAN1H:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E10:16"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<font color="Maroon" title="c1">A</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="jordan1h.html#E6:16"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


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