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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:17"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:17"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:17"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E4:17"/><i><font color="Green">E14</font></i>: 
 <a href="jordan1h.html#K1" target="_self">RealOrd</a>  <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="jordan1h.html#T11" target="_self">Th11</a>, <a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/>
<a NAME="E5:17"/><b>then </b>
 <a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="triang_1.html#T8">TRIANG_1:8</a></i>;<br/>
<a NAME="E6:17"/><b>then </b><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E1:17"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan1h.html#E2:17"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jordan1h.html#E3:17"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E7:17"/>
( <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan1h.html#E1:17"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan1h.html#E2:17"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E8:17"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="jordan1h.html#K1" target="_self">RealOrd</a> 
 
<b>by </b><i><a class="txt" href="jordan1h.html#E1:17"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan1h.html#E2:17"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jordan1h.html#E3:17"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan1h.html#E4:17"><i><font color="Green">E14</font></i></a>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E9:17"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="jordan1h.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E10:17"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan1h.html#E6:17"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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