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<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">g</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:4"/>
<font color="Maroon" title="c1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">g</font>
 ;<br/>

<a NAME="E2:4"/><b>then </b>
<font color="Maroon" title="c1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="goboard2.html#D2" title="GOBOARD2:def.2">GOBOARD2:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:4"/><i><font color="Green" title="E5">A1</font></i>: 
( <font color="Maroon" title="c3">x</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="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <font color="Maroon" title="c2">g</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:4"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5d.html#E4:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">j</font> = <font color="Maroon" title="c3">x</font> as    <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>  <b>by </b><i><a class="txt" href="jordan5d.html#E4:4"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/>
<a NAME="E7:4"/><i><font color="Green" title="E7">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <font color="Maroon" title="c2">g</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan5d.html#E5:4"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:4"/><b>then </b><i><font color="Green" title="E8">A4</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:4"/><b>then </b>
1 <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="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E10:4"/>
<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="jordan5d.html#E4:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="jordan5d.html#E7:4"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>

<a NAME="E11:4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <font color="Maroon" title="c2">g</font></span>)</span> <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="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5d.html#E8:4"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E12:4"/>
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan5d.html#E4:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="jordan5d.html#E7:4"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>


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