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<b>let </b><font color="Maroon" title="c1">a1</font>, <font color="Maroon" title="c2">a2</font> be  <a href="graph_2.html#V1" title="GRAPH_2:attr.1">TwoValued</a> <a href="graph_2.html#V2" title="GRAPH_2:attr.2">Alternating</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a2</font>
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">a2</font>
 <b>and </b><br/><a NAME="E3:33"/><i><font color="Green" title="E31">A3</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">a2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">a</font> be  <a href="graph_2.html#V1" title="GRAPH_2:attr.1">TwoValued</a> <a href="graph_2.html#V2" title="GRAPH_2:attr.2">Alternating</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E4:33"/><i><font color="Green" title="E32">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a1</font>
 <b>and </b><br/><a NAME="E5:33"/><i><font color="Green" title="E33">A5</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">a1</font>
 ;<br/>

<b>assume </b><a NAME="E6:33"/><i><font color="Green" title="E34">A6</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a1</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:33_1"/><i><font color="Green" title="E35">A7</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <b>and </b><br/><a NAME="E2:33_1"/><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon" title="c4">i</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> <font color="Maroon" title="c3">a</font>
 ;<br/><a NAME="E3:33_1"/><i><font color="Green" title="E37">A9</font></i>: 
<font color="Maroon" title="c4">i</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> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E1:33_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="graph_2.html#E2:33_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:33_1"/><i><font color="Green" title="E38">A10</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <font color="Maroon" title="c3">a</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <font color="Maroon" title="c1">a1</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <font color="Maroon" title="c2">a2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">X</font>, <font color="Maroon" title="c6">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:33_1"/><i><font color="Green" title="E39">A11</font></i>: 
( <font color="Maroon" title="c5">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">Y</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c6">Y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="ref" href="graph_2.html#T19" target="_self" title="GRAPH_2:th.19">Th19</a></i>;<br/><a NAME="E7:33_1"/>
( <font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</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="c3">a</font> &amp; <font color="Maroon" title="c1">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</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="c1">a1</font> &amp; <font color="Maroon" title="c2">a2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</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">a2</font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E2:33"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="graph_2.html#E4:33"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="graph_2.html#E3:33_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="graph_2.html#E4:33_1"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:33_1"/><b>then </b>
( ( <font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">X</font> or <font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Y</font> ) &amp; ( <font color="Maroon" title="c1">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">X</font> or <font color="Maroon" title="c1">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Y</font> ) &amp; ( <font color="Maroon" title="c2">a2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">X</font> or <font color="Maroon" title="c2">a2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Y</font> ) )
 <b>by </b><i><a class="txt" href="graph_2.html#E3:33"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="graph_2.html#E5:33"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="graph_2.html#E6:33_1"><i><font color="Green" title="E39">A11</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E9:33_1"/>
<font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E1:33"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="graph_2.html#E2:33"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="graph_2.html#E3:33"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="graph_2.html#E4:33"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="graph_2.html#E5:33"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="graph_2.html#E6:33"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="graph_2.html#E1:33_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="graph_2.html#E2:33_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="graph_2.html#T21" target="_self" title="GRAPH_2:th.21">Th21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:33"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a2</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E2:33"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="graph_2.html#E4:33"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


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