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<div class="add">

<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c3">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><a NAME="E1:34"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c4">e</font> =  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c2">F</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ] <b>means </b><font color="Maroon" title="c2">F</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default">$1 <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c2">F</font>;<br/>
<font color="Maroon" title="c2">F</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default">0 <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c2">F</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T72" title="FINSEQ_2:th.72">FINSEQ_2:72</a></i>
<br/>.= 
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c2">F</font>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:34"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="finsop_1.html#T11" title="FINSOP_1:th.11">FINSOP_1:11</a></i>
;<br/>
<a NAME="E3:34"/><b>then </b><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E4:34"/><i><font color="Green" title="E19">A3</font></i>: 
for <font color="Olive" title="b1">i</font> being   <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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><a NAME="E1:34_2"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<b>thus </b><font color="Maroon" title="c2">F</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c2">F</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p3">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T74" title="FINSEQ_2:th.74">FINSEQ_2:74</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p3">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c2">F</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:34"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="finsop_1.html#T5" title="FINSOP_1:th.5">FINSOP_1:5</a></i>
<br/>.= 
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c2">F</font>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:34"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="setwop_2.html#E1:34_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="setwiseo.html#T23" title="SETWISEO:th.23">SETWISEO:23</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:34"/>
for <font color="Olive" title="b1">i</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="setwop_2.html#E3:34"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="setwop_2.html#E4:34"><i><font color="Green" title="E19">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:34"/>
<font color="Maroon" title="c2">F</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c2">F</font>
 ;<br/>


</div>
