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<b>let </b><font color="Maroon">z1</font> be   <a href="binarith.html#NM1">Tuple</a> of 2,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>

<b>assume </b><a NAME="E1:11"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K11">*&gt;</a></span> <a href="binarith.html#K10">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K11">*&gt;</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:11"/><i><font color="Green">E43</font></i>: 
 <a href="binarith.html#K6">Binary</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">k1</font>,<font color="Maroon">k2</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T120">FINSEQ_2:120</a></i>;<br/>
<a NAME="E4:11"/>
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> ,<a href="margrel1.html#K8">TRUE</a> </span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a></i>;<br/>
<a NAME="E5:11"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; <font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/>
<b>then </b><i><font color="Green">E50</font></i>:  <a href="binari_2.html#K4" target="_self">Intval</a> <font color="Maroon">z1</font> = 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font></span>)</span> <a href="binop_2.html#K10">-</a> 4

;<br/>
<a NAME="E7:11"/><i><font color="Green">E51</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E8:11"/>
 <a href="finseq_1.html#K2">Seg</a> 1 <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<b>then </b><i><font color="Green">E52</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K6">Binary</a> <font color="Maroon">z1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D6">BINARITH:def 6</a></i>
<br/>.= 
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/>
<a NAME="E10:11"/>
1 <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E11:11"/><b>then </b>
1 <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E12:11"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="binarith.html#K6">Binary</a> <font color="Maroon">z1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="power.html#T29">POWER:29</a></i>;<br/>
<a NAME="E13:11"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="binarith.html#K6">Binary</a> <font color="Maroon">z1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/>
<a NAME="E14:11"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<b>then </b><i><font color="Green">E59</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K6">Binary</a> <font color="Maroon">z1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 = 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default">2 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#D6">BINARITH:def 6</a></i>
<br/>.= 
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/>
<a NAME="E16:11"/>
2 <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> 1
 
;<br/>
<a NAME="E17:11"/><b>then </b>
2 <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E18:11"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="binarith.html#K6">Binary</a> <font color="Maroon">z1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i>, <a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<a NAME="E19:11"/>
<span class="p1">(<span class="default"><a href="binarith.html#K6">Binary</a> <font color="Maroon">z1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/>
<b>then </b> <a href="binarith.html#K7">Absval</a> <font color="Maroon">z1</font> = 
<a href="binop_2.html#K47">addnat</a>  <a href="finsop_1.html#K2">$$</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">1,2</span><a href="finseq_4.html#K2">*&gt;</a></span>
<b>by </b><i>, , , , <a class="ref" href="binarith.html#D7">BINARITH:def 7</a></i>
<br/>.= 
<a href="binop_2.html#K47">addnat</a>  <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span> <a href="binarith.html#K10">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">2</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a></i>
<br/>.= 
<a href="binop_2.html#K47">addnat</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="finsop_1.html#K2">$$</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="finsop_1.html#K2">$$</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">2</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finsop_1.html#T6">FINSOP_1:6</a></i>
<br/>.= 
<a href="binop_2.html#K47">addnat</a>  <a href="binop_1.html#K2">.</a> 1,<span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="finsop_1.html#K2">$$</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default">2</span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>
<br/>.= 
<a href="binop_2.html#K47">addnat</a>  <a href="binop_1.html#K2">.</a> 1,2
<b>by </b><i><a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>
<br/>.= 
1 <a href="binop_2.html#K23">+</a> 2
<b>by </b><i><a class="ref" href="binop_2.html#D23">BINOP_2:def 23</a></i>
<br/>.= 
3

;<br/>
<b>hence </b><a NAME="E21:11"/>
 <a href="binari_2.html#K4" target="_self">Intval</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1
 <b>by </b><i/>;<br/>


</div>
