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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">a<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>  st  <a href="binarith.html#K9">Absval</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>;<br/>
<a NAME="E1:2"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="binarith.html#NM1">Tuple</a> of 1,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E2:2_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T117">FINSEQ_2:117</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E4:2_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T117">FINSEQ_2:117</a></i>;<br/>
<b>assume </b><a NAME="E5:2_1"/><i><font color="Green">E6</font></i>: 
 <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>assume </b><a NAME="E6:2_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 </span><b>by </b><i><a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="binari_3.html#E2:2_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="binari_3.html#E4:2_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="binari_3.html#E6:2_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><a NAME="E3:2_1_1_1"/><b>then </b>
(  <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="binari_3.html#E2:2_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="binari_3.html#E4:2_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="binari_3.html#E1:2_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="binarith.html#T41">BINARITH:41</a>, <a class="ref" href="binarith.html#T42">BINARITH:42</a></i>;<br/><b>hence </b><a NAME="E4:2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="binari_3.html#E5:2_1"><i><font color="Green">E6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><div class="add"><a NAME="E2:2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="binari_3.html#E2:2_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="binari_3.html#E4:2_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="binari_3.html#E6:2_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><a NAME="E3:2_1_1_2"/><b>then </b>
(  <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="binari_3.html#E2:2_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="binari_3.html#E4:2_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="binari_3.html#E1:2_1_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="binarith.html#T41">BINARITH:41</a>, <a class="ref" href="binarith.html#T42">BINARITH:42</a></i>;<br/><b>hence </b><a NAME="E4:2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="binari_3.html#E5:2_1"><i><font color="Green">E6</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:2"/><i><font color="Green">E4</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>  st  <a href="binarith.html#K9">Absval</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="binarith.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a> , <font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E3:2_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K12">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a> , <font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E5:2_2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K12">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/>
<b>assume </b><a NAME="E6:2_2"/><i><font color="Green">E8</font></i>: 
 <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<i><font color="Green">E9</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">c<sub>5</sub></font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
 <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="binari_3.html#E3:2_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="binarith.html#T46">BINARITH:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">c<sub>7</sub></font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E5:2_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="binari_3.html#E6:2_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="binarith.html#T46">BINARITH:46</a></i>
;<br/>
<a NAME="E8:2_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_2_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_2_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 </span><b>by </b><i><a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_2_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_2_1_1"/><b>then </b><i><font color="Green">E13</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">c<sub>7</sub></font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="binari_3.html#E1:2_2_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E3:2_2_2_1_1"/>
 <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">c<sub>5</sub></font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="binari_3.html#E1:2_2_2_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E4:2_2_2_1_1"/><b>then </b>
 <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="binari_3.html#E7:2_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="binari_3.html#E2:2_2_2_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><a NAME="E5:2_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="binari_3.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_2_1_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="binari_3.html#E1:2_2_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><a NAME="E3:2_2_2_1_2"/><b>then </b><i><font color="Green">E13</font></i>: 
 <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">c<sub>7</sub></font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E4:2_2_2_1_2"/>
 <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">c<sub>5</sub></font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="binari_3.html#E1:2_2_2_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a>, <a class="ref" href="margrel1.html#T38">MARGREL1:38</a></i>;<br/><a NAME="E5:2_2_2_1_2"/><b>then </b>
 <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="binari_3.html#E7:2_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="binari_3.html#E3:2_2_2_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><a NAME="E6:2_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="binari_3.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:2_2"/><b>then </b>
 <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="binari_3.html#E7:2_2"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="binari_3.html#E1:2_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="binari_3.html#E3:2_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="binari_3.html#E5:2_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="binari_3.html#E8:2_2"><i><font color="Green">E10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:2"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(<a class="txt" href="binari_3.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="binari_3.html#E2:2"><i><font color="Green">E4</font></i></a>);</i><br/>


</div>
