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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being   <a href="finseq_2.html#NM3" title="FINSEQ_2:NM.3">Tuple</a> of $1, <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>   st  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Olive" title="b2">F2</font> holds <br/><font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F2</font>;<br/>
<a NAME="E1:2"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 for <font color="Olive" title="b1">k</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">k</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
 for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being   <a href="finseq_2.html#NM3" title="FINSEQ_2:NM.3">Tuple</a> of <font color="Maroon" title="c1">k</font>, <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>   st  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Olive" title="b2">F2</font> holds <br/><font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">F1</font>, <font color="Maroon" title="c3">F2</font> be   <a href="finseq_2.html#NM3" title="FINSEQ_2:NM.3">Tuple</a> of <font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1, <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">F2</font> implies <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">T1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">k</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>, <font color="Maroon" title="c5">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T117" onmouseover="rs('finseq_2/T117')" onmouseout="rh()">FINSEQ_2:117</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:2_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
 <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">F2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">T2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">k</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>, <font color="Maroon" title="c7">d2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> <span class="kw"> such that </span><br/><a NAME="E6:2_1"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">T2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T117" onmouseover="rs('finseq_2/T117')" onmouseout="rh()">FINSEQ_2:117</a></span>;<br/>
<span class="lab"><font color="Green" title="E8">A6</font></span>: <span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c4">T1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c5">d1</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>)</span>)</span> = 
 <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">F1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="binarith.html#T20" onmouseover="rs('binarith/T20')" onmouseout="rh()">BINARITH:20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c6">T2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<font color="Maroon" title="c7">d2</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="binarith.html#T20" onmouseover="rs('binarith/T20')" onmouseout="rh()">BINARITH:20</a></span>
;<br/>
<a NAME="E8:2_1"/>
<font color="Maroon" title="c5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">d2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_1_2"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<font color="Maroon" title="c5">d1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">d2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_2_1"/>
( <font color="Maroon" title="c5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a>  or <font color="Maroon" title="c5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_2_1_1"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<font color="Maroon" title="c5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A9</font></span>: 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> (<font color="Maroon" title="c5">d1</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</a></span>;<br/><a NAME="E3:2_1_2_1_1"/>
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> (<font color="Maroon" title="c7">d2</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_2"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E1:2_1_2_1_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E4:2_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E2:2_1_2_1_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="binari_3.html#T1" target="_self" onmouseover="rs('binari_3/T1')" onmouseout="rh()">Th1</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_2_1_2"/><span class="lab"><font color="Green" title="E10">A10</font></span>: 
<font color="Maroon" title="c5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:2_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_2"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</a></span>;<br/><a NAME="E3:2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A11</font></span>: 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> (<font color="Maroon" title="c7">d2</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</a></span>;<br/><a NAME="E4:2_1_2_1_2"/>
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> (<font color="Maroon" title="c5">d1</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_2_1_2"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E5:2_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E3:2_1_2_1_2"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="ref" href="binari_3.html#T1" target="_self" onmouseover="rs('binari_3/T1')" onmouseout="rh()">Th1</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:2_1"/>
<font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="xcmplx_1.html#T2" onmouseover="rs('xcmplx_1/T2')" onmouseout="rh()">XCMPLX_1:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:2"/><span class="lab"><font color="Green" title="E4">A12</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">F1</font>, <font color="Maroon" title="c2">F2</font> be   <a href="finseq_2.html#NM3" title="FINSEQ_2:NM.3">Tuple</a> of 1, <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">F2</font> implies <font color="Maroon" title="c1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F2</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> <span class="kw"> such that </span><br/><a NAME="E2:2_2"/><span class="lab"><font color="Green" title="E5">A13</font></span>: 
<font color="Maroon" title="c1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T97" onmouseover="rs('finseq_2/T97')" onmouseout="rh()">FINSEQ_2:97</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:2_2"/><span class="lab"><font color="Green" title="E6">A14</font></span>: 
 <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">F2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F2</font></span><br/>

<span class="kw">assume </span><a NAME="E4:2_2"/><span class="lab"><font color="Green" title="E7">A15</font></span>: 
<font color="Maroon" title="c1">F1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">F2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">d2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> <span class="kw"> such that </span><br/><a NAME="E6:2_2"/><span class="lab"><font color="Green" title="E8">A16</font></span>: 
<font color="Maroon" title="c2">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T97" onmouseover="rs('finseq_2/T97')" onmouseout="rh()">FINSEQ_2:97</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_2_1"/>
( <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a>  or <font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_2_1_1"/><span class="lab"><font color="Green" title="E9">A17</font></span>: 
<font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A18</font></span>: 
 <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_2"><span class="lab"><font color="Green" title="E5">A13</font></span></a>, <a class="ref" href="binarith.html#T15" onmouseover="rs('binarith/T15')" onmouseout="rh()">BINARITH:15</a></span>;<br/><a NAME="E3:2_2_1_1"/>
<font color="Maroon" title="c4">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_2"><span class="lab"><font color="Green" title="E5">A13</font></span></a>, <a class="txt" href="#E6:2_2"><span class="lab"><font color="Green" title="E8">A16</font></span></a>, <a class="txt" href="#E4:2_2"><span class="lab"><font color="Green" title="E7">A15</font></span></a>, <a class="txt" href="#E1:2_2_1_1"><span class="lab"><font color="Green" title="E9">A17</font></span></a>, <a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:2_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_2"><span class="lab"><font color="Green" title="E8">A16</font></span></a>, <a class="txt" href="#E3:2_2"><span class="lab"><font color="Green" title="E6">A14</font></span></a>, <a class="txt" href="#E2:2_2_1_1"><span class="lab"><font color="Green" title="E10">A18</font></span></a>, <a class="ref" href="binarith.html#T16" onmouseover="rs('binarith/T16')" onmouseout="rh()">BINARITH:16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_2_1_2"/><span class="lab"><font color="Green" title="E9">A19</font></span>: 
<font color="Maroon" title="c3">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A20</font></span>: 
 <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_2"><span class="lab"><font color="Green" title="E5">A13</font></span></a>, <a class="ref" href="binarith.html#T16" onmouseover="rs('binarith/T16')" onmouseout="rh()">BINARITH:16</a></span>;<br/><a NAME="E3:2_2_1_2"/>
<font color="Maroon" title="c4">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_2"><span class="lab"><font color="Green" title="E5">A13</font></span></a>, <a class="txt" href="#E6:2_2"><span class="lab"><font color="Green" title="E8">A16</font></span></a>, <a class="txt" href="#E4:2_2"><span class="lab"><font color="Green" title="E7">A15</font></span></a>, <a class="txt" href="#E1:2_2_1_2"><span class="lab"><font color="Green" title="E9">A19</font></span></a>, <a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:2_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_2"><span class="lab"><font color="Green" title="E8">A16</font></span></a>, <a class="txt" href="#E3:2_2"><span class="lab"><font color="Green" title="E6">A14</font></span></a>, <a class="txt" href="#E2:2_2_1_2"><span class="lab"><font color="Green" title="E10">A20</font></span></a>, <a class="ref" href="binarith.html#T15" onmouseover="rs('binarith/T15')" onmouseout="rh()">BINARITH:15</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:2"/>
 for <font color="Olive" title="b1">n</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S10" onmouseover="rs('nat_1/S10')" onmouseout="rh()">NAT_1:sch 10</a>(<a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E4">A12</font></span></a>, <a class="txt" href="#E1:2"><span class="lab"><font color="Green" title="E3">A1</font></span></a>);</span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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