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<span class="kw">assume </span><a NAME="E1:83_1_1"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
 not <font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="nomin_1.html#D6" target="_self" onmouseover="rs('nomin_1/D6')" onmouseout="rh()">NOMIN_1:def 6</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" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> is    <a href="nomin_1.html#M1" title="NOMIN_1:mode.1">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">S</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E3:83_1_1"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
<font color="Maroon" title="c5">S</font> <a href="nomin_1.html#R2" title="NOMIN_1:pred.2">IsNDRankSeq</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E4:83_1_1"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nomin_1.html#D5" target="_self" onmouseover="rs('nomin_1/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E6:83_1_1"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 <span class="kw">and </span><br/><a NAME="E7:83_1_1"/><span class="lab"><font color="Green" title="E52">A6</font></span>: 
<font color="Maroon" title="c6">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_1_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E9:83_1_1"/><span class="lab"><font color="Green" title="E53">A7</font></span>: 
<font color="Maroon" title="c7">z</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="c5">S</font>
 <span class="kw">and </span><br/><a NAME="E10:83_1_1"/><span class="lab"><font color="Green" title="E54">A8</font></span>: 
<font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_1_1"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c7">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_1_1"><span class="lab"><font color="Green" title="E53">A7</font></span></a></span>;<br/>
<a NAME="E12:83_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_1_1"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_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="E1:83_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">z</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:83_1_1_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> is    <a href="nomin_1.html#M1" title="NOMIN_1:mode.1">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font></span><br/><div class="add"><a NAME="E2:83_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">w</font> being   <a href="nomin_1.html#NM2" title="NOMIN_1:NM.2">TypeSSNominativeData</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font> st <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E6:83_1_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="txt" href="#E10:83_1_1"><span class="lab"><font color="Green" title="E54">A8</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:83_1_1_1_1"/>
<font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> is    <a href="nomin_1.html#M1" title="NOMIN_1:mode.1">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E1:83_1_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</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="83_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:83_1_1_1_2"/><span class="lab"><font color="Green" title="E55">A9</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <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="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> is    <a href="nomin_1.html#M1" title="NOMIN_1:mode.1">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c9">lS</font> = <font color="Maroon" title="c8">z</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_1_1"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">S1</font> = <font color="Maroon" title="c5">S</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c9">lS</font>;<br/><a NAME="E3:83_1_1_1_2"/><span class="lab"><font color="Green" title="E56">A10</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c9">lS</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T60" onmouseover="rs('relat_1/T60')" onmouseout="rh()">RELAT_1:60</a></span>;<br/><a NAME="E4:83_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A11</font></span>: 
<font color="Maroon" title="c8">z</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="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_1_1"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:83_1_1_1_2"/>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">lS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1_1_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="ref" href="xreal_1.html#T14" onmouseover="rs('xreal_1/T14')" onmouseout="rh()">XREAL_1:14</a></span>;<br/><a NAME="E6:83_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A12</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">lS</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E7:83_1_1_1_2"/>
<font color="Maroon" title="c9">lS</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">z</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" onmouseover="rs('xreal_1/T10')" onmouseout="rh()">XREAL_1:10</a></span>;<br/><a NAME="E8:83_1_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c9">lS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">lS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_1_1_1_2"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/><a NAME="E9:83_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A13</font></span>: 
<font color="Maroon" title="c9">lS</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c9">lS</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_1_1_1_2"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E10:83_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">lS</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K1" title="NOMIN_1:func.1">NDSS</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">lS</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_1_1"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="txt" href="#E3:83_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E3:83_1_1_1_2"><span class="lab"><font color="Green" title="E56">A10</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">d</font> being   <a href="nomin_1.html#NM2" title="NOMIN_1:NM.2">TypeSSNominativeData</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">lS</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E12:83_1_1_1_2"/><span class="lab"><font color="Green" title="E60">A14</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_1_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="txt" href="#E10:83_1_1"><span class="lab"><font color="Green" title="E54">A8</font></span></a></span>;<br/><a NAME="E13:83_1_1_1_2"/><span class="lab"><font color="Green" title="E61">A15</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">lS</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E12:83_1_1_1_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</a></span>;<br/><a NAME="E14:83_1_1_1_2"/><span class="lab"><font color="Green" title="E62">A16</font></span>: 
(  not <font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> implies <font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:83_1_1_1_2_1"/><span class="lab"><font color="Green" title="E63">A17</font></span>: 
 not <font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</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="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_1_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="E1:83_1_1_1_2_1_1"/>
( <font color="Maroon" title="c9">lS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">lS</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_1_1_1_2"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:83_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">lS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/><div class="add"><a NAME="E2:83_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:83_1_1_1_2"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="txt" href="#E1:83_1_1_1_2_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E3:83_1_1_1_2_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">w</font> being   <a href="nomin_1.html#NM2" title="NOMIN_1:NM.2">TypeSSNominativeData</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font> st <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:83_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:83_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E64">A18</font></span>: 
<font color="Maroon" title="c9">lS</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <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="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">c</font> = <font color="Maroon" title="c9">lS</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><a NAME="E3:83_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c12">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A18</font></span></a>, <a class="ref" href="xreal_1.html#T14" onmouseover="rs('xreal_1/T14')" onmouseout="rh()">XREAL_1:14</a></span>;<br/><a NAME="E4:83_1_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A19</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><a NAME="E5:83_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E66">A20</font></span>: 
<font color="Maroon" title="c9">lS</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="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1_1_1_2"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E9:83_1_1_1_2"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E6:83_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c12">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">lS</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" onmouseover="rs('xreal_1/T10')" onmouseout="rh()">XREAL_1:10</a></span>;<br/><a NAME="E7:83_1_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">c</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="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E8:83_1_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A19</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:83_1_1_1_2_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K1" title="NOMIN_1:func.1">NDSS</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span>) &amp; <font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">lS</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E3:83_1_1_1_2"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E9:83_1_1_1_2"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E13:83_1_1_1_2"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="txt" href="#E1:83_1_1_1_2_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E10:83_1_1_1_2_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">w</font> being   <a href="nomin_1.html#NM2" title="NOMIN_1:NM.2">TypeSSNominativeData</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> st <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
 ;<br/><span class="kw">hence </span><a NAME="E11:83_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E15:83_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c9">lS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">lS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">lS</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T112" onmouseover="rs('finseq_3/T112')" onmouseout="rh()">FINSEQ_3:112</a></span>;<br/><a NAME="E16:83_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">lS</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c9">lS</font></span>)</span> &amp; <font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">lS</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E9:83_1_1_1_2"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E13:83_1_1_1_2"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E17:83_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c9">lS</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E18:83_1_1_1_2"/>
<font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> is    <a href="nomin_1.html#M1" title="NOMIN_1:mode.1">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E3:83_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E3:83_1_1_1_2"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E9:83_1_1_1_2"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E14:83_1_1_1_2"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="ref" href="nomin_1.html#T16" target="_self" onmouseover="rs('nomin_1/T16')" onmouseout="rh()">Th16</a>, <a class="ref" href="nomin_1.html#D5" target="_self" onmouseover="rs('nomin_1/D5')" onmouseout="rh()">Def5</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>
