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<a NAME="E1:84_1_1"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
 <a href="nomin_1.html#K4" title="NOMIN_1:func.4">ND_ex_1</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">D</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c4">D</font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_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="E1:84_1_1_1"/>
( <font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c4">D</font> is    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="nomin_1.html#D6" target="_self" onmouseover="rs('nomin_1/D6')" onmouseout="rh()">Def6</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:84_1_1_1_1"/>
<font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN: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="c3">v</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c4">D</font> is    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">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">hence </span><a NAME="E2:84_1_1_1_1"/>
<font color="Maroon" title="c3">v</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c4">D</font> is    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">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="#E2:84"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E1:84_1_1"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="nomin_1.html#T46" target="_self" onmouseover="rs('nomin_1/T46')" onmouseout="rh()">Th46</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="84_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:84_1_1_1_2"/>
<font color="Maroon" title="c4">D</font> is    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<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="c3">v</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c4">D</font> is    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">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">then </span><span class="kw">consider </span><font color="Maroon" title="c5">F</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E3:84_1_1_1_2"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
<font color="Maroon" title="c5">F</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="E4:84_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c5">F</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="E6:84_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A6</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 <span class="kw">and </span><br/><a NAME="E7:84_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A7</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">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84_1_1_1_2"><span class="lab"><font color="Green" title="E52">A5</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">reconsider </span><font color="Maroon" title="c7">Z</font> = <font color="Maroon" title="c6">Z</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84_1_1_1_2"><span class="lab"><font color="Green" title="E53">A6</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">D1</font> = <font color="Maroon" title="c4">D</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">Z</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84_1_1_1_2"><span class="lab"><font color="Green" title="E53">A6</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">V</font> = <font color="Maroon" title="c1">V</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84"><span class="lab"><font color="Green" title="E49">A2</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">v</font> = <font color="Maroon" title="c3">v</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84"><span class="lab"><font color="Green" title="E49">A2</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">d</font> = <font color="Maroon" title="c10">v</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c4">D</font>;<br/><a NAME="E12:84_1_1_1_2"/>
<font color="Maroon" title="c10">v</font> <a href="turing_1.html#K2" title="TURING_1:func.2">.--&gt;</a> <font color="Maroon" title="c8">D1</font> is   <a href="nomin_1.html#NM1" title="NOMIN_1:NM.1">NominativeSet</a> of <font color="Maroon" title="c9">V</font>,<font color="Maroon" title="c7">Z</font>
 ;<br/><a NAME="E13:84_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A8</font></span>: 
<font color="Maroon" title="c10">v</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c4">D</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="nomin_1.html#K1" title="NOMIN_1:func.1">NDSS</a> (<font color="Maroon" title="c9">V</font>,<font color="Maroon" title="c7">Z</font>)
 ;<br/><a NAME="E14:84_1_1_1_2"/><span class="lab"><font color="Green" title="E56">A9</font></span>: 
 <a href="nomin_1.html#K1" title="NOMIN_1:func.1">NDSS</a> (<font color="Maroon" title="c9">V</font>,<font color="Maroon" title="c7">Z</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="nomin_1.html#K1" title="NOMIN_1:func.1">NDSS</a> (<font color="Maroon" title="c9">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> <font color="Maroon" title="c7">Z</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nomin_1.html#T7" target="_self" onmouseover="rs('nomin_1/T7')" onmouseout="rh()">Th7</a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E16:84_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A10</font></span>: 
<font color="Maroon" title="c12">x</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">F</font>
 <span class="kw">and </span><br/><a NAME="E17:84_1_1_1_2"/><span class="lab"><font color="Green" title="E58">A11</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:84_1_1_1_2"><span class="lab"><font color="Green" title="E54">A7</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="c13">x</font> = <font color="Maroon" title="c12">x</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:84_1_1_1_2"><span class="lab"><font color="Green" title="E57">A10</font></span></a></span>;<br/><a NAME="E19:84_1_1_1_2"/><span class="lab"><font color="Green" title="E59">A12</font></span>: 
<font color="Maroon" title="c13">x</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">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:84_1_1_1_2"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">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="E21:84_1_1_1_2"/><span class="lab"><font color="Green" title="E60">A13</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E22:84_1_1_1_2"/><span class="lab"><font color="Green" title="E61">A14</font></span>: 
<font color="Maroon" title="c14">S</font> <a href="nomin_1.html#R2" title="NOMIN_1:pred.2">IsNDRankSeq</a> <font color="Maroon" title="c9">V</font>,<font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E23:84_1_1_1_2"/><span class="lab"><font color="Green" title="E62">A15</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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="c14">S</font> holds <br/><font color="Maroon" title="c14">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <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="c9">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"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84_1_1_1_2"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="ref" href="nomin_1.html#T26" target="_self" onmouseover="rs('nomin_1/T26')" onmouseout="rh()">Th26</a></span>;<br/><a NAME="E24:84_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E25:84_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A16</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <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="c14">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:84_1_1_1_2"><span class="lab"><font color="Green" title="E60">A13</font></span></a>, <a class="txt" href="#E19:84_1_1_1_2"><span class="lab"><font color="Green" title="E59">A12</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c14">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> = 
 <a href="nomin_1.html#K1" title="NOMIN_1:func.1">NDSS</a> (<font color="Maroon" title="c9">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"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:84_1_1_1_2"><span class="lab"><font color="Green" title="E62">A15</font></span></a></span>
<br/>.= 
 <a href="nomin_1.html#K1" title="NOMIN_1:func.1">NDSS</a> (<font color="Maroon" title="c9">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> <font color="Maroon" title="c7">Z</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:84_1_1_1_2"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E17:84_1_1_1_2"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="ref" href="finseq_3.html#T103" onmouseover="rs('finseq_3/T103')" onmouseout="rh()">FINSEQ_3:103</a></span>
;<br/><a NAME="E27:84_1_1_1_2"/><span class="kw">then </span>
 <a href="nomin_1.html#K1" title="NOMIN_1:func.1">NDSS</a> (<font color="Maroon" title="c9">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> <font color="Maroon" title="c7">Z</font></span>)</span>) <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="c14">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:84_1_1_1_2"><span class="lab"><font color="Green" title="E63">A16</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/><a NAME="E28:84_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">v</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <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="c14">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:84_1_1_1_2"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E14:84_1_1_1_2"><span class="lab"><font color="Green" title="E56">A9</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">hence </span><a NAME="E29:84_1_1_1_2"/>
<font color="Maroon" title="c3">v</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c4">D</font> is    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">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="#E22:84_1_1_1_2"><span class="lab"><font color="Green" title="E61">A14</font></span></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>

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