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<span class="kw">let </span><font color="Maroon" title="c1">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">  for <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">A</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b4">f</font> being  <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b3">n</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="Olive" title="b4">f</font> holds <br/><span class="p1">(<span class="default"><a href="nomin_1.html#K14" title="NOMIN_1:func.14">namingSeq</a> (<font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b4">f</font>,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> is    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">A</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">f</font> being  <font color="Maroon" title="c2">V</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</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="Olive" title="b2">f</font> holds <br/><span class="p1">(<span class="default"><a href="nomin_1.html#K14" title="NOMIN_1:func.14">namingSeq</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <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">  for <font color="Olive" title="b1">f</font> being  <font color="Maroon" title="c2">V</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</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="Olive" title="b1">f</font> holds <br/><span class="p1">(<span class="default"><a href="nomin_1.html#K14" title="NOMIN_1:func.14">namingSeq</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be  <font color="Maroon" title="c2">V</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</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> implies <span class="p1">(<span class="default"><a href="nomin_1.html#K14" title="NOMIN_1:func.14">namingSeq</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:86"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">and </span><br/><a NAME="E2:86"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<font color="Maroon" title="c4">n</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>
 ; <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"> <span class="p1">(<span class="default"><a href="nomin_1.html#K14" title="NOMIN_1:func.14">namingSeq</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">g</font> =  <a href="nomin_1.html#K14" title="NOMIN_1:func.14">namingSeq</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">a</font>);<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_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="E1:86_1"/>
( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c4">n</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="#E1:86"><span class="lab"><font color="Green" title="E51">A1</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="86_1_1"><span class="kw">suppose </span></a><a NAME="E1:86_1_1"/>
<font color="Maroon" title="c4">n</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"> <span class="p1">(<span class="default"><a href="nomin_1.html#K14" title="NOMIN_1:func.14">namingSeq</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font></span><br/><div class="add"><a NAME="E2:86_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="nomin_1.html#K14" title="NOMIN_1:func.14">namingSeq</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K13" title="NOMIN_1:func.13">naming</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<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>)</span>,<font color="Maroon" title="c1">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="nomin_1.html#D14" target="_self" onmouseover="rs('nomin_1/D14')" onmouseout="rh()">Def14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:86_1_1"/>
<span class="p1">(<span class="default"><a href="nomin_1.html#K14" title="NOMIN_1:func.14">namingSeq</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_2"><span class="kw">suppose </span></a><a NAME="E1:86_1_2"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<font color="Maroon" title="c4">n</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"> <span class="p1">(<span class="default"><a href="nomin_1.html#K14" title="NOMIN_1:func.14">namingSeq</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c4">n</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="E3:86_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="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_2"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E4:86_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A4</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="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a></span>;<br/><a NAME="E5:86_1_2"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K14" title="NOMIN_1:func.14">namingSeq</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="nomin_1.html#D14" target="_self" onmouseover="rs('nomin_1/D14')" onmouseout="rh()">Def14</a></span>;<br/><a NAME="E6:86_1_2"/>
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</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="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E7:86_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K14" title="NOMIN_1:func.14">namingSeq</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E5:86_1_2"><span class="lab"><font color="Green" title="E55">A5</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="E8:86_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="nomin_1.html#K14" title="NOMIN_1:func.14">namingSeq</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</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#K13" title="NOMIN_1:func.13">naming</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nomin_1.html#K14" title="NOMIN_1:func.14">namingSeq</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E4:86_1_2"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="nomin_1.html#D14" target="_self" onmouseover="rs('nomin_1/D14')" onmouseout="rh()">Def14</a></span>;<br/><span class="kw">hence </span><a NAME="E9:86_1_2"/>
<span class="p1">(<span class="default"><a href="nomin_1.html#K14" title="NOMIN_1:func.14">namingSeq</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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