<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</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">  for <font color="Olive" title="b1">lk</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">Key</font> being   <a href="matrix_0.html#NM2" title="MATRIX_0:NM.2">Matrix</a> of <font color="Olive" title="b1">lk</font>,6,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a><br/>  for <font color="Olive" title="b3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> <a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Olive" title="b2">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b3">k</font>) is    <a href="funct_7.html#M1" title="FUNCT_7:mode.1">FuncSequence</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> , <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">lk</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">Key</font> being   <a href="matrix_0.html#NM2" title="MATRIX_0:NM.2">Matrix</a> of <font color="Maroon" title="c2">lk</font>,6,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a><br/>  for <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> <a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Olive" title="b1">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">k</font>) is    <a href="funct_7.html#M1" title="FUNCT_7:mode.1">FuncSequence</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> , <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">Key</font> be   <a href="matrix_0.html#NM2" title="MATRIX_0:NM.2">Matrix</a> of <font color="Maroon" title="c2">lk</font>,6,<a href="numbers.html#NK1" title="NUMBERS:NK.1">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">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> <a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">k</font>) is    <a href="funct_7.html#M1" title="FUNCT_7:mode.1">FuncSequence</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> , <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">r</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"> ( <font color="Maroon" title="c4">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>) is    <a href="funct_7.html#M1" title="FUNCT_7:mode.1">FuncSequence</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> , <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:68"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>) is    <a href="funct_7.html#M1" title="FUNCT_7:mode.1">FuncSequence</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> , <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span><br/>

<a NAME="E2:68"/><span class="kw">then </span>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E3:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A2</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</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="E4:68"/>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E5:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A3</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</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="E6:68"/>
<font color="Maroon" title="c4">r</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="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D18" target="_self" onmouseover="rs('idea_1/D18')" onmouseout="rh()">Def18</a></span>;<br/>
<a NAME="E7:68"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:68"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E8:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A4</font></span>: 
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> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E9:68"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="idea_1.html#D18" target="_self" onmouseover="rs('idea_1/D18')" onmouseout="rh()">Def18</a></span>;<br/>
<a NAME="E10:68"/><span class="kw">then </span>
 not  <a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>
 
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E59">A5</font></span>:  <a href="funct_7.html#K5" title="FUNCT_7:func.5">firstdom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> = 
 <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#D6" onmouseover="rs('funct_7/D6')" onmouseout="rh()">FUNCT_7:def 6</a></span>
<br/>.= 
 <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a> <span class="p1">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c3">Key</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:68"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="ref" href="idea_1.html#D18" target="_self" onmouseover="rs('idea_1/D18')" onmouseout="rh()">Def18</a></span>
<br/>.= 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c3">Key</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span>

<br/>.= 
 <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
;<br/>
<a NAME="E12:68"/>
<font color="Maroon" title="c4">r</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="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D18" target="_self" onmouseover="rs('idea_1/D18')" onmouseout="rh()">Def18</a></span>;<br/>
<a NAME="E13:68"/><span class="kw">then </span>
<font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E14:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A6</font></span>: 
<font color="Maroon" title="c4">r</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"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E15:68"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="idea_1.html#D18" target="_self" onmouseover="rs('idea_1/D18')" onmouseout="rh()">Def18</a></span>;<br/>
<a NAME="E16:68"/><span class="kw">then </span>
 not  <a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>
 
;<br/>
<span class="kw">then </span> <a href="funct_7.html#K6" title="FUNCT_7:func.6">lastrng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> = 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <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> <span class="p3">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#D7" onmouseover="rs('funct_7/D7')" onmouseout="rh()">FUNCT_7:def 7</a></span>
<br/>.= 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D18" target="_self" onmouseover="rs('idea_1/D18')" onmouseout="rh()">Def18</a></span>
<br/>.= 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <span class="p1">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c3">Key</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:68"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="ref" href="idea_1.html#D18" target="_self" onmouseover="rs('idea_1/D18')" onmouseout="rh()">Def18</a></span>
<br/>.= 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c3">Key</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span>

;<br/>
<a NAME="E18:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A7</font></span>: 
 <a href="funct_7.html#K6" title="FUNCT_7:func.6">lastrng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<a NAME="E19:68"/>
 <a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>) is   <a href="funct_7.html#NM1" title="FUNCT_7:NM.1">FuncSequence</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#T39" target="_self" onmouseover="rs('idea_1/T39')" onmouseout="rh()">Th39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:68"/>
 <a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> (<font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">r</font>) is    <a href="funct_7.html#M1" title="FUNCT_7:mode.1">FuncSequence</a> of  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> , <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:68"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="txt" href="#E11:68"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="ref" href="funct_7.html#D9" onmouseover="rs('funct_7/D9')" onmouseout="rh()">FUNCT_7:def 9</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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