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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">lk</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">Key</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Olive" title="b1">lk</font>,6, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Olive" title="b2">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b3">r</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </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>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Olive" title="b2">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b3">r</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Key</font> being   <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">lk</font>,6, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Olive" title="b1">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">r</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </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>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Olive" title="b1">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">r</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">lk</font>,6, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">r</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </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>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">r</font></span>)</span>,<a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </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>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>  )</span><br/>







<a NAME="E1:75"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:75_1_1"/>
( <font color="Maroon" title="c4">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_1"/><span class="lab"><font color="Green" title="E59">A2</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#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </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><br/><div class="add"><a NAME="E2:75_1_1_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E3:75_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">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" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:75_1_1_1"/>
<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#K17" title="IDEA_1:func.17">IDEA_P_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#D17" target="_self" title="IDEA_1:def.17">Def17</a></span>;<br/><a NAME="E5:75_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN: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#K17" title="IDEA_1:func.17">IDEA_P_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="#E3:75_1_1_1"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E6:75_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A4</font></span>: 
<font color="Maroon" title="c4">r</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> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E7:75_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_1"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="idea_1.html#D17" target="_self" title="IDEA_1:def.17">Def17</a></span>;<br/><a NAME="E8:75_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A5</font></span>: 
not  <a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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#K7" title="FUNCT_7:func.7">lastrng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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#K17" title="IDEA_1:func.17">IDEA_P_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" title="FUNCT_7:def.7">FUNCT_7:def 7</a></span>
<br/>.= 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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#D17" target="_self" title="IDEA_1:def.17">Def17</a></span>
<br/>.= 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <span class="p1">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c4">r</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_1_1_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="idea_1.html#D17" target="_self" title="IDEA_1:def.17">Def17</a></span>
<br/>.= 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c4">r</font></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>

;<br/><a NAME="E10:75_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A6</font></span>: 
 <a href="funct_7.html#K7" title="FUNCT_7:func.7">lastrng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E11:75_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_7.html#K7" title="FUNCT_7:func.7">lastrng</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="txt" href="#E8:75_1_1_1"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="ref" href="funct_7.html#T61" title="FUNCT_7:th.61">FUNCT_7:61</a></span>;<br/><span class="kw">hence </span><a NAME="E12:75_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </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="txt" href="#E10:75_1_1_1"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="75_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_2"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </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><br/><div class="add"><a NAME="E2:75_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"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D17" target="_self" title="IDEA_1:def.17">Def17</a></span>;<br/><a NAME="E3:75_1_1_2"/><span class="kw">then </span>
 <a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E4:75_1_1_2"/><span class="kw">then </span>
 <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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="funct_7.html#T41" title="FUNCT_7:th.41">FUNCT_7:41</a></span>;<br/><span class="kw">hence </span><a NAME="E5:75_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </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" title="RELAT_1:def.19">RELAT_1:def 19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E2:75"/><span class="lab"><font color="Green" title="E59">A7</font></span>: 
 <a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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" title="IDEA_1:th.39">Th39</a></span>;<br/>
<a NAME="E3:75"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:75_2_1"/>
( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:75_2_1_1"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span><br/><div class="add"><a NAME="E2:75_2_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D17" target="_self" title="IDEA_1:def.17">Def17</a></span>;<br/><a NAME="E3:75_2_1_1"/><span class="kw">then </span>
 <a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E4:75_2_1_1"/><span class="kw">then </span>
 <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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="funct_7.html#T41" title="FUNCT_7:th.41">FUNCT_7:41</a></span>;<br/><span class="kw">hence </span><a NAME="E5:75_2_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="75_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:75_2_1_2"/><span class="lab"><font color="Green" title="E60">A8</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#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span><br/><div class="add"><a NAME="E2:75_2_1_2"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E3:75_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A9</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" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:75_2_1_2"/>
<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#K17" title="IDEA_1:func.17">IDEA_P_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#D17" target="_self" title="IDEA_1:def.17">Def17</a></span>;<br/><a NAME="E5:75_2_1_2"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN: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#K17" title="IDEA_1:func.17">IDEA_P_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="#E3:75_2_1_2"><span class="lab"><font color="Green" title="E61">A9</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E6:75_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A10</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN: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#K17" title="IDEA_1:func.17">IDEA_P_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" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E7:75_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_2_1_2"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="ref" href="idea_1.html#D17" target="_self" title="IDEA_1:def.17">Def17</a></span>;<br/><a NAME="E8:75_2_1_2"/><span class="kw">then </span>
not  <a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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">firstdom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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" title="FUNCT_7:def.6">FUNCT_7:def 6</a></span>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,1</span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_2_1_2"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="ref" href="idea_1.html#D17" target="_self" title="IDEA_1:def.17">Def17</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K15" title="IDEA_1:func.15">IDEA_P</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,1</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" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:75_2_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E2:75"><span class="lab"><font color="Green" title="E59">A7</font></span></a>, <a class="ref" href="funct_7.html#T64" title="FUNCT_7:th.64">FUNCT_7:64</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:75"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </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>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> <span class="p2">(<span class="default"><a href="idea_1.html#K17" title="IDEA_1:func.17">IDEA_P_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="idea_1.html#K14" title="IDEA_1:func.14">MESSAGES</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E1:75"><span class="lab"><font color="Green" title="E58">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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