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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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">x0</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K8" title="EUCLID_7:func.8">RN_Base</a> <font color="Maroon" title="c1">n</font> holds <br/> ex <font color="Olive" title="b3">l</font> being   <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Olive" title="b2">B</font> st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b3">l</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x0</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>; <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">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K8" title="EUCLID_7:func.8">RN_Base</a> <font color="Maroon" title="c1">n</font> holds <br/> ex <font color="Olive" title="b2">l</font> being   <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Olive" title="b1">B</font> st <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b2">l</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K8" title="EUCLID_7:func.8">RN_Base</a> <font color="Maroon" title="c1">n</font> implies  ex <font color="Olive" title="b1">l</font> being   <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">B</font> st <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">l</font> )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c2">x0</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T111" title="FINSEQ_2:th.111">FINSEQ_2:111</a></span>;<br/>
<a NAME="E2:96"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
 <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T111" title="FINSEQ_2:th.111">FINSEQ_2:111</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:96"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K8" title="EUCLID_7:func.8">RN_Base</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">l</font> being   <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">B</font> st <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">l</font></span><br/>

<a NAME="E4:96"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font> ) </span>}</span>  or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:96_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font></span><br/>

<a NAME="E2:96_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/>( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font> ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:96_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E50">A2</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/>


</div><span class="kw">end;</span></div>
<a NAME="E5:96"/>
<font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font> ) </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:96_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font> ) </span>}</span> </span><br/>

<a NAME="E2:96_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">i2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E50">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:96_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font> ) </span>}</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>
<a NAME="E6:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A4</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E7:96"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font> &amp; <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Olive" title="b1">x</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></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>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font> &amp; <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Olive" title="b1">x</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></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>}</span>  or <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:96_3"/>
<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font> &amp; <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Olive" title="b1">x</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></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>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font></span><br/>

<a NAME="E2:96_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> &amp;  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font> &amp; <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Olive" title="b1">x</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:96_3"/>
<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:96"><span class="lab"><font color="Green" title="E52">A4</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/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">B0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font> &amp; <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Olive" title="b1">x</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></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>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="lab"><font color="Green" title="E54">A6</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> = 
 <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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span></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 href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D12" target="_self" title="EUCLID_7:def.12">Def12</a></span>
;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font> implies  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i</font> ) );<br/>
<a NAME="E10:96"/><span class="lab"><font color="Green" title="E55">A7</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 
;<br/>
<a NAME="E11:96"/><span class="lab"><font color="Green" title="E56">A8</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</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> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</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> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:96_5"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</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> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E3:96_5"/><span class="lab"><font color="Green" title="E57">A9</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font>
 <span class="kw">and </span><br/><a NAME="E4:96_5"/><span class="lab"><font color="Green" title="E58">A10</font></span>: 
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i</font> &amp; <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c7">x2</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">i0</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:96_5"/><span class="lab"><font color="Green" title="E59">A11</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i0</font>
 <span class="kw">and </span><br/><a NAME="E7:96_5"/><span class="lab"><font color="Green" title="E60">A12</font></span>: 
<font color="Maroon" title="c8">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E8:96_5"/><span class="lab"><font color="Green" title="E61">A13</font></span>: 
<font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font>
 <span class="kw">and </span><br/><a NAME="E9:96_5"/>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c7">x2</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></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="#E4:96_5"><span class="lab"><font color="Green" title="E58">A10</font></span></a></span>;<br/>
<a NAME="E10:96_5"/>
<font color="Maroon" title="c8">i0</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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:96_5"><span class="lab"><font color="Green" title="E59">A11</font></span></a>, <a class="txt" href="#E7:96_5"><span class="lab"><font color="Green" title="E60">A12</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:96_5"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</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> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_5"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="txt" href="#E6:96_5"><span class="lab"><font color="Green" title="E59">A11</font></span></a>, <a class="txt" href="#E7:96_5"><span class="lab"><font color="Green" title="E60">A12</font></span></a>, <a class="txt" href="#E8:96_5"><span class="lab"><font color="Green" title="E61">A13</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/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c6">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">B0</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E13:96"/><span class="lab"><font color="Green" title="E57">A14</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E11:96"><span class="lab"><font color="Green" title="E56">A8</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( ( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font> implies for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i</font> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> ) &amp; ( not $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) );<br/>
<a NAME="E14:96"/><span class="lab"><font color="Green" title="E58">A15</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:96_6"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_6_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="E1:96_6_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font> or  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:96_6_1_1"/><span class="lab"><font color="Green" title="E59">A16</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E3:96_6_1_1"/><span class="lab"><font color="Green" title="E60">A17</font></span>: 
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">and </span><br/><a NAME="E4:96_6_1_1"/>
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i</font> &amp; <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c8">x2</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">y0</font> = <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c8">x2</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><a NAME="E6:96_6_1_1"/>
for <font color="Olive" title="b1">i2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i2</font> holds <br/><font color="Maroon" title="c9">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_6_1_1"><span class="lab"><font color="Green" title="E60">A17</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:96_6_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_6_1_1"><span class="lab"><font color="Green" title="E59">A16</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:96_6_1_2"/>
not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:96_6_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>] )
 ; <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">consider </span><font color="Maroon" title="c7">l2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E16:96"/><span class="lab"><font color="Green" title="E59">A18</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c7">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E14:96"><span class="lab"><font color="Green" title="E58">A15</font></span></a>);<br/></span>
<a NAME="E17:96"/><span class="lab"><font color="Green" title="E60">A19</font></span>: 
<font color="Maroon" title="c7">l2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<a NAME="E18:96"/>
for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>  st not <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font> holds <br/><font color="Maroon" title="c7">l2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">v</font> <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="txt" href="#E16:96"><span class="lab"><font color="Green" title="E59">A18</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">l3</font> = <font color="Maroon" title="c7">l2</font> as    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of  <a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E7:96"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E17:96"><span class="lab"><font color="Green" title="E60">A19</font></span></a>, <a class="ref" href="rlvect_2.html#D5" title="RLVECT_2:def.5">RLVECT_2:def 5</a></span>;<br/>
<a NAME="E20:96"/>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c8">l3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c8">l3</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:96_7"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c8">l3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font></span><br/>

<a NAME="E2:96_7"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> : <font color="Maroon" title="c8">l3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">v</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>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></span>;<br/>
<a NAME="E3:96_7"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> st <br/>( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> &amp; <font color="Maroon" title="c8">l3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E4:96_7"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:96"><span class="lab"><font color="Green" title="E59">A18</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/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">l0</font> = <font color="Maroon" title="c8">l3</font> as    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c5">B0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D8" title="RLVECT_2:def.8">RLVECT_2:def 8</a></span>;<br/>
<a NAME="E22:96"/><span class="lab"><font color="Green" title="E61">A20</font></span>: 
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c9">l0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D8" title="RLVECT_2:def.8">RLVECT_2:def 8</a></span>;<br/>
<a NAME="E23:96"/><span class="kw">then </span>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c9">l0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:96"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">l2</font> = <font color="Maroon" title="c9">l0</font> as    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">B</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D8" title="RLVECT_2:def.8">RLVECT_2:def 8</a></span>;<br/>
<a NAME="E25:96"/><span class="lab"><font color="Green" title="E62">A21</font></span>: 
<font color="Maroon" title="c5">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c9">l0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c9">l0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:96_8"/><span class="lab"><font color="Green" title="E63">A22</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c9">l0</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x6</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E3:96_8"/><span class="lab"><font color="Green" title="E64">A23</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x6</font>
 <span class="kw">and </span><br/><a NAME="E4:96_8"/><span class="lab"><font color="Green" title="E65">A24</font></span>: 
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c12">x6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i</font> &amp; <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c12">x6</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">x66</font> = <font color="Maroon" title="c12">x6</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T111" title="FINSEQ_2:th.111">FINSEQ_2:111</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">i8</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:96_8"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i8</font>
 <span class="kw">and </span><br/><a NAME="E8:96_8"/>
<font color="Maroon" title="c14">i8</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E9:96_8"/><span class="lab"><font color="Green" title="E66">A25</font></span>: 
<font color="Maroon" title="c12">x6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c14">i8</font>
 <span class="kw">and </span><br/><a NAME="E10:96_8"/>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c12">x6</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></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="#E4:96_8"><span class="lab"><font color="Green" title="E65">A24</font></span></a></span>;<br/>
<a NAME="E11:96_8"/>
<font color="Maroon" title="c9">l0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x66</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c14">i8</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:96"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="txt" href="#E1:96_8"><span class="lab"><font color="Green" title="E63">A22</font></span></a>, <a class="txt" href="#E3:96_8"><span class="lab"><font color="Green" title="E64">A23</font></span></a>, <a class="txt" href="#E4:96_8"><span class="lab"><font color="Green" title="E65">A24</font></span></a>, <a class="txt" href="#E9:96_8"><span class="lab"><font color="Green" title="E66">A25</font></span></a></span>;<br/>
<a NAME="E12:96_8"/><span class="kw">then </span>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> : <font color="Maroon" title="c9">l0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">v</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>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_8"><span class="lab"><font color="Green" title="E64">A23</font></span></a>, <a class="txt" href="#E4:96_8"><span class="lab"><font color="Green" title="E65">A24</font></span></a>, <a class="txt" href="#E9:96_8"><span class="lab"><font color="Green" title="E66">A25</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:96_8"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c9">l0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</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/>

<span class="kw">reconsider </span><font color="Maroon" title="c15">xx</font> = <font color="Maroon" title="c11">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_8"><span class="lab"><font color="Green" title="E63">A22</font></span></a>, <a class="ref" href="finseq_2.html#T111" title="FINSEQ_2:th.111">FINSEQ_2:111</a></span>;<br/>

</div><span class="kw">end;</span></div>
<a NAME="E26:96"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x1</font>, <font color="Maroon" title="c12">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c12">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x2</font> implies <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:96_9"/><span class="lab"><font color="Green" title="E63">A26</font></span>: 
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">and </span><br/><a NAME="E2:96_9"/><span class="lab"><font color="Green" title="E64">A27</font></span>: 
<font color="Maroon" title="c12">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">and </span><br/><a NAME="E3:96_9"/><span class="lab"><font color="Green" title="E65">A28</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2</font></span><br/>

<a NAME="E4:96_9"/><span class="lab"><font color="Green" title="E66">A29</font></span>: 
 ex <font color="Olive" title="b1">i2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c12">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:96"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="txt" href="#E2:96_9"><span class="lab"><font color="Green" title="E64">A27</font></span></a></span>;<br/>
<a NAME="E5:96_9"/>
 ex <font color="Olive" title="b1">i1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i1</font> &amp; <font color="Olive" title="b1">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:96"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="txt" href="#E1:96_9"><span class="lab"><font color="Green" title="E63">A26</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:96_9"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_9"><span class="lab"><font color="Green" title="E65">A28</font></span></a>, <a class="txt" href="#E4:96_9"><span class="lab"><font color="Green" title="E66">A29</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/>


</div><span class="kw">end;</span></div>
<a NAME="E27:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A30</font></span>: 
<font color="Maroon" title="c6">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/>
<a NAME="E28:96"/><span class="lab"><font color="Green" title="E64">A31</font></span>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c5">B0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:96_10"/><span class="lab"><font color="Green" title="E65">A32</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">B0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_10_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c11">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">B0</font>;<br/><span class="kw">assume </span><a NAME="E2:96_10_1"/>
<font color="Maroon" title="c5">B0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:96_10_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font>
 ;<br/><a NAME="E4:96_10_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> &amp;  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">i</font> &amp; <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Olive" title="b1">x</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 ;<br/><span class="kw">hence </span><a NAME="E5:96_10_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_10"><span class="lab"><font color="Green" title="E65">A32</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:96_10"/>
<font color="Maroon" title="c5">B0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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>
<a NAME="E29:96"/><span class="lab"><font color="Green" title="E65">A33</font></span>: 
for <font color="Olive" title="b1">i3</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i3</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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> &amp; not <font color="Olive" title="b1">i3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">i3</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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"> ( <font color="Maroon" title="c11">i3</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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> &amp; not <font color="Maroon" title="c11">i3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:96_11"/><span class="lab"><font color="Green" title="E66">A34</font></span>: 
<font color="Maroon" title="c11">i3</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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:96_11"/><span class="lab"><font color="Green" title="E67">A35</font></span>: 
not <font color="Maroon" title="c11">i3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span><br/>

<a NAME="E3:96_11"/><span class="lab"><font color="Green" title="E68">A36</font></span>: 
<font color="Maroon" title="c11">i3</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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_11"><span class="lab"><font color="Green" title="E66">A34</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:96_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A37</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E5:96_11"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D12" target="_self" title="EUCLID_7:def.12">Def12</a></span>;<br/>
<a NAME="E6:96_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A38</font></span>: 
<font color="Maroon" title="c11">i3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_11"><span class="lab"><font color="Green" title="E68">A36</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E7:96_11"/><span class="lab"><font color="Green" title="E71">A39</font></span>: 
not <font color="Maroon" title="c11">i3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_11"><span class="lab"><font color="Green" title="E67">A35</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E72">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="96_11_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:96_11_1"/>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c11">i3</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <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"> contradiction</span><br/><a NAME="E2:96_11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A41</font></span>: 
 <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c11">i3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_11"><span class="lab"><font color="Green" title="E69">A37</font></span></a>, <a class="txt" href="#E6:96_11"><span class="lab"><font color="Green" title="E70">A38</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">i5</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:96_11_1"/><span class="lab"><font color="Green" title="E74">A42</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c11">i3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i5</font>
 <span class="kw">and </span><br/><a NAME="E5:96_11_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i5</font>
 <span class="kw">and </span><br/><a NAME="E6:96_11_1"/>
<font color="Maroon" title="c12">i5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E7:96_11_1"/><span class="lab"><font color="Green" title="E75">A43</font></span>: 
 <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c11">i3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c12">i5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:96"><span class="lab"><font color="Green" title="E57">A14</font></span></a></span>;<br/><a NAME="E8:96_11_1"/><span class="lab"><font color="Green" title="E76">A44</font></span>: 
 <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c11">i3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:96"><span class="lab"><font color="Green" title="E64">A31</font></span></a>, <a class="txt" href="#E2:96_11_1"><span class="lab"><font color="Green" title="E73">A41</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E9:96_11_1"/>
<font color="Maroon" title="c11">i3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_11"><span class="lab"><font color="Green" title="E69">A37</font></span></a>, <a class="txt" href="#E6:96_11"><span class="lab"><font color="Green" title="E70">A38</font></span></a>, <a class="txt" href="#E7:96_11_1"><span class="lab"><font color="Green" title="E75">A43</font></span></a>, <a class="ref" href="euclid_7.html#T27" target="_self" title="EUCLID_7:th.27">Th27</a></span>;<br/><span class="kw">hence </span><a NAME="E10:96_11_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:96_11"><span class="lab"><font color="Green" title="E71">A39</font></span></a>, <a class="txt" href="#E4:96_11_1"><span class="lab"><font color="Green" title="E74">A42</font></span></a>, <a class="txt" href="#E8:96_11_1"><span class="lab"><font color="Green" title="E76">A44</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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>
<a NAME="E9:96_11"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c11">i3</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c11">i3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_11"><span class="lab"><font color="Green" title="E69">A37</font></span></a>, <a class="txt" href="#E6:96_11"><span class="lab"><font color="Green" title="E70">A38</font></span></a>, <a class="ref" href="euclid_7.html#D12" target="_self" title="EUCLID_7:def.12">Def12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:96_11"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:96_11"><span class="lab"><font color="Green" title="E72">A40</font></span></a>, <a class="ref" href="euclid_4.html#T3" title="EUCLID_4:th.3">EUCLID_4:3</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>
<a NAME="E30:96"/><span class="lab"><font color="Green" title="E66">A45</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></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="E31:96"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></span>;<br/>
<a NAME="E32:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A46</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:96"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E33:96"/><span class="lab"><font color="Green" title="E68">A47</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T182" title="RELAT_1:th.182">RELAT_1:182</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:96"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="txt" href="#E9:96"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="euclid_7.html#T2" target="_self" title="EUCLID_7:th.2">Th2</a></span>
;<br/>
<a NAME="E35:96"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> is   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:96"><span class="lab"><font color="Green" title="E66">A45</font></span></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">F2</font> = <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E32:96"><span class="lab"><font color="Green" title="E67">A46</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">F2</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:96"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="relat_1.html#T182" title="RELAT_1:th.182">RELAT_1:182</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:96"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="ref" href="euclid_7.html#T2" target="_self" title="EUCLID_7:th.2">Th2</a></span>
;<br/>
<a NAME="E38:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A48</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></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="E39:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A49</font></span>: 
 <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> <font color="Maroon" title="c11">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></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/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">F3</font> = <font color="Maroon" title="c11">F2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T111" title="FINSEQ_2:th.111">FINSEQ_2:111</a></span>;<br/>
<span class="lab"><font color="Green" title="E71">A50</font></span>: <font color="Maroon" title="c2">x0</font> = 
 <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#T33" target="_self" title="EUCLID_7:th.33">Th33</a></span>
<br/>.= 
 <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c12">F3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:96"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="txt" href="#E9:96"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="txt" href="#E29:96"><span class="lab"><font color="Green" title="E65">A33</font></span></a>, <a class="ref" href="euclid_7.html#T26" target="_self" title="EUCLID_7:th.26">Th26</a>, <a class="ref" href="finseq_3.html#T99" title="FINSEQ_3:th.99">FINSEQ_3:99</a></span>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c11">F2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#T45" target="_self" title="EUCLID_7:th.45">Th45</a></span>
;<br/>
<a NAME="E42:96"/><span class="lab"><font color="Green" title="E72">A51</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></span>;<br/>
<a NAME="E43:96"/><span class="lab"><font color="Green" title="E73">A52</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></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="E44:96"/><span class="lab"><font color="Green" title="E74">A53</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">F2</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="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:96"><span class="lab"><font color="Green" title="E69">A48</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E45:96"/><span class="lab"><font color="Green" title="E75">A54</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:96"><span class="lab"><font color="Green" title="E64">A31</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E46:96"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:96"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/>
<a NAME="E47:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A55</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:96"><span class="lab"><font color="Green" title="E72">A51</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T182" title="RELAT_1:th.182">RELAT_1:182</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:96"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:96"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="ref" href="euclid_7.html#T2" target="_self" title="EUCLID_7:th.2">Th2</a></span>
;<br/>
<a NAME="E49:96"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> is   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:96"><span class="lab"><font color="Green" title="E73">A52</font></span></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">F0</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:96"><span class="lab"><font color="Green" title="E76">A55</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">F0</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T182" title="RELAT_1:th.182">RELAT_1:182</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:96"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:96"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="ref" href="euclid_7.html#T2" target="_self" title="EUCLID_7:th.2">Th2</a></span>
;<br/>
<a NAME="E52:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A56</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">F0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></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="E53:96"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:96"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/>
<span class="kw">then </span> <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">F0</font> = 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:96"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:96"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>
;<br/>
<a NAME="E55:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A57</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">F0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c9">l0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:96"><span class="lab"><font color="Green" title="E75">A54</font></span></a>, <a class="txt" href="#E22:96"><span class="lab"><font color="Green" title="E61">A20</font></span></a>, <a class="txt" href="#E25:96"><span class="lab"><font color="Green" title="E62">A21</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E56:96"/><span class="lab"><font color="Green" title="E79">A58</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</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="c11">F2</font> holds <br/><font color="Maroon" title="c11">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">F0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_18"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c14">i7</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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"> ( <font color="Maroon" title="c15">i</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="c11">F2</font> implies <font color="Maroon" title="c11">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">F0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font></span>)</span> )</span><br/>

<a NAME="E2:96_18"/><span class="lab"><font color="Green" title="E80">A59</font></span>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T99" title="FINSEQ_3:th.99">FINSEQ_3:99</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:96_18"/>
<font color="Maroon" title="c15">i</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="c11">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">F0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font></span>)</span></span><br/>

<a NAME="E4:96_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A60</font></span>: 
<font color="Maroon" title="c15">i</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> <font color="Maroon" title="c11">F2</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="E5:96_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A61</font></span>: 
<font color="Maroon" title="c15">i</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="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:96"><span class="lab"><font color="Green" title="E74">A53</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:96_18"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">i2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">r</font> =  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T111" title="FINSEQ_2:th.111">FINSEQ_2:111</a></span>;<br/>
<a NAME="E9:96_18"/>
<font color="Maroon" title="c16">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_18"><span class="lab"><font color="Green" title="E82">A61</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:96_18"/><span class="lab"><font color="Green" title="E83">A62</font></span>: 
<font color="Maroon" title="c18">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">and </span><br/><a NAME="E12:96_18"/><span class="lab"><font color="Green" title="E84">A63</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:96"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E13:96_18"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:96"><span class="lab"><font color="Green" title="E64">A31</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">r2</font> = <font color="Maroon" title="c18">x2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:96_18"><span class="lab"><font color="Green" title="E83">A62</font></span></a></span>;<br/>
<a NAME="E15:96_18"/><span class="lab"><font color="Green" title="E85">A64</font></span>: 
 ex <font color="Olive" title="b1">i22</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i22</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i22</font> &amp; <font color="Olive" title="b1">i22</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c19">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i22</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:96"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="txt" href="#E11:96_18"><span class="lab"><font color="Green" title="E83">A62</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">i4</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E17:96_18"/><span class="lab"><font color="Green" title="E86">A65</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i4</font>
 <span class="kw">and </span><br/><a NAME="E18:96_18"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i4</font>
 <span class="kw">and </span><br/><a NAME="E19:96_18"/>
<font color="Maroon" title="c20">i4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E20:96_18"/><span class="lab"><font color="Green" title="E87">A66</font></span>: 
<font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c20">i4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:96_18"><span class="lab"><font color="Green" title="E84">A63</font></span></a></span>;<br/>
<a NAME="E21:96_18"/><span class="lab"><font color="Green" title="E88">A67</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:96"><span class="lab"><font color="Green" title="E64">A31</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<font color="Maroon" title="c13">F0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_18"><span class="lab"><font color="Green" title="E82">A61</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
 <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:96"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="txt" href="#E11:96_18"><span class="lab"><font color="Green" title="E83">A62</font></span></a>, <a class="txt" href="#E12:96_18"><span class="lab"><font color="Green" title="E84">A63</font></span></a>, <a class="txt" href="#E15:96_18"><span class="lab"><font color="Green" title="E85">A64</font></span></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></span>
;<br/>
<a NAME="E23:96_18"/><span class="kw">then </span>
 <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">F0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:96"><span class="lab"><font color="Green" title="E77">A56</font></span></a>, <a class="txt" href="#E39:96"><span class="lab"><font color="Green" title="E70">A49</font></span></a>, <a class="txt" href="#E4:96_18"><span class="lab"><font color="Green" title="E81">A60</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E24:96_18"/><span class="kw">then </span>
 <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> : <font color="Maroon" title="c9">l0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">v</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>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:96"><span class="lab"><font color="Green" title="E78">A57</font></span></a>, <a class="ref" href="rlvect_2.html#D6" title="RLVECT_2:def.6">RLVECT_2:def 6</a></span>;<br/>
<a NAME="E25:96_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A68</font></span>: 
 ex <font color="Olive" title="b1">v0</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> st <br/>(  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v0</font> &amp; <font color="Maroon" title="c9">l0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">v0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
;<br/>
<a NAME="E26:96_18"/><span class="kw">then </span>
 <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:96"><span class="lab"><font color="Green" title="E59">A18</font></span></a></span>;<br/>
<a NAME="E27:96_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A69</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:96"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="txt" href="#E21:96_18"><span class="lab"><font color="Green" title="E88">A67</font></span></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></span>;<br/>
<a NAME="E28:96_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A70</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_18"><span class="lab"><font color="Green" title="E82">A61</font></span></a>, <a class="txt" href="#E12:96_18"><span class="lab"><font color="Green" title="E84">A63</font></span></a>, <a class="txt" href="#E15:96_18"><span class="lab"><font color="Green" title="E85">A64</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/>
<a NAME="E29:96_18"/><span class="lab"><font color="Green" title="E92">A71</font></span>: 
<font color="Maroon" title="c16">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:96"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="txt" href="#E5:96_18"><span class="lab"><font color="Green" title="E82">A61</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E30:96_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A72</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E31:96_18"/><span class="lab"><font color="Green" title="E94">A73</font></span>: 
<font color="Maroon" title="c16">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:96_18"><span class="lab"><font color="Green" title="E92">A71</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E32:96_18"/><span class="kw">then </span>
<font color="Maroon" title="c20">i4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:96_18"><span class="lab"><font color="Green" title="E93">A72</font></span></a>, <a class="txt" href="#E20:96_18"><span class="lab"><font color="Green" title="E87">A66</font></span></a>, <a class="ref" href="euclid_7.html#T27" target="_self" title="EUCLID_7:th.27">Th27</a></span>;<br/>
<a NAME="E33:96_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A74</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_18"><span class="lab"><font color="Green" title="E82">A61</font></span></a>, <a class="txt" href="#E17:96_18"><span class="lab"><font color="Green" title="E86">A65</font></span></a>, <a class="txt" href="#E2:96_18"><span class="lab"><font color="Green" title="E80">A59</font></span></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></span>;<br/>
<a NAME="E34:96_18"/><span class="lab"><font color="Green" title="E96">A75</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:96"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="txt" href="#E11:96_18"><span class="lab"><font color="Green" title="E83">A62</font></span></a>, <a class="txt" href="#E12:96_18"><span class="lab"><font color="Green" title="E84">A63</font></span></a>, <a class="txt" href="#E15:96_18"><span class="lab"><font color="Green" title="E85">A64</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E35:96_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A76</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p5">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_18"><span class="lab"><font color="Green" title="E80">A59</font></span></a>, <a class="txt" href="#E27:96_18"><span class="lab"><font color="Green" title="E90">A69</font></span></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></span>;<br/>
<a NAME="E36:96_18"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_18"><span class="lab"><font color="Green" title="E80">A59</font></span></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/>
<a NAME="E37:96_18"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:96_18"><span class="lab"><font color="Green" title="E96">A75</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E38:96_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A77</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font></span>)</span></span>)</span> <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="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96_18"><span class="lab"><font color="Green" title="E80">A59</font></span></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/>
<font color="Maroon" title="c9">l0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font></span>)</span> = 
<font color="Maroon" title="c9">l0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:96"><span class="lab"><font color="Green" title="E77">A56</font></span></a>, <a class="txt" href="#E44:96"><span class="lab"><font color="Green" title="E74">A53</font></span></a>, <a class="txt" href="#E4:96_18"><span class="lab"><font color="Green" title="E81">A60</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c9">l0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:96_18"><span class="lab"><font color="Green" title="E95">A74</font></span></a>, <a class="txt" href="#E38:96_18"><span class="lab"><font color="Green" title="E98">A77</font></span></a>, <a class="txt" href="#E35:96_18"><span class="lab"><font color="Green" title="E97">A76</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:96"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="txt" href="#E30:96_18"><span class="lab"><font color="Green" title="E93">A72</font></span></a>, <a class="txt" href="#E31:96_18"><span class="lab"><font color="Green" title="E94">A73</font></span></a>, <a class="txt" href="#E25:96_18"><span class="lab"><font color="Green" title="E89">A68</font></span></a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c9">l0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">F0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font></span>)</span> = 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c4">x1</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c16">i2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:96"><span class="lab"><font color="Green" title="E77">A56</font></span></a>, <a class="txt" href="#E44:96"><span class="lab"><font color="Green" title="E74">A53</font></span></a>, <a class="txt" href="#E4:96_18"><span class="lab"><font color="Green" title="E81">A60</font></span></a>, <a class="txt" href="#E28:96_18"><span class="lab"><font color="Green" title="E91">A70</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:96_18"><span class="lab"><font color="Green" title="E93">A72</font></span></a>, <a class="txt" href="#E31:96_18"><span class="lab"><font color="Green" title="E94">A73</font></span></a>, <a class="ref" href="euclid_7.html#D12" target="_self" title="EUCLID_7:def.12">Def12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c4">x1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_18"><span class="lab"><font color="Green" title="E82">A61</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E41:96_18"/>
<font color="Maroon" title="c11">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">F0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">F0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font></span>)</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>
<span class="kw">reconsider </span><font color="Maroon" title="c14">F01</font> = <font color="Maroon" title="c13">F0</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">rf</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">l1</font> = <font color="Maroon" title="c9">l0</font> as    <a href="rlvect_2.html#M1" title="RLVECT_2:mode.1">Linear_Combination</a> of  <a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">X</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font>;<br/>
<a NAME="E60:96"/><span class="lab"><font color="Green" title="E80">A78</font></span>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T99" title="FINSEQ_3:th.99">FINSEQ_3:99</a></span>;<br/>
<a NAME="E61:96"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">F0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:96"><span class="lab"><font color="Green" title="E77">A56</font></span></a>, <a class="txt" href="#E39:96"><span class="lab"><font color="Green" title="E70">A49</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E62:96"/><span class="kw">then </span>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l0</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c13">F0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:96"><span class="lab"><font color="Green" title="E71">A50</font></span></a>, <a class="txt" href="#E56:96"><span class="lab"><font color="Green" title="E79">A58</font></span></a>, <a class="ref" href="rlvect_2.html#D9" title="RLVECT_2:def.9">RLVECT_2:def 9</a></span>;<br/>
<a NAME="E63:96"/><span class="kw">then </span>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c10">l2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:96"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="txt" href="#E60:96"><span class="lab"><font color="Green" title="E80">A78</font></span></a>, <a class="txt" href="#E55:96"><span class="lab"><font color="Green" title="E78">A57</font></span></a>, <a class="ref" href="rlvect_2.html#D10" title="RLVECT_2:def.10">RLVECT_2:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E64:96"/>
 ex <font color="Olive" title="b1">l</font> being   <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c3">B</font> st <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">l</font>
 ; <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>
