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

<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">j</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font> implies <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font>
 <span class="kw">and </span><br/><a NAME="E2:57"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E3:57"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">x1</font> =  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">x2</font> =  <a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font>;<br/>
<span class="lab"><font color="Green" title="E38">A4</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</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"><font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="c2">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>
;<br/>
<span class="lab"><font color="Green" title="E39">A5</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#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</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#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</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="c2">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr2.html#T74" title="MATRIXR2:th.74">MATRIXR2:74</a></span>
;<br/>
<span class="lab"><font color="Green" title="E40">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#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</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#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</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="c2">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr2.html#T74" title="MATRIXR2:th.74">MATRIXR2:74</a></span>
;<br/>
<a NAME="E7:57"/><span class="lab"><font color="Green" title="E41">A7</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="euclid_7.html#K1" title="EUCLID_7:func.1">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</a></span>;<br/>
<a NAME="E8:57"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E42">A8</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="euclid_7.html#K1" title="EUCLID_7:func.1">&lt;:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span></span>)</span> = 
<span class="p1"><a href="euclid_7.html#K1" title="EUCLID_7:func.1">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="relset_1.html#T38" title="RELSET_1:th.38">RELSET_1:38</a></span>
;<br/>
<a NAME="E10:57"/>
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>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="euclid_7.html#K1" title="EUCLID_7:func.1">&lt;:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_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>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font></span>)</span> implies <span class="p1">(<span class="default"><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="euclid_7.html#K1" title="EUCLID_7:func.1">&lt;:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:57_5"/><span class="lab"><font color="Green" title="E43">A9</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font></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="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="euclid_7.html#K1" title="EUCLID_7:func.1">&lt;:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">nx</font> = <font color="Maroon" title="c6">x</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="lab"><font color="Green" title="E44">A10</font></span>: <span class="p1">(<span class="default"><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="euclid_7.html#K1" title="EUCLID_7:func.1">&lt;:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_7.html#K1" title="EUCLID_7:func.1">&lt;:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E9:57"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E1:57_5"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<br/>.= 
<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">nx</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">nx</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E7:57"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E1:57_5"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="ref" href="funct_3.html#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</a></span>
;<br/>
<a NAME="E4:57_5"/><span class="lab"><font color="Green" title="E45">A11</font></span>: 
<font color="Maroon" title="c7">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E1:57_5"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E5:57_5"/><span class="lab"><font color="Green" title="E46">A12</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">nx</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E1:57_5"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_5_2"><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="E1:57_5_2"/>
( <font color="Maroon" title="c7">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> or <font color="Maroon" title="c7">nx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_5_2_1"><span class="kw">suppose </span></a><a NAME="E1:57_5_2_1"/><span class="lab"><font color="Green" title="E47">A13</font></span>: 
<font color="Maroon" title="c7">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font>
 ; <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="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="euclid_7.html#K1" title="EUCLID_7:func.1">&lt;:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/><div class="add"><a NAME="E2:57_5_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A14</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="matrixr2.html#T75" title="MATRIXR2:th.75">MATRIXR2:75</a></span>;<br/><a NAME="E3:57_5_2_1"/><span class="lab"><font color="Green" title="E49">A15</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">nx</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="#E1:57"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E1:57_5_2_1"><span class="lab"><font color="Green" title="E47">A13</font></span></a>, <a class="ref" href="matrixr2.html#T76" title="MATRIXR2:th.76">MATRIXR2:76</a></span>;<br/><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">nx</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">nx</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> = 
<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">nx</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">nx</font></span>)</span>

<br/>.= 
1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_5_2_1"><span class="lab"><font color="Green" title="E49">A15</font></span></a>, <a class="txt" href="#E2:57_5_2_1"><span class="lab"><font color="Green" title="E48">A14</font></span></a>, <a class="ref" href="binop_2.html#D11" title="BINOP_2:def.11">BINOP_2:def 11</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><span class="kw">hence </span><a NAME="E5:57_5_2_1"/>
<span class="p1">(<span class="default"><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="euclid_7.html#K1" title="EUCLID_7:func.1">&lt;:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E1:57_5"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="#E3:57_5"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="ref" href="finseq_2.html#T71" title="FINSEQ_2:th.71">FINSEQ_2:71</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="57_5_2_2"><span class="kw">suppose </span></a><a NAME="E1:57_5_2_2"/><span class="lab"><font color="Green" title="E47">A16</font></span>: 
<font color="Maroon" title="c7">nx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">i</font>
 ; <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="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="euclid_7.html#K1" title="EUCLID_7:func.1">&lt;:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c8">r</font> = <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">nx</font> 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="E3:57_5_2_2"/><span class="lab"><font color="Green" title="E48">A17</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">nx</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="#E5:57_5"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="txt" href="#E4:57_5"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="txt" href="#E1:57_5_2_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="ref" href="matrixr2.html#T76" title="MATRIXR2:th.76">MATRIXR2:76</a></span>;<br/><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">nx</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">nx</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> = 
<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">nx</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">nx</font></span>)</span>

<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_5_2_2"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="ref" href="binop_2.html#D11" title="BINOP_2:def.11">BINOP_2:def 11</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><span class="kw">hence </span><a NAME="E5:57_5_2_2"/>
<span class="p1">(<span class="default"><a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="euclid_7.html#K1" title="EUCLID_7:func.1">&lt;:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E1:57_5"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="#E3:57_5"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="ref" href="finseq_2.html#T71" title="FINSEQ_2:th.71">FINSEQ_2:71</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E11:57"/><span class="kw">then </span>
<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="euclid_7.html#K1" title="EUCLID_7:func.1">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E9:57"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/>
<a NAME="E12:57"/><span class="kw">then </span>
<a href="binop_2.html#K35" title="BINOP_2:func.35">multreal</a>  <a href="funcop_1.html#K3" title="FUNCOP_1:func.3">.:</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="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span> <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="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D3" title="FUNCOP_1:def.3">FUNCOP_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:57"/>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">j</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T111" title="RVSUM_1:th.111">RVSUM_1:111</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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