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<span class="kw">let </span><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">x2</font>, <font color="Maroon" title="c3">y1</font>, <font color="Maroon" title="c4">y2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</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="c2">x2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">y1</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="c4">y2</font> implies  <a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</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="c2">x2</font>
 <span class="kw">and </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">y1</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="c4">y2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span></span><br/>

<a NAME="E3:16"/><span class="lab"><font color="Green" title="E16">A3</font></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"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span>
 
<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/>
<a NAME="E4:16"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</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"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</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 NAME="E5:16"/>
(  <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> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:16"/><span class="kw">then </span>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>;<br/>
<a NAME="E7:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A5</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"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span></span>)</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"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</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"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="funcop_1.html#T69" title="FUNCOP_1:th.69">FUNCOP_1:69</a></span>;<br/>
<a NAME="E8:16"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E9:16"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</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"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</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 NAME="E10:16"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E11:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A7</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="rvsum_1.html#D9" title="RVSUM_1:def.9">RVSUM_1:def 9</a></span>;<br/>
<a NAME="E12:16"/><span class="lab"><font color="Green" title="E21">A8</font></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> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span>
 
<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/>
<a NAME="E13:16"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
 <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/>
<a NAME="E14:16"/><span class="kw">then </span>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">y2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>;<br/>
<a NAME="E15:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A10</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"><font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span></span>)</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> <font color="Maroon" title="c3">y1</font></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> <font color="Maroon" title="c4">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="funcop_1.html#T69" title="FUNCOP_1:th.69">FUNCOP_1:69</a></span>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">y2</font> = 
 <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="c3">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">y1</font>
<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="E17:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A11</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="rvsum_1.html#D9" title="RVSUM_1:def.9">RVSUM_1:def 9</a></span>;<br/>
<a NAME="E18:16"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</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> <font color="Maroon" title="c3">y1</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="E19:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span> <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="c3">y1</font>
 
<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="E20:16"/><span class="lab"><font color="Green" title="E26">A13</font></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> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span>
 
<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/>
<a NAME="E21:16"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c1">x1</font></span>)</span>,<span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">x2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:16"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>;<br/>
<a NAME="E22:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A14</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"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="euclid_7.html#K1" title="EUCLID_7:func.1">:&gt;</a></span></span>)</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> <font color="Maroon" title="c1">x1</font></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> <font color="Maroon" title="c2">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="ref" href="funcop_1.html#T69" title="FUNCOP_1:th.69">FUNCOP_1:69</a></span>;<br/>
<a NAME="E23:16"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">y1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">x2</font> = 
 <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="c1">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">x1</font>
<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="E25:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A16</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E22:16"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="rvsum_1.html#D9" title="RVSUM_1:def.9">RVSUM_1:def 9</a></span>;<br/>
<a NAME="E26:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A17</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</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> <font color="Maroon" title="c1">x1</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="E27:16"/><span class="lab"><font color="Green" title="E31">A18</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_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"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">i</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>)</span>)</span> implies <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16_3"/><span class="lab"><font color="Green" title="E32">A19</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">and </span><br/><a NAME="E2:16_3"/><span class="lab"><font color="Green" title="E33">A20</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/>

<a NAME="E3:16_3"/>
<font color="Maroon" title="c5">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> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="txt" href="#E2:16_3"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:16_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A21</font></span>: 
<font color="Maroon" title="c5">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="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</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 NAME="E5:16_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E6:16_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A22</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/>
<a NAME="E7:16_3"/>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E23:16"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E4:16_3"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E8:16_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A23</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/>
<a NAME="E9:16_3"/><span class="lab"><font color="Green" title="E37">A24</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">y1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:16"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E4:16_3"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:16_3_1_1"/>
( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font> or <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_3_1_1_1"/><span class="lab"><font color="Green" title="E38">A25</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/><div class="add"><a NAME="E2:16_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A26</font></span>: 
<font color="Maroon" title="c5">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="c1">x1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E3:16_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A27</font></span>: 
<font color="Maroon" title="c5">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="c1">x1</font>
 <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="E4:16_3_1_1_1"/>
<font color="Maroon" title="c5">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="c2">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16_3_1_1_1"><span class="lab"><font color="Green" title="E39">A26</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="E5:16_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A28</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E6:16_3_1_1_1"/><span class="lab"><font color="Green" title="E42">A29</font></span>: 
<font color="Maroon" title="c5">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="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:16"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E2:16_3_1_1_1"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A30</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_3_1_1_1"><span class="lab"><font color="Green" title="E42">A29</font></span></a>, <a class="ref" href="mfold_2.html#T13" target="_self" title="MFOLD_2:th.13">Th13</a></span>
;<br/><a NAME="E8:16_3_1_1_1"/>
( <font color="Maroon" title="c1">x1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E1:16_3"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="txt" href="#E1:16_3_1_1_1"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><span class="kw">hence </span><a NAME="E9:16_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_3"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="txt" href="#E8:16_3"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="txt" href="#E3:16_3_1_1_1"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="txt" href="#E5:16_3_1_1_1"><span class="lab"><font color="Green" title="E41">A28</font></span></a>, <a class="txt" href="#E7:16_3_1_1_1"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_3_1_1_2"/><span class="lab"><font color="Green" title="E38">A31</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/><div class="add"><a NAME="E2:16_3_1_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E23:16"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E4:16_3"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E3:16_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A32</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><a NAME="E4:16_3_1_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E4:16_3"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E5:16_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span>;<br/><a NAME="E6:16_3_1_1_2"/><span class="lab"><font color="Green" title="E41">A34</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_1_1_2"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E7:16_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A35</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span></span>)</span>
 ;<br/><a NAME="E8:16_3_1_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">y1</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_3"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></span>;<br/><a NAME="E9:16_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A36</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_1_1_2"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E10:16_3_1_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_1_1_2"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="txt" href="#E6:16_3_1_1_2"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><a NAME="E11:16_3_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E12:16_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A37</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E13:16_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A38</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span> <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="c3">y1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_3_1_1_2"><span class="lab"><font color="Green" title="E43">A36</font></span></a></span>;<br/><a NAME="E14:16_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A39</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></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="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:16"><span class="lab"><font color="Green" title="E24">A11</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="E15:16_3_1_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E7:16_3_1_1_2"><span class="lab"><font color="Green" title="E42">A35</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A40</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_3_1_1_2"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">y1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_3_1_1_2"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="ref" href="mfold_2.html#T13" target="_self" title="MFOLD_2:th.13">Th13</a></span>
;<br/><a NAME="E17:16_3_1_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></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> <font color="Maroon" title="c3">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:16_3_1_1_2"><span class="lab"><font color="Green" title="E45">A38</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="E18:16_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A41</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_3_1_1_2"><span class="lab"><font color="Green" title="E42">A35</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E19:16_3_1_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span> <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="c3">y1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16_3_1_1_2"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="txt" href="#E9:16_3_1_1_2"><span class="lab"><font color="Green" title="E43">A36</font></span></a></span>;<br/><a NAME="E20:16_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A42</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></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> <font color="Maroon" title="c4">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</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="E21:16_3_1_1_2"/>
( <font color="Maroon" title="c3">y1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">y2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E12:16_3_1_1_2"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="txt" href="#E9:16_3_1_1_2"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><span class="kw">hence </span><a NAME="E22:16_3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E7:16_3_1_1_2"><span class="lab"><font color="Green" title="E42">A35</font></span></a>, <a class="txt" href="#E20:16_3_1_1_2"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E18:16_3_1_1_2"><span class="lab"><font color="Green" title="E48">A41</font></span></a>, <a class="txt" href="#E5:16_3_1_1_2"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="txt" href="#E3:16_3_1_1_2"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="txt" href="#E16:16_3_1_1_2"><span class="lab"><font color="Green" title="E47">A40</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:16_3"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_3"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="mfold_2.html#T13" target="_self" title="MFOLD_2:th.13">Th13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>)</span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
;<br/>
<a NAME="E29:16"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E19:16"><span class="lab"><font color="Green" title="E25">A12</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="E30:16"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>)</span>)</span> <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"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E31:16"/>
 <a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">x2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">y2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>)</span>)</span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> (<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:16"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
