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

<span class="kw">let </span><font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c2">f2</font>, <font color="Maroon" title="c3">f3</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">t</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <span class="p1">(<span class="default"><a href="euclid_8.html#K6" title="EUCLID_8:func.6">VFunc</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">f3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a> </span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> </span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> </span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">t</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_8.html#K6" title="EUCLID_8:func.6">VFunc</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">f3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a> </span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> </span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> </span>)</span></span><br/>







<a NAME="E1:68"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
( <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3 <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="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E2:68"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
( <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3 <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="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E3:68"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
( <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<span class="lab"><font color="Green" title="E29">A5</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a> </span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> </span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> </span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="euclid_8.html#E2"><span class="lab"><font color="Green" title="E2">Lm3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="euclid_8.html#E2"><span class="lab"><font color="Green" title="E2">Lm3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> 1</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="euclid_8.html#E2"><span class="lab"><font color="Green" title="E2">Lm3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>

;<br/>
<a NAME="E5:68"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
( <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font> &amp; <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3 <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="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E6:68"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
( <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font> &amp; <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3 <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="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E7:68"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
( <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<span class="lab"><font color="Green" title="E33">A9</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a> </span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> </span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> </span>)</span> = 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E5:68"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E6:68"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="euclid_8.html#E3"><span class="lab"><font color="Green" title="E3">Lm4</font></span></a></span>
<br/>.= 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>

;<br/>
<a NAME="E9:68"/><span class="lab"><font color="Green" title="E34">A10</font></span>: 
( <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font> &amp; <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font> &amp; <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3 <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="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a> </span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> </span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> </span>)</span> = 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E8:68"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E9:68"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="euclid_8.html#E3"><span class="lab"><font color="Green" title="E3">Lm4</font></span></a></span>
<br/>.= 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>

;<br/>
<span class="kw">hence </span><a NAME="E11:68"/>
<span class="p1">(<span class="default"><a href="euclid_8.html#K6" title="EUCLID_8:func.6">VFunc</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">f3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a> </span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> </span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_8.html#D5" target="_self" title="EUCLID_8:def.5">Def1</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>
