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<b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:41"/><i><font color="Green">E30</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3
 ;<br/>

<b>set </b><font color="Maroon">g</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">h</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font>;<br/>
<a NAME="E2:41"/><i><font color="Green">E31</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E3:41"/>
(  <a href="gr_cy_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="turing_1.html#K4" target="_self">Prefix</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="gr_cy_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="turing_1.html#K4" target="_self">Prefix</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span>,2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> )
 <b>by </b><i/>;<br/>

<a NAME="E4:41"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/>
<a NAME="E5:41"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<i><font color="Green">E35</font></i>: <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="turing_1.html#E2:41"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_7.html#T3">FINSEQ_7:3</a></i>
;<br/>
<a NAME="E7:41"/><i><font color="Green">E75</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>then </b><i><font color="Green">E114</font></i>: <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2 = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E2:41"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="finseq_7.html#T3">FINSEQ_7:3</a></i>
;<br/>
<i><font color="Green">E116</font></i>: <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 3 = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 3</span>)</span>
<b>by </b><i>, <a class="txt" href="turing_1.html#E2:41"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_7.html#T3">FINSEQ_7:3</a></i>
;<br/>
<b>consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:41"/><i><font color="Green">E117</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 3 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<i><font color="Green">E118</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> = 
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="txt" href="turing_1.html#E2:41"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
4 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/>
<a NAME="E14:41"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon">p3</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q3</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E16:41"/><i><font color="Green">E121</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> 3 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q3</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T26">FINSEQ_2:26</a></i>;<br/>
<a NAME="E17:41"/>
 <a href="finseq_1.html#K2">Seg</a> 3 <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p3</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:41"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a>  <a href="turing_1.html#K4" target="_self">Prefix</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span>,3
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T33">FINSEQ_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon">p4</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q4</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E20:41"/><i><font color="Green">E124</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p4</font> <a href="hidden.html#R1">=</a> 4 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q4</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T26">FINSEQ_2:26</a></i>;<br/>
<a NAME="E21:41"/>
 <a href="finseq_1.html#K2">Seg</a> 4 <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p4</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E22:41"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">p4</font> <a href="hidden.html#R1">=</a>  <a href="turing_1.html#K4" target="_self">Prefix</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span>,4
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T33">FINSEQ_1:33</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x3</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E26:41"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:41"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/>
<a NAME="E27:41"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="txt" href="turing_1.html#E4:41"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/>
<a NAME="E28:41"/>
<font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2
 
<b>by </b><i><a class="txt" href="turing_1.html#E5:41"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/>
<a NAME="E29:41"/><b>then </b>
<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="turing_1.html#T2" target="_self">Th2</a>, , <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<b>hence </b> <a href="gr_cy_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="turing_1.html#K4" target="_self">Prefix</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span>,3</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x3</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>

;<br/><br/>

<b>reconsider </b><font color="Maroon">x4</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 3 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E32:41"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">p4</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="txt" href="turing_1.html#E1:41"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/>
<a NAME="E33:41"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">p4</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="txt" href="turing_1.html#E4:41"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/>
<a NAME="E34:41"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">p4</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2
 
<b>by </b><i><a class="txt" href="turing_1.html#E5:41"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/>
<a NAME="E35:41"/>
<font color="Maroon">p4</font> <a href="funct_1.html#K1">.</a> 4 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 3
 
<b>by </b><i><a class="txt" href="turing_1.html#E6:41"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/>
<a NAME="E36:41"/><b>then </b>
<font color="Maroon">p4</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K10">&lt;*</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span><a href="finseq_4.html#K10">*&gt;</a></span>
 
<b>by </b><i>, , , , <a class="ref" href="finseq_4.html#T91">FINSEQ_4:91</a></i>;<br/>
<b>hence </b> <a href="gr_cy_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="turing_1.html#K4" target="_self">Prefix</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">s</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span>,4</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x3</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x4</font>
<b>by </b><i><a class="ref" href="turing_1.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span>

;<br/><br/>


</div>
