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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">F</font>, <font color="Olive">G</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <br/> for <font color="Olive">s</font> being  <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">G</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="partfun1.html#K1">*</a> <font color="Olive">s</font> &amp; not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">F</font> holds <br/> <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Olive">G</font>;<br/>
<a NAME="E1:33"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">G</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> 1;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33_1"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E2:33_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E3:33_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E4:33_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E5:33_1"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="finseq_1.html#K2">Seg</a> 1 <b>by </b><i><a class="ref" href="finseq_2.html#T28">FINSEQ_2:28</a></i>;<br/>
<a NAME="E7:33_1"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:33_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:33_1"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:33_1"/>
<font color="Maroon">F</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> 1, <a href="numbers.html#K6">ExtREAL</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T30">FINSEQ_2:30</a></i>;<br/>
<a NAME="E11:33_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T37">FINSEQ_2:37</a></i>;<br/>
<a NAME="E12:33_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="supinf_2.html#K13">.</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E13:33_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="supinf_2.html#K13">.</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E14:33_1"/><b>then </b>
<font color="Maroon">G</font> <a href="supinf_2.html#K13">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:33_1"/><b>then </b>
<font color="Maroon">G</font> <a href="supinf_2.html#K13">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T25">FUNCT_1:25</a></i>;<br/>
<b>hence </b><a NAME="E16:33_1"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> or  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">G</font> )
 <b>by </b><i>, , <a class="txt" href="convfun1.html#E1:33"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:33"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:33_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>thus </b><a NAME="E2:33_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">G</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33_2_1"/><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E2:33_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E3:33_2_1"/><i><font color="Green">E76</font></i>: 
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E4:33_2_1"/><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:33_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E6:33_2_1"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:33_2_1"/><b>then </b><i><font color="Green">E79</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s'</font> = <font color="Maroon">s</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <b>by </b><i>, <a class="ref" href="finseq_2.html#T28">FINSEQ_2:28</a></i>;<br/>
<a NAME="E9:33_2_1"/><i><font color="Green">E102</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:33"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:33_2_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">F</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="numbers.html#K6">ExtREAL</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T30">FINSEQ_2:30</a></i>;<br/>
<a NAME="E11:33_2_1"/><b>then </b><i><font color="Green">E105</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T37">FINSEQ_2:37</a></i>;<br/>
<a NAME="E12:33_2_1"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:33_2_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:33"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E14:33_2_1"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:33"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span><b> such that </b><br/><a NAME="E17:33_2_1"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/>( ( <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> implies <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> ) &amp; ( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> implies <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 ) &amp; ( <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">q</font> implies <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> <a href="binop_2.html#K10">-</a> 1 ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E18:33_2_1"/><i><font color="Green">E109</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">p</font> <a href="funct_2.html#K2">"</a>  as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <b>by </b><i>, <a class="ref" href="finseq_2.html#T28">FINSEQ_2:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p2</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i>, <a class="ref" href="finseq_2.html#T36">FINSEQ_2:36</a></i>;<br/>
<a NAME="E21:33_2_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E22:33_2_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E23:33_2_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E24:33_2_1"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T37">FINSEQ_2:37</a></i>;<br/>
<a NAME="E25:33_2_1"/><b>then </b><i><font color="Green">E126</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">s'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <b>by </b><i>, </i>;<br/>
<a NAME="E27:33_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E28:33_2_1"/><b>then </b><i><font color="Green">E127</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E29:33_2_1"/><b>then </b><i><font color="Green">E128</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p1'</font> = <font color="Maroon">p1</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <b>by </b><i>, <a class="ref" href="finseq_2.html#T28">FINSEQ_2:28</a></i>;<br/>
<a NAME="E31:33_2_1"/><i><font color="Green">E129</font></i>: 
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E32:33_2_1"/><b>then </b><i><font color="Green">E132</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E33:33_2_1"/><i><font color="Green">E133</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E34:33_2_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">p1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:33_2_1_1"/><i><font color="Green">E135</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E2:33_2_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:33_2_1_1"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>, <a href="numbers.html#K6">ExtREAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T30">FINSEQ_2:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">H1</font> = <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">p1'</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T36">FINSEQ_2:36</a></i>;<br/>
<a NAME="E5:33_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:33_2_1_1"/><b>then </b><i><font color="Green">E137</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T37">FINSEQ_2:37</a></i>;<br/>
<a NAME="E7:33_2_1_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">p1</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_2_1_1_1"/><i><font color="Green">E143</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E2:33_2_1_1_1"/><b>then </b><i><font color="Green">E144</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>;<br/>
<a NAME="E3:33_2_1_1_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:33_2_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:33_2_1_1_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">p1</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E6:33_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">s'</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>;<br/>
<a NAME="E7:33_2_1_1_1"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">p1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E8:33_2_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E9:33_2_1_1_1"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">p1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">p1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E11:33_2_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:33_2_1_1_1"/><b>then </b><i><font color="Green">E153</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">p1</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>;<br/>
<a NAME="E13:33_2_1_1_1"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">H</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E14:33_2_1_1_1"/>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:33"><i><font color="Green">E34</font></i></a>, , , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E15:33_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">p1</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="convfun1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/>
<b>hence </b><a NAME="E16:33_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">p1</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:33_2_1_1"/>
<font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">p1</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:33_2_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E36:33_2_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E37:33_2_1"/><b>then </b>
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T25">FUNCT_1:25</a></i>;<br/>
<a NAME="E38:33_2_1"/><b>then </b><i><font color="Green">E156</font></i>: 
 <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="convfun1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E39:33_2_1"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E40:33_2_1"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/>
<a NAME="E41:33_2_1"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E42:33_2_1"/><b>then </b><i><font color="Green">E158</font></i>: 
( <font color="Maroon">H</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">G</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E43:33_2_1"/>
<font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E44:33_2_1"/><b>then </b>
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="supinf_2.html#K13">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="supinf_2.html#K13">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> )
 
<b>by </b><i>, , , <a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/>
<a NAME="E45:33_2_1"/><b>then </b><i><font color="Green">E159</font></i>: 
(  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E46:33_2_1"/><i><font color="Green">E160</font></i>: 
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T25">FUNCT_1:25</a></i>;<br/>
<font color="Maroon">H</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font> = 
<font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
<br/>.= 
<font color="Maroon">F</font>
<b>by </b><i>, <a class="ref" href="relat_1.html#T78">RELAT_1:78</a></i>
;<br/>
<b>hence </b><a NAME="E48:33_2_1"/>
 <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">G</font>
 <b>by </b><i>, , , , , , , , </i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E3:33"/><i><font color="Green">E161</font></i>: 
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(, );<br/></i>
<a NAME="E4:33"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">G</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> 0;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33_3"/><i><font color="Green">E168</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:33_3"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E3:33_3"/><i><font color="Green">E170</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T44">RELAT_1:44</a></i>;<br/>
<a NAME="E4:33_3"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:33_3"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E6:33_3"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:33_3"/><b>then </b>
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K6">ExtREAL</a>  &amp; <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K6">ExtREAL</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<b>hence </b><a NAME="E8:33_3"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> or  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">G</font> )
 ;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">G</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>;<br/>



<a NAME="E5:33"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_4_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_4_1_1"><b>suppose </b></a><a NAME="E1:33_4_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_4_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_4_1_2"><b>suppose </b></a><a NAME="E1:33_4_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_4_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:33"/><i><font color="Green">E172</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</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="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E7:33"/>
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">s</font> &amp; not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> )
 ;<br/>

<b>hence </b><a NAME="E8:33"/>
 <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/>


</div>
