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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="unialg_1.html#V2">quasi_total</a>  <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:117"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="comput_1.html#R1" target="_self">is_primitive-recursively_expressed_by</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:117"/><i><font color="Green">E82</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E4:117"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E5:117"/><i><font color="Green">E84</font></i>: 
( <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E6:117"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> implies  <a href="finseq_3.html#K2">Del</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; (  <a href="finseq_3.html#K2">Del</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> implies ( <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> ) ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> ) ) ) )
 <b>by </b><i><a class="ref" href="comput_1.html#D12" target="_self">Def12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V2">quasi_total</a>  <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="comput_1.html#E5:117"><i><font color="Green">E84</font></i></a>, <a class="ref" href="comput_1.html#T21" target="_self">Th21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:117_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1"><b>suppose </b></a><a NAME="E1:117_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:117_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="unialg_1.html#V2">quasi_total</a> &amp; ( not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> implies  not <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> ) )
 <b>by </b><i><a class="txt" href="comput_1.html#E1:117"><i><font color="Green">E81</font></i></a>, <a class="ref" href="comput_1.html#T71" target="_self">Th71</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_2"><b>suppose </b></a><a NAME="E1:117_1_2"/><i><font color="Green">E86</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:117_1_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:117_1_2"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E3:117_1_2"><i><font color="Green">E87</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:117_1_2"/><i><font color="Green">E88</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:117_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E7:117_1_2"/><i><font color="Green">E89</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E1:117_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="comput_1.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E8:117_1_2"/><i><font color="Green">E90</font></i>: 
 <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E4:117"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E5:117"><i><font color="Green">E84</font></i></a>, <a class="txt" href="comput_1.html#E3:117_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="comput_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E9:117_1_2"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="unialg_1.html#V2">quasi_total</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D2">UNIALG_1:def 2</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:117_1_2_1"/><i><font color="Green">E91</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E2:117_1_2_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:117_1_2_1"/><i><font color="Green">E93</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:117"><i><font color="Green">E82</font></i></a>, <a class="txt" href="comput_1.html#E2:117_1_2_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>10</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E4:117_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E5:117_1_2_1"/><b>then </b>
 <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E4:117"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E5:117"><i><font color="Green">E84</font></i></a>, <a class="txt" href="comput_1.html#E1:117_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="comput_1.html#E3:117_1_2_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="comput_1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E6:117_1_2_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="txt" href="comput_1.html#E6:117"><i><font color="Green">E85</font></i></a>, <a class="txt" href="comput_1.html#E7:117_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="comput_1.html#E1:117_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="comput_1.html#E3:117_1_2_1"><i><font color="Green">E93</font></i></a></i>;<br/>
<div><i><font color="Green">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:117_1_2_1_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>11</sub></font>]
 ;<br/><a NAME="E2:117_1_2_1_1"/><i><font color="Green">E97</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T25" target="_self">Th25</a></i>;<br/><i><font color="Green">E98</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E1:117_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="comput_1.html#E3:117_1_2_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="jordan2b.html#T12">JORDAN2B:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><a NAME="E4:117_1_2_1_1"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="comput_1.html#E1:117_1_2_1_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E7:117_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E8:117_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> is    <a href="subset_1.html#M1">Element</a> of  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E5:117"><i><font color="Green">E84</font></i></a>, <a class="txt" href="comput_1.html#E2:117_1_2_1_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="comput_1.html#E3:117_1_2_1_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/><a NAME="E9:117_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>hence </b><a NAME="E10:117_1_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="comput_1.html#E6:117"><i><font color="Green">E85</font></i></a>, <a class="txt" href="comput_1.html#E1:117_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="comput_1.html#E3:117_1_2_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="comput_1.html#E1:117_1_2_1_1"><i><font color="Green">E96</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:117_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="comput_1.html#E6:117_1_2_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="comput_1.html#E7:117_1_2_1"><i><font color="Green">E95</font></i></a>);<br/></i>
<a NAME="E9:117_1_2_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E10:117_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E4:117"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E1:117_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="comput_1.html#E3:117_1_2_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:117_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E12:117_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E9:117_1_2_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="funct_7.html#T37">FUNCT_7:37</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:117_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="unialg_1.html#V2">quasi_total</a> &amp; ( not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> implies  not <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> ) )
 <b>by </b><i><a class="txt" href="comput_1.html#E6:117"><i><font color="Green">E85</font></i></a>, <a class="txt" href="comput_1.html#E6:117_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="comput_1.html#E7:117_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="comput_1.html#E8:117_1_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
