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<b>let </b><font color="Maroon" title="c1">g</font> be    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>  <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:115"/><i><font color="Green" title="E78">A1</font></i>: 
<font color="Maroon" title="c1">g</font> <a href="comput_1.html#R1" title="COMPUT_1:pred.1">is_primitive-recursively_expressed_by</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c4">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:115"/><i><font color="Green" title="E79">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E4:115"/><i><font color="Green" title="E80">A3</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> )
 <b>and </b><br/><a NAME="E5:115"/><i><font color="Green" title="E81">A4</font></i>: 
( <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f2</font> )
 <b>and </b><br/><a NAME="E6:115"/><i><font color="Green" title="E82">A5</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> holds <br/>( ( <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">g</font> implies  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font> ) &amp; (  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font> implies <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">g</font> ) &amp; ( <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">g</font> implies <font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">i</font></span>)</span> ) &amp; ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( ( <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">g</font> implies ( <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">g</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Olive" title="b2">n</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Olive" title="b2">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f2</font> ) ) &amp; ( <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">g</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Olive" title="b2">n</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Olive" title="b2">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f2</font> implies <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">g</font> ) &amp; ( <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">g</font> implies <font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<span class="p2">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Olive" title="b2">n</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Olive" title="b2">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> ) ) ) )
 <b>by </b><i><a class="ref" href="comput_1.html#D12" target="_self" title="COMPUT_1:def.12">Def12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">f2'</font> = <font color="Maroon" title="c3">f2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>  <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="comput_1.html#E5:115"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="comput_1.html#T21" target="_self" title="COMPUT_1:th.21">Th21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:115_1"/>
( <font color="Maroon" title="c2">f1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c2">f1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1"><b>suppose </b></a><a NAME="E1:115_1_1"/>
<font color="Maroon" title="c2">f1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:115_1_1"/>
( <font color="Maroon" title="c1">g</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a> &amp; ( not <font color="Maroon" title="c2">f1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c1">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )
 <b>by </b><i><a class="txt" href="comput_1.html#E1:115"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="comput_1.html#T71" target="_self" title="COMPUT_1:th.71">Th71</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_2"><b>suppose </b></a><a NAME="E1:115_1_2"/><i><font color="Green" title="E83">A6</font></i>: 
not <font color="Maroon" title="c2">f1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c7">pp</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:115_1_2"/><i><font color="Green" title="E84">A7</font></i>: 
<font color="Maroon" title="c7">pp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:115_1_2"/>
<font color="Maroon" title="c7">pp</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c5">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E3:115_1_2"><i><font color="Green" title="E84">A7</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c7">pp</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E6:115_1_2"/><i><font color="Green" title="E85">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:115_1_2"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><a NAME="E7:115_1_2"/><i><font color="Green" title="E86">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E1:115_1_2"><i><font color="Green" title="E83">A6</font></i></a>, <a class="ref" href="comput_1.html#T25" target="_self" title="COMPUT_1:th.25">Th25</a></i>;<br/><a NAME="E8:115_1_2"/><i><font color="Green" title="E87">A10</font></i>: 
 <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E4:115"><i><font color="Green" title="E80">A3</font></i></a>, <a class="txt" href="comput_1.html#E5:115"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="comput_1.html#E3:115_1_2"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="comput_1.html#T12" target="_self" title="COMPUT_1:th.12">Th12</a></i>;<br/><a NAME="E9:115_1_2"/>
<font color="Maroon" title="c1">g</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D2" title="UNIALG_1:def.2">UNIALG_1:def 2</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:115_1_2_1"/><i><font color="Green" title="E88">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">y</font>
 <b>and </b><br/><a NAME="E2:115_1_2_1"/><i><font color="Green" title="E89">A12</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">g</font>
 ;<br/>

<a NAME="E3:115_1_2_1"/><i><font color="Green" title="E90">A13</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:115"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="comput_1.html#E2:115_1_2_1"><i><font color="Green" title="E89">A12</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon" title="c10">y</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">g</font>;<br/>
<a NAME="E4:115_1_2_1"/>
<font color="Maroon" title="c10">y</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E5:115_1_2_1"/><b>then </b>
 <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E4:115"><i><font color="Green" title="E80">A3</font></i></a>, <a class="txt" href="comput_1.html#E5:115"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="comput_1.html#E1:115_1_2_1"><i><font color="Green" title="E88">A11</font></i></a>, <a class="txt" href="comput_1.html#E3:115_1_2_1"><i><font color="Green" title="E90">A13</font></i></a>, <a class="ref" href="comput_1.html#T12" target="_self" title="COMPUT_1:th.12">Th12</a></i>;<br/>
<a NAME="E6:115_1_2_1"/><b>then </b><i><font color="Green" title="E91">A14</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="comput_1.html#E6:115"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="comput_1.html#E7:115_1_2"><i><font color="Green" title="E86">A9</font></i></a>, <a class="txt" href="comput_1.html#E1:115_1_2_1"><i><font color="Green" title="E88">A11</font></i></a>, <a class="txt" href="comput_1.html#E3:115_1_2_1"><i><font color="Green" title="E90">A13</font></i></a></i>;<br/>
<div><i><font color="Green" title="E92">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:115_1_2_1_1"/><i><font color="Green" title="E93">A16</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">k</font>]
 ;<br/><a NAME="E2:115_1_2_1_1"/><i><font color="Green" title="E94">A17</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T25" target="_self" title="COMPUT_1:th.25">Th25</a></i>;<br/><i><font color="Green" title="E95">A18</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E1:115_1_2_1"><i><font color="Green" title="E88">A11</font></i></a>, <a class="txt" href="comput_1.html#E3:115_1_2_1"><i><font color="Green" title="E90">A13</font></i></a>, <a class="ref" href="jordan2b.html#T12" title="JORDAN2B:th.12">JORDAN2B:12</a></i>
<br/>.= 
<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
;<br/><a NAME="E4:115_1_2_1_1"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font> )
 <b>by </b><i><a class="txt" href="comput_1.html#E1:115_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">gyk</font> = <font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c11">k</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c13">gyik</font> = <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c12">gyk</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E7:115_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c13">gyik</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><a NAME="E8:115_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f2</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E5:115"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="comput_1.html#E2:115_1_2_1_1"><i><font color="Green" title="E94">A17</font></i></a>, <a class="txt" href="comput_1.html#E3:115_1_2_1_1"><i><font color="Green" title="E95">A18</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/><a NAME="E9:115_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f2'</font>
 ;<br/><b>hence </b><a NAME="E10:115_1_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="comput_1.html#E6:115"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="comput_1.html#E1:115_1_2_1"><i><font color="Green" title="E88">A11</font></i></a>, <a class="txt" href="comput_1.html#E3:115_1_2_1"><i><font color="Green" title="E90">A13</font></i></a>, <a class="txt" href="comput_1.html#E1:115_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:115_1_2_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="comput_1.html#E6:115_1_2_1"><i><font color="Green" title="E91">A14</font></i></a>, <a class="txt" href="comput_1.html#E7:115_1_2_1"><i><font color="Green" title="E92">A15</font></i></a>);<br/></i>
<a NAME="E9:115_1_2_1"/><b>then </b><i><font color="Green" title="E93">A19</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">g</font>
 
;<br/>
<a NAME="E10:115_1_2_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">y</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E4:115"><i><font color="Green" title="E80">A3</font></i></a>, <a class="txt" href="comput_1.html#E1:115_1_2_1"><i><font color="Green" title="E88">A11</font></i></a>, <a class="txt" href="comput_1.html#E3:115_1_2_1"><i><font color="Green" title="E90">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:115_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E12:115_1_2_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">g</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E9:115_1_2_1"><i><font color="Green" title="E93">A19</font></i></a>, <a class="ref" href="funct_7.html#T37" title="FUNCT_7:th.37">FUNCT_7:37</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:115_1_2"/>
( <font color="Maroon" title="c1">g</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a> &amp; ( not <font color="Maroon" title="c2">f1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  not <font color="Maroon" title="c1">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )
 <b>by </b><i><a class="txt" href="comput_1.html#E6:115"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="comput_1.html#E6:115_1_2"><i><font color="Green" title="E85">A8</font></i></a>, <a class="txt" href="comput_1.html#E7:115_1_2"><i><font color="Green" title="E86">A9</font></i></a>, <a class="txt" href="comput_1.html#E8:115_1_2"><i><font color="Green" title="E87">A10</font></i></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
