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<span class="kw">let </span><font color="Maroon" title="c1">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   <a href="comput_1.html#NV2" title="COMPUT_1:NV.2">to-naturals</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/>for <font color="Olive" title="b3">g</font> being   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">g</font> <a href="comput_1.html#R1" title="COMPUT_1:pred.1">is_primitive-recursively_expressed_by</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">i</font> holds <br/><font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">i</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   <a href="comput_1.html#NV2" title="COMPUT_1:NV.2">to-naturals</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies for <font color="Olive" title="b1">g</font> being   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">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="c1">i</font> holds <br/><font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:112"/><span class="lab"><font color="Green" title="E69">A1</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:112"/><span class="lab"><font color="Green" title="E70">A2</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font> being   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">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="c1">i</font> holds <br/><font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">g</font> be    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">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="c1">i</font> implies <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">n</font> = <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;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">h</font> =  <a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font>;<br/>
<span class="kw">given </span><font color="Maroon" title="c7">n9</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> <span class="kw"> such that </span><a NAME="E4:112"/><span class="lab"><font color="Green" title="E71">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">n9</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> 
 <span class="kw">and </span><br/><a NAME="E5:112"/>
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E6:112"/>
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n9</font>
 <span class="kw">and </span><br/><a NAME="E7:112"/><span class="lab"><font color="Green" title="E72">A4</font></span>: 
<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="c7">n9</font>
 <span class="kw">and </span><br/><a NAME="E8:112"/>
<font color="Maroon" title="c7">n9</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>
 <span class="kw">and </span><br/><a NAME="E9:112"/><span class="lab"><font color="Green" title="E73">A5</font></span>: 
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="c7">n9</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="c1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">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="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">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="c1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> implies <font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_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="c1">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="recdef_1.html#K1" title="RECDEF_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="c1">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="c1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">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="c1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">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="c1">i</font>,<font color="Olive" title="b2">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_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="c1">i</font>,<font color="Olive" title="b2">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">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="c1">i</font>,<font color="Olive" title="b2">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_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="c1">i</font>,<font color="Olive" title="b2">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">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="c1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> implies <font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_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="c1">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="recdef_1.html#K1" title="RECDEF_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="c1">i</font>,<font color="Olive" title="b2">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_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="c1">i</font>,<font color="Olive" title="b2">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> ) ) ) )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="comput_1.html#D12" target="_self" title="COMPUT_1:def.12">COMPUT_1:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span><br/>

<div><span class="lab"><font color="Green" title="E74">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">p</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<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</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> implies <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> ) &amp; ( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> implies <font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">p</font> ) )</span><br/><span class="kw">defpred </span><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> ] <span class="kw">means </span>( ( <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> implies <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> ) &amp; ( <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> implies <font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,$1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,$1</span>)</span> ) );<br/><span class="kw">set </span><font color="Maroon" title="c9">k</font> = <font color="Maroon" title="c8">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font>;<br/><a NAME="E1:112_1"/><span class="lab"><font color="Green" title="E75">A7</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T40" title="FUNCT_7:th.40">FUNCT_7:40</a></span>;<br/><a NAME="E2:112_1"/><span class="lab"><font color="Green" title="E76">A8</font></span>: 
 <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> <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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/><a NAME="E3:112_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A9</font></span>: 
<font color="Maroon" title="c1">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="c8">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:112_1"/><span class="lab"><font color="Green" title="E78">A10</font></span>: 
for <font color="Olive" title="b1">m</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>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">m</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">m</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:112_1_1"/><span class="lab"><font color="Green" title="E79">A11</font></span>: 
( <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> iff <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E2:112_1_1"/><span class="lab"><font color="Green" title="E80">A12</font></span>: 
( <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> implies <font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c10">m</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E3:112_1_1"/>
( <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> iff ( <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c10">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:112"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E9:112"><span class="lab"><font color="Green" title="E73">A5</font></span></a>, <a class="txt" href="#E2:112_1"><span class="lab"><font color="Green" title="E76">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:112_1_1"/>
( <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> iff <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:112_1"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="txt" href="#E1:112_1_1"><span class="lab"><font color="Green" title="E79">A11</font></span></a>, <a class="txt" href="#E2:112_1_1"><span class="lab"><font color="Green" title="E80">A12</font></span></a>, <a class="txt" href="comput_1.html#E81"><span class="lab"><font color="Green" title="E65">Lm6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> implies <font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">m</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> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )</span><br/>

<a NAME="E5:112_1_1"/><span class="lab"><font color="Green" title="E81">A13</font></span>: 
( <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> iff ( <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c10">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f2</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:112_1"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="txt" href="comput_1.html#E81"><span class="lab"><font color="Green" title="E65">Lm6</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:112_1_1"/><span class="lab"><font color="Green" title="E82">A14</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">m</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> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/>

<a NAME="E7:112_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">m</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="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c10">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:112"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E9:112"><span class="lab"><font color="Green" title="E73">A5</font></span></a>, <a class="txt" href="#E2:112_1"><span class="lab"><font color="Green" title="E76">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:112_1_1"/>
<font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">m</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> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:112"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E9:112"><span class="lab"><font color="Green" title="E73">A5</font></span></a>, <a class="txt" href="#E2:112_1"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="txt" href="#E3:112_1"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="txt" href="#E1:112_1_1"><span class="lab"><font color="Green" title="E79">A11</font></span></a>, <a class="txt" href="#E2:112_1_1"><span class="lab"><font color="Green" title="E80">A12</font></span></a>, <a class="txt" href="#E5:112_1_1"><span class="lab"><font color="Green" title="E81">A13</font></span></a>, <a class="txt" href="#E6:112_1_1"><span class="lab"><font color="Green" title="E82">A14</font></span></a>, <a class="txt" href="comput_1.html#E81"><span class="lab"><font color="Green" title="E65">Lm6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:112_1"/><span class="lab"><font color="Green" title="E79">A15</font></span>: 
( <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> iff  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:112_1"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="txt" href="comput_1.html#E81"><span class="lab"><font color="Green" title="E65">Lm6</font></span></a></span>;<br/><a NAME="E6:112_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> implies ( <font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">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="recdef_1.html#K1" title="RECDEF_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="Maroon" title="c8">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">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="recdef_1.html#K1" title="RECDEF_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="Maroon" title="c8">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:112"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E9:112"><span class="lab"><font color="Green" title="E73">A5</font></span></a>, <a class="txt" href="#E2:112_1"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="txt" href="#E3:112_1"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="txt" href="comput_1.html#E81"><span class="lab"><font color="Green" title="E65">Lm6</font></span></a></span>;<br/><a NAME="E7:112_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A16</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:112"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E9:112"><span class="lab"><font color="Green" title="E73">A5</font></span></a>, <a class="txt" href="#E2:112_1"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="txt" href="#E5:112_1"><span class="lab"><font color="Green" title="E79">A15</font></span></a></span>;<br/><a NAME="E8:112_1"/>
for <font color="Olive" title="b1">m</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">m</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E7:112_1"><span class="lab"><font color="Green" title="E80">A16</font></span></a>, <a class="txt" href="#E4:112_1"><span class="lab"><font color="Green" title="E78">A10</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E9:112_1"/>
( ( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> implies <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> ) &amp; ( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> implies <font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">p</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1"><span class="lab"><font color="Green" title="E75">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E11:112"/><span class="lab"><font color="Green" title="E75">A17</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<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</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#T60" target="_self" title="COMPUT_1:th.60">Th60</a></span>;<br/>
<a NAME="E12:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A18</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E7:112"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E10:112"><span class="lab"><font color="Green" title="E74">A6</font></span></a>, <a class="ref" href="subset_1.html#T8" title="SUBSET_1:th.8">SUBSET_1:8</a></span>;<br/>
<a NAME="E13:112"/><span class="kw">then </span>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> holds <br/><font color="Maroon" title="c4">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:112"><span class="lab"><font color="Green" title="E75">A17</font></span></a>, <a class="txt" href="#E10:112"><span class="lab"><font color="Green" title="E74">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:112"/>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:112"><span class="lab"><font color="Green" title="E76">A18</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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