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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="firebrick">::  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="margrel1.html#V2" title="MARGREL1:attr.2">homogeneous</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b3">p</font> being    <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="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Olive" title="b1">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>  st <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="Olive" title="b3">p</font> &amp; <font color="Olive" title="b1">f1</font> is  <a href="comput_1.html#V2" title="COMPUT_1:attr.2">len-total</a>  holds <br/><span class="p1">(<span class="default"><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>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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="Olive" title="b1">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="b3">p</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></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="margrel1.html#V2" title="MARGREL1:attr.2">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being    <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="margrel1.html#K19" title="MARGREL1:func.19">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>  st <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="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c2">f1</font> is  <a href="comput_1.html#V2" title="COMPUT_1:attr.2">len-total</a>  holds <br/><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="Olive" title="b1">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="Olive" title="b1">p</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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="margrel1.html#K19" title="MARGREL1:func.19">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c4">p</font> &amp; <font color="Maroon" title="c2">f1</font> is  <a href="comput_1.html#V2" title="COMPUT_1:attr.2">len-total</a>  implies <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="c4">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="c4">p</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:105"/><span class="lab"><font color="Green" title="E67">A1</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="c4">p</font>
 <span class="kw">and </span><br/><a NAME="E2:105"/><span class="lab"><font color="Green" title="E68">A2</font></span>: 
<font color="Maroon" title="c2">f1</font> is  <a href="comput_1.html#V2" title="COMPUT_1:attr.2">len-total</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c4">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="c4">p</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span><br/>

<a NAME="E3:105"/><span class="lab"><font color="Green" title="E69">A3</font></span>: 
 <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">i</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T105" title="FINSEQ_3:th.105">FINSEQ_3:105</a></span>;<br/>
<a NAME="E4:105"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">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="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E5:105"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c2">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="finseq_3.html#T109" title="FINSEQ_3:th.109">FINSEQ_3:109</a></span>;<br/>
<a NAME="E6:105"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A4</font></span>: 
 <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">i</font>) is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:105"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="ref" href="finseq_2.html#T92" title="FINSEQ_2:th.92">FINSEQ_2:92</a></span>;<br/>
<a NAME="E7:105"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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="margrel1.html#K19" title="MARGREL1:func.19">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="comput_1.html#T23" target="_self" title="COMPUT_1:th.23">Th26</a></span>;<br/>
<a NAME="E8:105"/><span class="kw">then </span>
<font color="Maroon" title="c4">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E6:105"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="txt" href="comput_1.html#E74"><span class="lab"><font color="Green" title="E65">Lm6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:105"/>
<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="c4">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="c4">p</font>,<font color="Maroon" title="c1">i</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="comput_1.html#E74"><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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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