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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">F2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b3">h</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">F</font>  st <font color="Olive" title="b2">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Olive" title="b3">h</font> holds <br/> <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Olive" title="b2">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Olive" title="b1">F</font></span><br/><b>let </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">F2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">h</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>  st <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Olive" title="b1">h</font> holds <br/> <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c2">F</font></span><br/><b>let </b><font color="Maroon" title="c4">h</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c4">h</font> implies  <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c2">F</font> )</span><br/>





<b>assume </b><a NAME="E1:49"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c4">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c2">F</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>suppose </b></a><a NAME="E1:49_1_1"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c2">F</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">gF</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:49_1_1"/><i><font color="Green" title="E31">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</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="c5">gF</font> &amp; <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">gF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c5">gF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">gF</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) &amp;  <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">gF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span> )
 <b>by </b><i><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></i>;<br/><a NAME="E4:49_1_1"/><i><font color="Green" title="E32">A4</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E1:49_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:49_1_1"/><b>then </b><i><font color="Green" title="E33">A5</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:49_1_1"/><b>then </b><i><font color="Green" title="E34">A6</font></i>: 
1 <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="c2">F</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:49_1_1"/><i><font color="Green" title="E35">A7</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E5:49_1_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:49_1_1"/><i><font color="Green" title="E36">A8</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E6:49_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E9:49_1_1"/><i><font color="Green" title="E37">A9</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:49_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h</font>
 <b>by </b><i><a class="txt" href="#E8:49_1_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><a NAME="E11:49_1_1"/><b>then </b><i><font color="Green" title="E38">A10</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">F</font> )
 <b>by </b><i><a class="txt" href="#E8:49_1_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="#E1:49"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a>, <a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></i>;<br/><a NAME="E12:49_1_1"/><b>then </b><i><font color="Green" title="E39">A11</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E8:49_1_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E13:49_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:49_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="#E8:49_1_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="#E11:49_1_1"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:49_1_1"/><b>then </b><i><font color="Green" title="E40">A12</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">h2</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>  st $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st $1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">i</font> ) holds <br/><font color="Maroon" title="c5">gF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Olive" title="b1">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><a NAME="E15:49_1_1"/>
for <font color="Olive" title="b1">h2</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>  st ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">i</font> ) holds <br/><font color="Maroon" title="c5">gF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Olive" title="b1">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">h2</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> ) implies <font color="Maroon" title="c5">gF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c6">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:49_1_1_1"/><i><font color="Green" title="E41">A13</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">gF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c6">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E2:49_1_1_1"/><i><font color="Green" title="E42">A14</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E6:49_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:49_1_1_1"/><i><font color="Green" title="E43">A15</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<div><i><font color="Green" title="E44">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_1_1"/><i><font color="Green" title="E45">A17</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:49_1_1_1_1"/><i><font color="Green" title="E46">A18</font></i>: 
( <font color="Maroon" title="c7">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="c6">h2</font> &amp; <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="#E6:49_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="#E3:49_1_1_1"><i><font color="Green" title="E43">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">nx</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E3:49_1_1_1_1"><i><font color="Green" title="E46">A18</font></i></a>, <a class="txt" href="#E2:49_1_1_1"><i><font color="Green" title="E42">A14</font></i></a></i>;<br/><a NAME="E5:49_1_1_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">nx</font> &amp; <font color="Maroon" title="c8">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> )
 <b>by </b><i><a class="txt" href="#E3:49_1_1_1_1"><i><font color="Green" title="E46">A18</font></i></a>, <a class="txt" href="#E9:49_1_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:49_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c8">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">nx</font> &amp; <font color="Maroon" title="c8">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> ) )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E7:49_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:49_1_1_1"><i><font color="Green" title="E41">A13</font></i></a>, <a class="txt" href="#E1:49_1_1_1_1"><i><font color="Green" title="E45">A17</font></i></a>, <a class="txt" href="#E3:49_1_1_1_1"><i><font color="Green" title="E46">A18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c6">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="#E6:49_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="#E2:49_1_1_1"><i><font color="Green" title="E42">A14</font></i></a>, <a class="txt" href="#E4:49_1_1_1"><i><font color="Green" title="E44">A16</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E6:49_1_1_1"/>
<font color="Maroon" title="c5">gF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c6">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E3:49_1_1"><i><font color="Green" title="E31">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E16:49_1_1"/><b>then </b><i><font color="Green" title="E41">A19</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/><a NAME="E17:49_1_1"/><i><font color="Green" title="E42">A20</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:49_1_1_2"/><i><font color="Green" title="E43">A21</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:49_1_1_2"/>
for <font color="Olive" title="b1">h2</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>  st <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">i</font> ) holds <br/><font color="Maroon" title="c5">gF</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Olive" title="b1">h2</font></span>)</span></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">h2</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c7">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> ) implies <font color="Maroon" title="c5">gF</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:49_1_1_2_1"/><i><font color="Green" title="E44">A22</font></i>: 
( <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c7">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">gF</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E2:49_1_1_2_1"/><i><font color="Green" title="E45">A23</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E6:49_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:49_1_1_2_1"/><b>then </b><i><font color="Green" title="E46">A24</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">h2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E4:49_1_1_2_1"/><i><font color="Green" title="E47">A25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E1:49_1_1_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:49_1_1_2_1"/><i><font color="Green" title="E48">A26</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E6:49_1_1_2_1"/><i><font color="Green" title="E49">A27</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E1:49_1_1_2_1"><i><font color="Green" title="E44">A22</font></i></a></i>;<br/>
<a NAME="E7:49_1_1_2_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E8:49_1_1_2_1"/><b>then </b><i><font color="Green" title="E50">A28</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E1:49_1_1_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:49_1_1_2_1"/><i><font color="Green" title="E51">A29</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E10:49_1_1_2_1"/><i><font color="Green" title="E52">A30</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E11:49_1_1_2_1"/><b>then </b><i><font color="Green" title="E53">A31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:49_1_1_2_1"><i><font color="Green" title="E47">A25</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:49_1_1_2_1"/><i><font color="Green" title="E54">A32</font></i>: 
( <font color="Maroon" title="c8">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="c7">h2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="txt" href="#E9:49_1_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1"><i><font color="Green" title="E45">A23</font></i></a>, <a class="txt" href="#E3:49_1_1_2_1"><i><font color="Green" title="E46">A24</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E14:49_1_1_2_1"/><i><font color="Green" title="E55">A33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E11:49_1_1_2_1"><i><font color="Green" title="E53">A31</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">nx</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E13:49_1_1_2_1"><i><font color="Green" title="E54">A32</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1"><i><font color="Green" title="E45">A23</font></i></a></i>;<br/>
<a NAME="E16:49_1_1_2_1"/><i><font color="Green" title="E56">A34</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">nx</font> &amp; <font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> )
 
<b>by </b><i><a class="txt" href="#E13:49_1_1_2_1"><i><font color="Green" title="E54">A32</font></i></a>, <a class="txt" href="#E9:49_1_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E17:49_1_1_2_1"/><i><font color="Green" title="E57">A35</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:49_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="#E9:49_1_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:49_1_1_2_1"/>
<font color="Maroon" title="c7">h2</font> is   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>
 
<b>by </b><i><a class="txt" href="#E9:49_1_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1"><i><font color="Green" title="E45">A23</font></i></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">h2r</font> = <font color="Maroon" title="c7">h2</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>  <b>by </b><i><a class="txt" href="#E2:49_1_1_2_1"><i><font color="Green" title="E45">A23</font></i></a>, <a class="txt" href="#E3:49_1_1_2_1"><i><font color="Green" title="E46">A24</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E20:49_1_1_2_1"/><i><font color="Green" title="E58">A36</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">h2r</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="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E17:49_1_1_2_1"><i><font color="Green" title="E57">A35</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">h2b</font> =  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c10">h2r</font>,<font color="Maroon" title="c9">nx</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</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="E22:49_1_1_2_1"/><i><font color="Green" title="E59">A37</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">h2b</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">h2r</font>
 
<b>by </b><i><a class="ref" href="finseq_7.html#T20" title="FINSEQ_7:th.20">FINSEQ_7:20</a></i>;<br/>
<a NAME="E23:49_1_1_2_1"/><b>then </b>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">h2b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">h2r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E60">A38</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">h2b</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">h2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
<b>by </b><i><a class="txt" href="#E7:49_1_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<i><font color="Green" title="E61">A39</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">h2b</font> = 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">h2</font>
<b>by </b><i><a class="ref" href="finseq_7.html#T24" title="FINSEQ_7:th.24">FINSEQ_7:24</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">h2d</font> = <font color="Maroon" title="c11">h2b</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>, <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font> <b>by </b><i><a class="txt" href="#E24:49_1_1_2_1"><i><font color="Green" title="E60">A38</font></i></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/>
<a NAME="E27:49_1_1_2_1"/><i><font color="Green" title="E62">A40</font></i>: 
<font color="Maroon" title="c12">h2d</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<b>by </b><i><a class="txt" href="#E25:49_1_1_2_1"><i><font color="Green" title="E61">A39</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E28:49_1_1_2_1"/>
<font color="Maroon" title="c12">h2d</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="int_5.html#T39" title="INT_5:th.39">INT_5:39</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">h4</font> = <font color="Maroon" title="c12">h2d</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font> <b>by </b><i><a class="txt" href="#E27:49_1_1_2_1"><i><font color="Green" title="E62">A40</font></i></a></i>;<br/>
<a NAME="E30:49_1_1_2_1"/><i><font color="Green" title="E63">A41</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">h4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E6:49_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E31:49_1_1_2_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">h4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 
;<br/>
<b>then </b><i><font color="Green" title="E64">A42</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">h4</font>
<b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
<b>by </b><i><a class="txt" href="#E6:49_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E33:49_1_1_2_1"/><b>then </b>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E34:49_1_1_2_1"/><b>then </b><i><font color="Green" title="E65">A43</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <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="c2">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E35:49_1_1_2_1"/><i><font color="Green" title="E66">A44</font></i>: 
<font color="Maroon" title="c13">h4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h2r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="euclid_7.html#T11" target="_self" title="EUCLID_7:th.11">Th13</a>, <a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E16:49_1_1_2_1"><i><font color="Green" title="E56">A34</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="#E10:49_1_1_2_1"><i><font color="Green" title="E52">A30</font></i></a></i>;<br/>
<i><font color="Green" title="E67">A45</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">h2</font>
<b>by </b><i><a class="txt" href="#E3:49_1_1_2_1"><i><font color="Green" title="E46">A24</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
<b>by </b><i><a class="txt" href="#E6:49_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E37:49_1_1_2_1"/><b>then </b><i><font color="Green" title="E68">A46</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E38:49_1_1_2_1"/><b>then </b><i><font color="Green" title="E69">A47</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E39:49_1_1_2_1"/><i><font color="Green" title="E70">A48</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <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="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E37:49_1_1_2_1"><i><font color="Green" title="E68">A46</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_2_1_5"/>
( <font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_5_1"><b>suppose </b></a><a NAME="E1:49_1_1_2_1_5_1"/><i><font color="Green" title="E71">A49</font></i>: 
<font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">gF</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:49_1_1_2_1_5_1"/><b>then </b><i><font color="Green" title="E72">A50</font></i>: 
<font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E5:49_1_1_2_1"><i><font color="Green" title="E48">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:49_1_1_2_1_5_1"/><i><font color="Green" title="E73">A51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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="c13">h4</font>
 <b>by </b><i><a class="txt" href="#E11:49_1_1_2_1"><i><font color="Green" title="E53">A31</font></i></a>, <a class="txt" href="#E24:49_1_1_2_1"><i><font color="Green" title="E60">A38</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:49_1_1_2_1_5_1"/><i><font color="Green" title="E74">A52</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">nx</font> &amp; <font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">h2r</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">h2r</font> )
 <b>by </b><i><a class="txt" href="#E9:49_1_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="#E13:49_1_1_2_1"><i><font color="Green" title="E54">A32</font></i></a>, <a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:49_1_1_2_1_5_1"/><b>then </b><i><font color="Green" title="E75">A53</font></i>: 
<font color="Maroon" title="c11">h2b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="euclid_7.html#T11" target="_self" title="EUCLID_7:th.11">Th13</a>, <a class="txt" href="#E13:49_1_1_2_1"><i><font color="Green" title="E54">A32</font></i></a></i>;<br/><a NAME="E6:49_1_1_2_1_5_1"/><i><font color="Green" title="E76">A54</font></i>: 
<font color="Maroon" title="c2">F</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">nx</font>
 <b>by </b><i><a class="txt" href="#E13:49_1_1_2_1"><i><font color="Green" title="E54">A32</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E7:49_1_1_2_1_5_1"/><i><font color="Green" title="E77">A55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E4:49_1_1_2_1"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="#E10:49_1_1_2_1"><i><font color="Green" title="E52">A30</font></i></a>, <a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E8:49_1_1_2_1_5_1"/><i><font color="Green" title="E78">A56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">nx</font>
 <b>by </b><i><a class="txt" href="#E16:49_1_1_2_1"><i><font color="Green" title="E56">A34</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E9:49_1_1_2_1_5_1"/><i><font color="Green" title="E79">A57</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">h4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E6:49_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E10:49_1_1_2_1_5_1"/><i><font color="Green" title="E80">A58</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c13">h4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1_5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> implies <font color="Maroon" title="c13">h4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font> )</span><br/>

<b>assume </b><a NAME="E1:49_1_1_2_1_5_1_1"/><i><font color="Green" title="E81">A59</font></i>: 
( <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">h4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font></span><br/>

<a NAME="E2:49_1_1_2_1_5_1_1"/><b>then </b><i><font color="Green" title="E82">A60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_2_1_5_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c14">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_2_1_5_1_1_1_1"/><i><font color="Green" title="E83">A61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">h4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font></span><br/><div class="add"><a NAME="E2:49_1_1_2_1_5_1_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:49_1_1_2_1_5_1_1_1_1"/><b>then </b><i><font color="Green" title="E84">A62</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="#E1:49_1_1_2_1_5_1_1"><i><font color="Green" title="E81">A59</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then </b><font color="Maroon" title="c13">h4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> = 
<font color="Maroon" title="c11">h2b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="txt" href="#E22:49_1_1_2_1"><i><font color="Green" title="E59">A37</font></i></a>, <a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_1"><i><font color="Green" title="E81">A59</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon" title="c10">h2r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1"><i><font color="Green" title="E71">A49</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_1"><i><font color="Green" title="E81">A59</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_1_1_1"><i><font color="Green" title="E83">A61</font></i></a>, <a class="txt" href="#E3:49_1_1_2_1_5_1_1_1_1"><i><font color="Green" title="E84">A62</font></i></a>, <a class="ref" href="finseq_7.html#T32" title="FINSEQ_7:th.32">FINSEQ_7:32</a></i>
<br/>.= 
<font color="Maroon" title="c7">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="txt" href="#E3:49_1_1_2_1_5_1_1_1_1"><i><font color="Green" title="E84">A62</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_1"><i><font color="Green" title="E81">A59</font></i></a>, <a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="txt" href="#E1:49_1_1_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_1_1_1"><i><font color="Green" title="E83">A61</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_1"><i><font color="Green" title="E81">A59</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:49_1_1_2_1_5_1_1_1_1"/>
<font color="Maroon" title="c13">h4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_2_1_5_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c14">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">h4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font></span><br/><div class="add"><a NAME="E2:49_1_1_2_1_5_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E2:49_1_1_2_1_5_1_1"><i><font color="Green" title="E82">A60</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_2_1_5_1_1_1_2"/>
<font color="Maroon" title="c13">h4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="#E13:49_1_1_2_1"><i><font color="Green" title="E54">A32</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1_5_1"><i><font color="Green" title="E74">A52</font></i></a>, <a class="ref" href="euclid_7.html#T11" target="_self" title="EUCLID_7:th.11">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E11:49_1_1_2_1_5_1"/><i><font color="Green" title="E81">A63</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>;<br/><a NAME="E12:49_1_1_2_1_5_1"/><i><font color="Green" title="E82">A64</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>;<br/><a NAME="E13:49_1_1_2_1_5_1"/><i><font color="Green" title="E83">A65</font></i>: 
for <font color="Olive" title="b1">i4</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i4</font> &amp; <font color="Olive" title="b1">i4</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">nx</font> holds <br/><span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1_5_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i4</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i4</font> &amp; <font color="Maroon" title="c14">i4</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">nx</font> implies <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i4</font> )</span><br/>

<b>assume </b><a NAME="E1:49_1_1_2_1_5_1_2"/><i><font color="Green" title="E84">A66</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i4</font> &amp; <font color="Maroon" title="c14">i4</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">nx</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i4</font></span><br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">nx</font> implies <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span> );<br/>
<a NAME="E2:49_1_1_2_1_5_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E3:49_1_1_2_1_5_1_2"/><b>then </b><i><font color="Green" title="E85">A67</font></i>: 
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E4:49_1_1_2_1_5_1_2"/><i><font color="Green" title="E86">A68</font></i>: 
<font color="Maroon" title="c10">h2r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h2r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="#E4:49_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E5:49_1_1_2_1_5_1_2"/><i><font color="Green" title="E87">A69</font></i>: 
<font color="Maroon" title="c11">h2b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">h2b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="#E4:49_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E22:49_1_1_2_1"><i><font color="Green" title="E59">A37</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E6:49_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="#E36:49_1_1_2_1"><i><font color="Green" title="E67">A45</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">h4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E4:49_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_2"><i><font color="Green" title="E84">A66</font></i></a>, <a class="txt" href="#E3:49_1_1_2_1_5_1_2"><i><font color="Green" title="E85">A67</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1_5_1_2"><i><font color="Green" title="E86">A68</font></i></a>, <a class="txt" href="#E5:49_1_1_2_1_5_1_2"><i><font color="Green" title="E87">A69</font></i></a>, <a class="ref" href="finseq_7.html#T32" title="FINSEQ_7:th.32">FINSEQ_7:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="#E6:49_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="#E32:49_1_1_2_1"><i><font color="Green" title="E64">A42</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>
;<br/>
<a NAME="E8:49_1_1_2_1_5_1_2"/><b>then </b><i><font color="Green" title="E88">A70</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E9:49_1_1_2_1_5_1_2"/><i><font color="Green" title="E89">A71</font></i>: 
for <font color="Olive" title="b1">k3</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k3</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1_5_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">k3</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:49_1_1_2_1_5_1_2_3"/><i><font color="Green" title="E90">A72</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:49_1_1_2_1_5_1_2_3"/><i><font color="Green" title="E91">A73</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E3:49_1_1_2_1_5_1_2_3"/>
( 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">nx</font> implies <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1_5_1_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1_1_2_1_5_1_2_3_1"/><i><font color="Green" title="E92">A74</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">nx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/>

<a NAME="E2:49_1_1_2_1_5_1_2_3_1"/><b>then </b><i><font color="Green" title="E93">A75</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E16:49_1_1_2_1"><i><font color="Green" title="E56">A34</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:49_1_1_2_1_5_1_2_3_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E4:49_1_1_2_1_5_1_2_3_1"/><b>then </b><i><font color="Green" title="E94">A76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E93">A75</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:49_1_1_2_1_5_1_2_3_1"/><i><font color="Green" title="E95">A77</font></i>: 
<font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E6:49_1_1_2_1_5_1_2_3_1"/><b>then </b><i><font color="Green" title="E96">A78</font></i>: 
<font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">nx</font>
 
<b>by </b><i><a class="txt" href="#E1:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E92">A74</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:49_1_1_2_1_5_1_2_3_1"/><b>then </b><i><font color="Green" title="E97">A79</font></i>: 
<font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E16:49_1_1_2_1"><i><font color="Green" title="E56">A34</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:49_1_1_2_1_5_1_2_3_1"/><i><font color="Green" title="E98">A80</font></i>: 
<font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E16:49_1_1_2_1"><i><font color="Green" title="E56">A34</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E6:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E96">A78</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:49_1_1_2_1_5_1_2_3_1"/><i><font color="Green" title="E99">A81</font></i>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E12:49_1_1_2_1_5_1"><i><font color="Green" title="E82">A64</font></i></a>, <a class="txt" href="#E7:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E97">A79</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E10:49_1_1_2_1_5_1_2_3_1"/><i><font color="Green" title="E100">A82</font></i>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E11:49_1_1_2_1_5_1"><i><font color="Green" title="E81">A63</font></i></a>, <a class="txt" href="#E8:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E98">A80</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E11:49_1_1_2_1_5_1_2_3_1"/><i><font color="Green" title="E101">A83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E2:49_1_1_2_1_5_1"><i><font color="Green" title="E72">A50</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E92">A74</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:49_1_1_2_1_5_1_2_3_1"/><i><font color="Green" title="E102">A84</font></i>: 
<font color="Maroon" title="c10">h2r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h2r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E93">A75</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E13:49_1_1_2_1_5_1_2_3_1"/><i><font color="Green" title="E103">A85</font></i>: 
<font color="Maroon" title="c11">h2b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">h2b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E22:49_1_1_2_1"><i><font color="Green" title="E59">A37</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E93">A75</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E14:49_1_1_2_1_5_1_2_3_1"/><i><font color="Green" title="E104">A86</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E94">A76</font></i></a>, <a class="txt" href="#E36:49_1_1_2_1"><i><font color="Green" title="E67">A45</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E105">A87</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <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="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">h4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E92">A74</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E93">A75</font></i></a>, <a class="txt" href="#E11:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E101">A83</font></i></a>, <a class="txt" href="#E12:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E102">A84</font></i></a>, <a class="txt" href="#E13:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E103">A85</font></i></a>, <a class="ref" href="finseq_7.html#T32" title="FINSEQ_7:th.32">FINSEQ_7:32</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <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="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E14:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E104">A86</font></i></a>, <a class="txt" href="#E32:49_1_1_2_1"><i><font color="Green" title="E64">A42</font></i></a>, <a class="txt" href="#E36:49_1_1_2_1"><i><font color="Green" title="E67">A45</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/>
<a NAME="E16:49_1_1_2_1_5_1_2_3_1"/><i><font color="Green" title="E106">A88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <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="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E93">A75</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_2_3"><i><font color="Green" title="E91">A73</font></i></a>, <a class="txt" href="#E7:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E97">A79</font></i></a>, <a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_2_3"><i><font color="Green" title="E90">A72</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E92">A74</font></i></a>, <a class="txt" href="#E5:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E95">A77</font></i></a>, <a class="txt" href="#E9:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E99">A81</font></i></a>, <a class="txt" href="#E10:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E100">A82</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E93">A75</font></i></a>, <a class="txt" href="#E15:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E105">A87</font></i></a>, <a class="txt" href="#E16:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E106">A88</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>
;<br/>
<b>hence </b><a NAME="E18:49_1_1_2_1_5_1_2_3_1"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_2_3"><i><font color="Green" title="E91">A73</font></i></a>, <a class="txt" href="#E7:49_1_1_2_1_5_1_2_3_1"><i><font color="Green" title="E97">A79</font></i></a>, <a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:49_1_1_2_1_5_1_2_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:49_1_1_2_1_5_1_2"/><i><font color="Green" title="E90">A89</font></i>: 
for <font color="Olive" title="b1">k3</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k3</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E8:49_1_1_2_1_5_1_2"><i><font color="Green" title="E88">A70</font></i></a>, <a class="txt" href="#E9:49_1_1_2_1_5_1_2"><i><font color="Green" title="E89">A71</font></i></a>);<br/></i>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i4</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i4</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="#E1:49_1_1_2_1_5_1_2"><i><font color="Green" title="E84">A66</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
<br/>.= 
<font color="Maroon" title="c14">i4</font>

;<br/>
<b>hence </b><a NAME="E12:49_1_1_2_1_5_1_2"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i4</font>
 <b>by </b><i><a class="txt" href="#E1:49_1_1_2_1_5_1_2"><i><font color="Green" title="E84">A66</font></i></a>, <a class="txt" href="#E10:49_1_1_2_1_5_1_2"><i><font color="Green" title="E90">A89</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E14:49_1_1_2_1_5_1"/><i><font color="Green" title="E84">A90</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1_5_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:49_1_1_2_1_5_1_3"/><i><font color="Green" title="E85">A91</font></i>: 
( <font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span></span><br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( <font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1 &amp; <font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span> );<br/>
<a NAME="E2:49_1_1_2_1_5_1_3"/><i><font color="Green" title="E86">A92</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>;<br/>
<a NAME="E3:49_1_1_2_1_5_1_3"/><i><font color="Green" title="E87">A93</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1_5_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_5_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_2_1_5_1_3_1_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">nx</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">nx</font> )
 </span><b>by </b><i><a class="txt" href="#E16:49_1_1_2_1"><i><font color="Green" title="E56">A34</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_5_1_3_1_1_1"><b>case </b></a><a NAME="E1:49_1_1_2_1_5_1_3_1_1_1"/><i><font color="Green" title="E88">A94</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">nx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span></span><br/><div class="add"><b>then </b><i><font color="Green" title="E89">A95</font></i>: <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font>
<b>by </b><i><a class="txt" href="#E16:49_1_1_2_1"><i><font color="Green" title="E56">A34</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_3_1_1_1"><i><font color="Green" title="E88">A94</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
;<br/><a NAME="E3:49_1_1_2_1_5_1_3_1_1_1"/><i><font color="Green" title="E90">A96</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="#E1:49_1_1_2_1_5_1_3_1_1_1"><i><font color="Green" title="E88">A94</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_3"><i><font color="Green" title="E86">A92</font></i></a>, <a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a>, <a class="txt" href="#E4:49_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E6:49_1_1"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="#E30:49_1_1_2_1"><i><font color="Green" title="E63">A41</font></i></a>, <a class="txt" href="#E35:49_1_1_2_1"><i><font color="Green" title="E66">A44</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_3_1_1_1"><i><font color="Green" title="E88">A94</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E14:49_1_1_2_1"><i><font color="Green" title="E55">A33</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1"><i><font color="Green" title="E45">A23</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E4:49_1_1_2_1"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="#E10:49_1_1_2_1"><i><font color="Green" title="E52">A30</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:49_1_1_2_1_5_1_3_1_1_1"><i><font color="Green" title="E90">A96</font></i></a>, <a class="ref" href="euclidlp.html#T7" title="EUCLIDLP:th.7">EUCLIDLP:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1" title="EUCLID_4:th.1">EUCLID_4:1</a></i>
;<br/><b>hence </b><a NAME="E6:49_1_1_2_1_5_1_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:49_1_1_2_1_5_1_3_1_1_1"><i><font color="Green" title="E89">A95</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_5_1_3_1_1_2"><b>case </b></a><a NAME="E1:49_1_1_2_1_5_1_3_1_1_2"/><i><font color="Green" title="E88">A97</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">nx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span></span><br/><div class="add"><a NAME="E2:49_1_1_2_1_5_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E89">A98</font></i>: 
<font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">nx</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>;<br/><a NAME="E3:49_1_1_2_1_5_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E88">A97</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E4:49_1_1_2_1_5_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E90">A99</font></i>: 
<font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:49_1_1_2_1_5_1_3_1_1_2"/>
<font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E6:49_1_1_2_1_5_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E91">A100</font></i>: 
<font color="Maroon" title="c9">nx</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:49_1_1_2_1_5_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E92">A101</font></i>: 
<font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1_5_1"><i><font color="Green" title="E74">A52</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E89">A98</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:49_1_1_2_1_5_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E93">A102</font></i>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E3:49_1_1_2_1_5_1_3"><i><font color="Green" title="E87">A93</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E90">A99</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E9:49_1_1_2_1_5_1_3_1_1_2"/><i><font color="Green" title="E94">A103</font></i>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_3"><i><font color="Green" title="E86">A92</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E90">A99</font></i></a>, <a class="txt" href="#E7:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E92">A101</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><i><font color="Green" title="E95">A104</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">nx</font>
<b>by </b><i><a class="txt" href="#E16:49_1_1_2_1"><i><font color="Green" title="E56">A34</font></i></a>, <a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E13:49_1_1_2_1"><i><font color="Green" title="E54">A32</font></i></a>, <a class="txt" href="#E30:49_1_1_2_1"><i><font color="Green" title="E63">A41</font></i></a>, <a class="txt" href="#E35:49_1_1_2_1"><i><font color="Green" title="E66">A44</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E2:49_1_1_2_1"><i><font color="Green" title="E45">A23</font></i></a>, <a class="txt" href="#E14:49_1_1_2_1"><i><font color="Green" title="E55">A33</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E4:49_1_1_2_1"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="#E10:49_1_1_2_1"><i><font color="Green" title="E52">A30</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
;<br/><i><font color="Green" title="E96">A105</font></i>: <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E89">A98</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E90">A99</font></i></a>, <a class="txt" href="#E7:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E92">A101</font></i></a>, <a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span>
<b>by </b><i><a class="txt" href="#E13:49_1_1_2_1_5_1"><i><font color="Green" title="E83">A65</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E89">A98</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E90">A99</font></i></a>, <a class="txt" href="#E6:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E91">A100</font></i></a>, <a class="txt" href="#E8:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E93">A102</font></i></a>, <a class="txt" href="#E9:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E94">A103</font></i></a></i>
;<br/><span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E89">A98</font></i></a>, <a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E90">A99</font></i></a>, <a class="txt" href="#E7:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E92">A101</font></i></a>, <a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E89">A98</font></i></a>, <a class="txt" href="#E10:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E95">A104</font></i></a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1_5_1"><i><font color="Green" title="E74">A52</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_3"><i><font color="Green" title="E86">A92</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T63" title="RVSUM_1:th.63">RVSUM_1:63</a></i>
;<br/><b>hence </b><a NAME="E14:49_1_1_2_1_5_1_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E11:49_1_1_2_1_5_1_3_1_1_2"><i><font color="Green" title="E96">A105</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E5:49_1_1_2_1_5_1_3"/><b>then </b><i><font color="Green" title="E88">A106</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E6:49_1_1_2_1_5_1_3"/><i><font color="Green" title="E89">A107</font></i>: 
for <font color="Olive" title="b1">k3</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k3</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1_5_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">k3</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:49_1_1_2_1_5_1_3_2"/><i><font color="Green" title="E90">A108</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:49_1_1_2_1_5_1_3_2"/>
( <font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1_5_1_3_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1_1_2_1_5_1_3_2_1"/><i><font color="Green" title="E91">A109</font></i>: 
( <font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span></span><br/>

<a NAME="E2:49_1_1_2_1_5_1_3_2_1"/><i><font color="Green" title="E92">A110</font></i>: 
<font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E3:49_1_1_2_1_5_1_3_2_1"/><i><font color="Green" title="E93">A111</font></i>: 
<font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E16:49_1_1_2_1"><i><font color="Green" title="E56">A34</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:49_1_1_2_1_5_1_3_2_1"/><i><font color="Green" title="E94">A112</font></i>: 
<font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E5:49_1_1_2_1_5_1_3_2_1"/><b>then </b><i><font color="Green" title="E95">A113</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E93">A111</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:49_1_1_2_1_5_1_3_2_1"/><i><font color="Green" title="E96">A114</font></i>: 
<font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:49_1_1_2_1"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_3"><i><font color="Green" title="E86">A92</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E91">A109</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:49_1_1_2_1_5_1_3_2_1"/>
<font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E1:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E91">A109</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E94">A112</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:49_1_1_2_1_5_1_3_2_1"/><b>then </b><i><font color="Green" title="E97">A115</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font> &amp; <font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1_5_1"><i><font color="Green" title="E74">A52</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E92">A110</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:49_1_1_2_1_5_1_3_2_1"/><b>then </b><i><font color="Green" title="E98">A116</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E10:49_1_1_2_1_5_1_3_2_1"/><i><font color="Green" title="E99">A117</font></i>: 
<font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E11:49_1_1_2_1_5_1_3_2_1"/><i><font color="Green" title="E100">A118</font></i>: 
<font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E12:49_1_1_2_1_5_1_3_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E9:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E98">A116</font></i></a>, <a class="txt" href="#E5:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E95">A113</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:49_1_1_2_1_5_1_3_2_1"/><b>then </b><i><font color="Green" title="E101">A119</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E102">A120</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <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="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">h4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E32:49_1_1_2_1"><i><font color="Green" title="E64">A42</font></i></a>, <a class="txt" href="#E36:49_1_1_2_1"><i><font color="Green" title="E67">A45</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h2b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E22:49_1_1_2_1"><i><font color="Green" title="E59">A37</font></i></a>, <a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E9:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E98">A116</font></i></a>, <a class="txt" href="#E5:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E95">A113</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h2r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E91">A109</font></i></a>, <a class="txt" href="#E5:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E95">A113</font></i></a>, <a class="txt" href="#E9:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E98">A116</font></i></a>, <a class="txt" href="#E10:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E99">A117</font></i></a>, <a class="txt" href="#E11:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E100">A118</font></i></a>, <a class="ref" href="finseq_7.html#T32" title="FINSEQ_7:th.32">FINSEQ_7:32</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h2r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E9:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E98">A116</font></i></a>, <a class="txt" href="#E5:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E95">A113</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <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="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E13:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E101">A119</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/>
<a NAME="E15:49_1_1_2_1_5_1_3_2_1"/><i><font color="Green" title="E103">A121</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <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="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E9:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E98">A116</font></i></a>, <a class="txt" href="#E5:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E95">A113</font></i></a>, <a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<i><font color="Green" title="E104">A122</font></i>: <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a>, <a class="txt" href="#E8:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E97">A115</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:49_1_1_2_1_5_1_3"><i><font color="Green" title="E87">A93</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_3_2"><i><font color="Green" title="E90">A108</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E91">A109</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E94">A112</font></i></a>, <a class="txt" href="#E8:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E97">A115</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c16">f1</font> = <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">f2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c18">f3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c19">f4</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a>, <a class="txt" href="#E8:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E97">A115</font></i></a>, <a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E95">A113</font></i></a>, <a class="txt" href="#E9:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E98">A116</font></i></a>, <a class="txt" href="#E14:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E102">A120</font></i></a>, <a class="txt" href="#E15:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E103">A121</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">f1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c18">f3</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c17">f2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c19">f4</font>
<b>by </b><i><a class="txt" href="#E5:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E95">A113</font></i></a>, <a class="txt" href="#E6:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E96">A114</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">f1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c17">f2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c18">f3</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c19">f4</font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T19" title="RFUNCT_1:th.19">RFUNCT_1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">f1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c17">f2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c19">f4</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c18">f3</font>
<b>by </b><i><a class="ref" href="rfunct_1.html#T19" title="RFUNCT_1:th.19">RFUNCT_1:19</a></i>
;<br/>
<b>hence </b><a NAME="E23:49_1_1_2_1_5_1_3_2_1"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E16:49_1_1_2_1_5_1_3_2_1"><i><font color="Green" title="E104">A122</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:49_1_1_2_1_5_1_3_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:49_1_1_2_1_5_1_3"/><i><font color="Green" title="E90">A123</font></i>: 
for <font color="Olive" title="b1">k3</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k3</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E5:49_1_1_2_1_5_1_3"><i><font color="Green" title="E88">A106</font></i></a>, <a class="txt" href="#E6:49_1_1_2_1_5_1_3"><i><font color="Green" title="E89">A107</font></i></a>);<br/></i>
<a NAME="E8:49_1_1_2_1_5_1_3"/><i><font color="Green" title="E91">A124</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c9">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">nx</font>
 
<b>by </b><i><a class="txt" href="#E1:49_1_1_2_1_5_1_3"><i><font color="Green" title="E85">A91</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>;<br/>
<a NAME="E9:49_1_1_2_1_5_1_3"/><b>then </b>
( <font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c9">nx</font></span>)</span> &amp; <font color="Maroon" title="c9">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c9">nx</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="#E1:49_1_1_2_1_5_1_3"><i><font color="Green" title="E85">A91</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:49_1_1_2_1_5_1_3"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:49_1_1_2_1_5_1_3"><i><font color="Green" title="E90">A123</font></i></a>, <a class="txt" href="#E8:49_1_1_2_1_5_1_3"><i><font color="Green" title="E91">A124</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E15:49_1_1_2_1_5_1"/><i><font color="Green" title="E85">A125</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> holds <br/><span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1_5_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> implies <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> )</span><br/>

<b>assume </b><a NAME="E1:49_1_1_2_1_5_1_4"/><i><font color="Green" title="E86">A126</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span><br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> implies <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span> );<br/>
<a NAME="E2:49_1_1_2_1_5_1_4"/><i><font color="Green" title="E87">A127</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>;<br/>
<a NAME="E3:49_1_1_2_1_5_1_4"/><i><font color="Green" title="E88">A128</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:49_1_1_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="#E37:49_1_1_2_1"><i><font color="Green" title="E68">A46</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E4:49_1_1_2_1_5_1_4"/><i><font color="Green" title="E89">A129</font></i>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:49_1_1_2_1"><i><font color="Green" title="E48">A26</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1"><i><font color="Green" title="E71">A49</font></i></a>, <a class="txt" href="#E14:49_1_1_2_1_5_1"><i><font color="Green" title="E84">A90</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:49_1_1_2_1_5_1_4"><i><font color="Green" title="E88">A128</font></i></a>, <a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E6:49_1_1_2_1"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_4"><i><font color="Green" title="E87">A127</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1_5_1_4"><i><font color="Green" title="E89">A129</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_7.html#T6" target="_self" title="EUCLID_7:th.6">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:49_1_1_2_1_5_1"><i><font color="Green" title="E73">A51</font></i></a>, <a class="txt" href="#E5:49_1_1_2_1_5_1"><i><font color="Green" title="E75">A53</font></i></a>, <a class="txt" href="#E6:49_1_1_2_1_5_1"><i><font color="Green" title="E76">A54</font></i></a>, <a class="txt" href="#E7:49_1_1_2_1_5_1"><i><font color="Green" title="E77">A55</font></i></a>, <a class="txt" href="#E8:49_1_1_2_1_5_1"><i><font color="Green" title="E78">A56</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E6:49_1_1_2_1_5_1_4"/><b>then </b><i><font color="Green" title="E90">A130</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="#E1:49_1_1_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="#E9:49_1_1_2_1"><i><font color="Green" title="E51">A29</font></i></a>, <a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a>, <a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>;<br/>
<a NAME="E7:49_1_1_2_1_5_1_4"/><i><font color="Green" title="E91">A131</font></i>: 
for <font color="Olive" title="b1">k3</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k3</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1_5_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">k3</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:49_1_1_2_1_5_1_4_2"/><i><font color="Green" title="E92">A132</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:49_1_1_2_1_5_1_4_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> implies <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1_5_1_4_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1_1_2_1_5_1_4_2_1"/><i><font color="Green" title="E93">A133</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/>

<a NAME="E2:49_1_1_2_1_5_1_4_2_1"/><i><font color="Green" title="E94">A134</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E3:49_1_1_2_1_5_1_4_2_1"/><b>then </b><i><font color="Green" title="E95">A135</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E1:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E93">A133</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:49_1_1_2_1_5_1_4_2_1"/><i><font color="Green" title="E96">A136</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E5:49_1_1_2_1_5_1_4_2_1"/><i><font color="Green" title="E97">A137</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E6:49_1_1_2_1_5_1_4_2_1"/><b>then </b><i><font color="Green" title="E98">A138</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font>
 
<b>by </b><i><a class="txt" href="#E4:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E96">A136</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:49_1_1_2_1_5_1_4_2_1"/><i><font color="Green" title="E99">A139</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E8:49_1_1_2_1_5_1_4_2_1"/><i><font color="Green" title="E100">A140</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E5:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E97">A137</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E94">A134</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:49_1_1_2_1_5_1_4_2_1"/><i><font color="Green" title="E101">A141</font></i>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E95">A135</font></i></a>, <a class="txt" href="#E11:49_1_1_2_1_5_1"><i><font color="Green" title="E81">A63</font></i></a>, <a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a>, <a class="txt" href="#E6:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E98">A138</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E10:49_1_1_2_1_5_1_4_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E38:49_1_1_2_1"><i><font color="Green" title="E69">A47</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E93">A133</font></i></a>, <a class="txt" href="#E7:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E99">A139</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:49_1_1_2_1_5_1_4_2_1"/><b>then </b><i><font color="Green" title="E102">A142</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E103">A143</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">h4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E32:49_1_1_2_1"><i><font color="Green" title="E64">A42</font></i></a>, <a class="txt" href="#E36:49_1_1_2_1"><i><font color="Green" title="E67">A45</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h2b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E22:49_1_1_2_1"><i><font color="Green" title="E59">A37</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E93">A133</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h2r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_1"><i><font color="Green" title="E72">A50</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E93">A133</font></i></a>, <a class="txt" href="#E8:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E100">A140</font></i></a>, <a class="ref" href="finseq_7.html#T32" title="FINSEQ_7:th.32">FINSEQ_7:32</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h2r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E20:49_1_1_2_1"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E93">A133</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E11:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E102">A142</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/>
<a NAME="E13:49_1_1_2_1_5_1_4_2_1"/><i><font color="Green" title="E104">A144</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E93">A133</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a>, <a class="txt" href="#E6:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E98">A138</font></i></a>, <a class="txt" href="#E3:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E95">A135</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E12:49_1_1_2_1_5_1"><i><font color="Green" title="E82">A64</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_4_2"><i><font color="Green" title="E92">A132</font></i></a>, <a class="txt" href="#E3:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E95">A135</font></i></a>, <a class="txt" href="#E6:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E98">A138</font></i></a>, <a class="txt" href="#E9:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E101">A141</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E93">A133</font></i></a>, <a class="txt" href="#E12:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E103">A143</font></i></a>, <a class="txt" href="#E13:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E104">A144</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>
;<br/>
<b>hence </b><a NAME="E15:49_1_1_2_1_5_1_4_2_1"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a>, <a class="txt" href="#E6:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E98">A138</font></i></a>, <a class="txt" href="#E3:49_1_1_2_1_5_1_4_2_1"><i><font color="Green" title="E95">A135</font></i></a>, <a class="txt" href="#E34:49_1_1_2_1"><i><font color="Green" title="E65">A43</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:49_1_1_2_1_5_1_4_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:49_1_1_2_1_5_1_4"/><i><font color="Green" title="E92">A145</font></i>: 
for <font color="Olive" title="b1">k3</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k3</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E6:49_1_1_2_1_5_1_4"><i><font color="Green" title="E90">A130</font></i></a>, <a class="txt" href="#E7:49_1_1_2_1_5_1_4"><i><font color="Green" title="E91">A131</font></i></a>);<br/></i>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E1:49_1_1_2_1_5_1_4"><i><font color="Green" title="E86">A126</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>
<br/>.= 
<font color="Maroon" title="c14">i</font>

;<br/>
<b>hence </b><a NAME="E10:49_1_1_2_1_5_1_4"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="#E1:49_1_1_2_1_5_1_4"><i><font color="Green" title="E86">A126</font></i></a>, <a class="txt" href="#E8:49_1_1_2_1_5_1_4"><i><font color="Green" title="E92">A145</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E16:49_1_1_2_1_5_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">h4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 ;<br/><a NAME="E17:49_1_1_2_1_5_1"/><b>then </b><i><font color="Green" title="E86">A146</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">h4</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/><a NAME="E18:49_1_1_2_1_5_1"/><b>then </b><i><font color="Green" title="E87">A147</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">h4</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E19:49_1_1_2_1_5_1"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E9:49_1_1_2_1_5_1"><i><font color="Green" title="E79">A57</font></i></a>, <a class="txt" href="#E17:49_1_1_2_1_5_1"><i><font color="Green" title="E86">A146</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E20:49_1_1_2_1_5_1"/><b>then </b><i><font color="Green" title="E88">A148</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <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="c2">F</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E21:49_1_1_2_1_5_1"/><i><font color="Green" title="E89">A149</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:49_1_1_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="#E9:49_1_1_2_1_5_1"><i><font color="Green" title="E79">A57</font></i></a>, <a class="txt" href="#E18:49_1_1_2_1_5_1"><i><font color="Green" title="E87">A147</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E22:49_1_1_2_1_5_1"/><i><font color="Green" title="E90">A150</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E10:49_1_1_2_1"><i><font color="Green" title="E52">A30</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="#E20:49_1_1_2_1_5_1"><i><font color="Green" title="E88">A148</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E23:49_1_1_2_1_5_1"/><i><font color="Green" title="E91">A151</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E10:49_1_1_2_1"><i><font color="Green" title="E52">A30</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1"><i><font color="Green" title="E47">A25</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E24:49_1_1_2_1_5_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>;<br/><a NAME="E25:49_1_1_2_1_5_1"/><b>then </b><i><font color="Green" title="E92">A152</font></i>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E21:49_1_1_2_1_5_1"><i><font color="Green" title="E89">A149</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E26:49_1_1_2_1_5_1"/><i><font color="Green" title="E93">A153</font></i>: 
<font color="Maroon" title="c5">gF</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">gF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E3:49_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="#E9:49_1_1_2_1"><i><font color="Green" title="E51">A29</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><font color="Maroon" title="c5">gF</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">gF</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:49_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="#E9:49_1_1_2_1"><i><font color="Green" title="E51">A29</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:49_1_1_2"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="#E8:49_1_1_2_1"><i><font color="Green" title="E50">A28</font></i></a>, <a class="txt" href="#E10:49_1_1_2_1_5_1"><i><font color="Green" title="E80">A58</font></i></a>, <a class="txt" href="#E25:49_1_1_2_1_5_1"><i><font color="Green" title="E92">A152</font></i></a>, <a class="txt" href="#E26:49_1_1_2_1_5_1"><i><font color="Green" title="E93">A153</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:49_1_1_2_1_5_1"><i><font color="Green" title="E73">A51</font></i></a>, <a class="txt" href="#E5:49_1_1_2_1_5_1"><i><font color="Green" title="E75">A53</font></i></a>, <a class="txt" href="#E22:49_1_1_2_1_5_1"><i><font color="Green" title="E90">A150</font></i></a>, <a class="txt" href="#E23:49_1_1_2_1_5_1"><i><font color="Green" title="E91">A151</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c13">h4</font></span>)</span></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E21:49_1_1_2_1_5_1"><i><font color="Green" title="E89">A149</font></i></a>, <a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>
;<br/><b>hence </b><a NAME="E28:49_1_1_2_1_5_1"/>
<font color="Maroon" title="c5">gF</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E4:49_1_1_2_1"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="#E15:49_1_1_2_1_5_1"><i><font color="Green" title="E85">A125</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_5_2"><b>suppose </b></a><a NAME="E1:49_1_1_2_1_5_2"/>
<font color="Maroon" title="c9">nx</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">gF</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:49_1_1_2_1_5_2"/><b>then </b><i><font color="Green" title="E71">A154</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">nx</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_2_1_5_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="c9">nx</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">nx</font> )
 </span><b>by </b><i><a class="txt" href="#E2:49_1_1_2_1_5_2"><i><font color="Green" title="E71">A154</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_5_2_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_2_1_5_2_1_1"/><i><font color="Green" title="E72">A155</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="c9">nx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">gF</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:49_1_1_2_1_5_2_1_1"/><i><font color="Green" title="E73">A156</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:49_1_1_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="#E37:49_1_1_2_1"><i><font color="Green" title="E68">A46</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:49_1_1_2_1_5_2_1_1"/><i><font color="Green" title="E74">A157</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E10:49_1_1_2_1"><i><font color="Green" title="E52">A30</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1"><i><font color="Green" title="E47">A25</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E4:49_1_1_2_1_5_2_1_1"/><i><font color="Green" title="E75">A158</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c7">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1_5_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> implies <font color="Maroon" title="c7">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font> )</span><br/>

<b>assume </b><a NAME="E1:49_1_1_2_1_5_2_1_1_1"/><i><font color="Green" title="E76">A159</font></i>: 
( <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font></span><br/>

<a NAME="E2:49_1_1_2_1_5_2_1_1_1"/><b>then </b><i><font color="Green" title="E77">A160</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_5_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_2_1_5_2_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="c14">i</font> )
 </span><b>by </b><i><a class="txt" href="#E2:49_1_1_2_1_5_2_1_1_1"><i><font color="Green" title="E77">A160</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_5_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_2_1_5_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font></span><br/><div class="add"><b>hence </b><a NAME="E2:49_1_1_2_1_5_2_1_1_1_1_1"/>
<font color="Maroon" title="c7">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="#E1:49_1_1_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_2_1_1_1"><i><font color="Green" title="E76">A159</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_5_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_2_1_5_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="c14">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font></span><br/><div class="add"><b>hence </b><a NAME="E2:49_1_1_2_1_5_2_1_1_1_1_2"/>
<font color="Maroon" title="c7">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="#E13:49_1_1_2_1"><i><font color="Green" title="E54">A32</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_2_1_1"><i><font color="Green" title="E72">A155</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:49_1_1_2_1_5_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>;<br/><a NAME="E6:49_1_1_2_1_5_2_1_1"/><b>then </b><i><font color="Green" title="E76">A161</font></i>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E6:49_1_1_2_1"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E7:49_1_1_2_1_5_2_1_1"/><i><font color="Green" title="E77">A162</font></i>: 
<font color="Maroon" title="c5">gF</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">gF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E3:49_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_2_1_1"><i><font color="Green" title="E73">A156</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E8:49_1_1_2_1_5_2_1_1"/><i><font color="Green" title="E78">A163</font></i>: 
<font color="Maroon" title="c2">F</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">nx</font>
 <b>by </b><i><a class="txt" href="#E13:49_1_1_2_1"><i><font color="Green" title="E54">A32</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><font color="Maroon" title="c5">gF</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">gF</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:49_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="#E2:49_1_1_2_1_5_2_1_1"><i><font color="Green" title="E73">A156</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:49_1_1_2"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="#E8:49_1_1_2_1"><i><font color="Green" title="E50">A28</font></i></a>, <a class="txt" href="#E4:49_1_1_2_1_5_2_1_1"><i><font color="Green" title="E75">A158</font></i></a>, <a class="txt" href="#E6:49_1_1_2_1_5_2_1_1"><i><font color="Green" title="E76">A161</font></i></a>, <a class="txt" href="#E7:49_1_1_2_1_5_2_1_1"><i><font color="Green" title="E77">A162</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:49_1_1_2_1"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="#E10:49_1_1_2_1"><i><font color="Green" title="E52">A30</font></i></a>, <a class="txt" href="#E39:49_1_1_2_1"><i><font color="Green" title="E70">A48</font></i></a>, <a class="txt" href="#E1:49_1_1_2_1_5_2_1_1"><i><font color="Green" title="E72">A155</font></i></a>, <a class="txt" href="#E3:49_1_1_2_1_5_2_1_1"><i><font color="Green" title="E74">A157</font></i></a>, <a class="txt" href="#E8:49_1_1_2_1_5_2_1_1"><i><font color="Green" title="E78">A163</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E2:49_1_1_2_1_5_2_1_1"><i><font color="Green" title="E73">A156</font></i></a>, <a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def12</a></i>
;<br/><b>hence </b><a NAME="E10:49_1_1_2_1_5_2_1_1"/>
<font color="Maroon" title="c5">gF</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_5_2_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_2_1_5_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">nx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">gF</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:49_1_1_2_1_5_2_1_2"/>
<font color="Maroon" title="c5">gF</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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c7">h2</font></span>)</span></span>)</span> <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="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E1:49_1_1_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="#E13:49_1_1_2_1"><i><font color="Green" title="E54">A32</font></i></a>, <a class="txt" href="#E16:49_1_1_2_1"><i><font color="Green" title="E56">A34</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:49_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E18:49_1_1"/><i><font color="Green" title="E43">A164</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E16:49_1_1"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="#E17:49_1_1"><i><font color="Green" title="E42">A20</font></i></a>);<br/></i><a NAME="E19:49_1_1"/><i><font color="Green" title="E44">A165</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="#E4:49_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>;<br/><a NAME="E20:49_1_1"/><b>then </b>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c4">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font>
 ;<br/><b>then </b><font color="Maroon" title="c5">gF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span>
<b>by </b><i><a class="txt" href="#E18:49_1_1"><i><font color="Green" title="E43">A164</font></i></a>, <a class="txt" href="#E19:49_1_1"><i><font color="Green" title="E44">A165</font></i></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid_7.html#K5" title="EUCLID_7:func.5">accum</a> <font color="Maroon" title="c3">F2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:49"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E12:49_1_1"><i><font color="Green" title="E39">A11</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c3">F2</font>
<b>by </b><i><a class="txt" href="#E14:49_1_1"><i><font color="Green" title="E40">A12</font></i></a>, <a class="ref" href="euclid_7.html#T22" target="_self" title="EUCLID_7:th.22">Th24</a></i>
;<br/><b>hence </b><a NAME="E22:49_1_1"/>
 <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E3:49_1_1"><i><font color="Green" title="E31">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2"><b>suppose </b></a><a NAME="E1:49_1_2"/><i><font color="Green" title="E30">A166</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c2">F</font></span><br/><div class="add"><a NAME="E2:49_1_2"/><b>then </b><i><font color="Green" title="E31">A167</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:49_1_2"/><i><font color="Green" title="E32">A168</font></i>: 
 <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="#E1:49_1_2"><i><font color="Green" title="E30">A166</font></i></a>, <a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></i>;<br/><a NAME="E4:49_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 ;<br/><a NAME="E5:49_1_2"/><b>then </b><i><font color="Green" title="E33">A169</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="euclid_7.html#K2" title="EUCLID_7:func.2">(*)</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/><a NAME="E6:49_1_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E2:49_1_2"><i><font color="Green" title="E31">A167</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:49_1_2"/><b>then </b>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">F2</font>
 <b>by </b><i><a class="txt" href="#E1:49"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E5:49_1_2"><i><font color="Green" title="E33">A169</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:49_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</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="c3">F2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E9:49_1_2"/>
 <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K4" title="EUCLID_7:func.4">Sum</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E1:49_1_2"><i><font color="Green" title="E30">A166</font></i></a>, <a class="txt" href="#E3:49_1_2"><i><font color="Green" title="E32">A168</font></i></a>, <a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
