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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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 <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">F</font></span>)</span>  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#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b2">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b1">F</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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 <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span>  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#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c2">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">h</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" 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#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c2">F</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:48"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c2">F</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_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="48_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c2">F</font></span><br/><div class="add"><a NAME="E2:48_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:48_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A4</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E4:48_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A5</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E5:48_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E6:48_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/><a NAME="E7:48_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A7</font></span>: 
 <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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/><a NAME="E8:48_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A8</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">gF</font> =  <a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</font>;<br/><a NAME="E9:48_1_1"/><span class="lab"><font color="Green" title="E35">A9</font></span>: 
 <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> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>;<br/><a NAME="E10:48_1_1"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
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/><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</font></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#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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">h2</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span>  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/><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</font></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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="E11:48_1_1"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E12:48_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E5:48_1_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E7:48_1_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E13:48_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A14</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E14:48_1_1"/><span class="lab"><font color="Green" title="E39">A15</font></span>: 
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E15:48_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A16</font></span>: 
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/><a NAME="E16:48_1_1"/><span class="lab"><font color="Green" title="E41">A17</font></span>: 
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E17:48_1_1"/><span class="lab"><font color="Green" title="E42">A18</font></span>: 
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="48_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:48_1_1_1"/><span class="lab"><font color="Green" title="E43">A19</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">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="E2:48_1_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 <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span>  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/><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="48_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A20</font></span>: 
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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E2:48_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A21</font></span>: 
<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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">h2</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" 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 <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:48_1_1_1_1"/><span class="lab"><font color="Green" title="E46">A22</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:48_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A23</font></span>: 
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="E5:48_1_1_1_1"/><span class="lab"><font color="Green" title="E48">A24</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E6:48_1_1_1_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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<a NAME="E7:48_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A25</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:48_1_1_1_1"/><span class="lab"><font color="Green" title="E50">A26</font></span>: 
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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<a NAME="E9:48_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A27</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E10:48_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A28</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E11:48_1_1_1_1"/><span class="lab"><font color="Green" title="E53">A29</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E12:48_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A30</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">h2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E55">A31</font></span>:  <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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">h2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
;<br/>
<a NAME="E14:48_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A32</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E15:48_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A33</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E16:48_1_1_1_1"/><span class="lab"><font color="Green" title="E58">A34</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:48_1_1_1_1"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E17:48_1_1_1_1"/>
<font color="Maroon" title="c7">h2</font> is   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E11:48_1_1_1_1"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1_1_1"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="txt" href="#E12:48_1_1_1_1"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E19:48_1_1_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E11:48_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E20:48_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A35</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E22:48_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A36</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">h2</font>
 <span class="kw">and </span><br/><a NAME="E23:48_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A37</font></span>: 
<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="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E11:48_1_1_1_1"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="txt" href="#E12:48_1_1_1_1"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E9:48_1_1_1_1"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">nx</font> = <font color="Maroon" title="c9">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1_1_1"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="txt" href="#E22:48_1_1_1_1"><span class="lab"><font color="Green" title="E60">A36</font></span></a></span>;<br/>
<a NAME="E25:48_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A38</font></span>: 
<font color="Maroon" title="c10">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E22:48_1_1_1_1"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">h2b</font> =  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <font color="Maroon" title="c8">h2r</font>,<font color="Maroon" title="c10">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/>
<span class="lab"><font color="Green" title="E63">A39</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">h2b</font> = 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">h2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_7.html#T24" title="FINSEQ_7:th.24">FINSEQ_7:24</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>
;<br/>
<a NAME="E28:48_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A40</font></span>: 
 <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="c8">h2r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_7.html#T20" title="FINSEQ_7:th.20">FINSEQ_7:20</a></span>;<br/>
<a NAME="E29:48_1_1_1_1"/><span class="kw">then </span>
 <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="c8">h2r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E65">A41</font></span>:  <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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">h2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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 <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:48_1_1_1_1"><span class="lab"><font color="Green" title="E63">A39</font></span></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></span>;<br/>
<a NAME="E32:48_1_1_1_1"/><span class="lab"><font color="Green" title="E66">A42</font></span>: 
<font color="Maroon" title="c12">h2d</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_5.html#T39" title="INT_5:th.39">INT_5:39</a></span>;<br/>
<a NAME="E33:48_1_1_1_1"/>
<font color="Maroon" title="c12">h2d</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:48_1_1_1_1"><span class="lab"><font color="Green" title="E63">A39</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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 <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:48_1_1_1_1"><span class="lab"><font color="Green" title="E66">A42</font></span></a></span>;<br/>
<a NAME="E35:48_1_1_1_1"/><span class="lab"><font color="Green" title="E67">A43</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">nx</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E22:48_1_1_1_1"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E36:48_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A44</font></span>: 
<font color="Maroon" title="c13">h4</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E25:48_1_1_1_1"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="ref" href="euclid_7.html#T11" target="_self" title="EUCLID_7:th.11">Th11</a></span>;<br/>
<a NAME="E37:48_1_1_1_1"/><span class="lab"><font color="Green" title="E69">A45</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E38:48_1_1_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E70">A46</font></span>:  <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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">h4</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
;<br/>
<a NAME="E40:48_1_1_1_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E41:48_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A47</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_5"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_1_1_1_5"/>
( <font color="Maroon" title="c10">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="c10">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="48_1_1_1_1_5_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E72">A48</font></span>: 
<font color="Maroon" title="c10">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="E2:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E73">A49</font></span>: 
<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="c8">h2r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E74">A50</font></span>: 
<font color="Maroon" title="c10">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="c8">h2r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E22:48_1_1_1_1"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E4:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E75">A51</font></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"><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="c10">nx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:48_1_1_1_1"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E23:48_1_1_1_1"><span class="lab"><font color="Green" title="E61">A37</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><a NAME="E5:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E76">A52</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>;<br/><a NAME="E6:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E77">A53</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">nx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E22:48_1_1_1_1"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E7:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E78">A54</font></span>: 
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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E8:48_1_1_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A55</font></span>: 
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:48_1_1_1_1"><span class="lab"><font color="Green" title="E61">A37</font></span></a>, <a class="txt" href="#E6:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E77">A53</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E74">A50</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E73">A49</font></span></a>, <a class="ref" href="euclid_7.html#T11" target="_self" title="EUCLID_7:th.11">Th11</a></span>;<br/><a NAME="E9:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E80">A56</font></span>: 
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="48_1_1_1_1_5_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48_1_1_1_1_5_1_1"/><span class="lab"><font color="Green" title="E81">A57</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:48_1_1_1_1_5_1_1"/><span class="lab"><font color="Green" title="E82">A58</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E3:48_1_1_1_1_5_1_1"/><span class="lab"><font color="Green" title="E83">A59</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E81">A57</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_5_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_1_1_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="48_1_1_1_1_5_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E84">A60</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:48_1_1_1_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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:48_1_1_1_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A61</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E81">A57</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E28:48_1_1_1_1"><span class="lab"><font color="Green" title="E64">A40</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E82">A58</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c8">h2r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E72">A48</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E81">A57</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E82">A58</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E84">A60</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E85">A61</font></span></a>, <a class="ref" href="finseq_7.html#T32" title="FINSEQ_7:th.32">FINSEQ_7:32</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E82">A58</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E85">A61</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c14">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_1_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E82">A58</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E84">A60</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E5:48_1_1_1_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_5_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:48_1_1_1_1_5_1_1_1_2"/><span class="kw">then </span>
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E83">A59</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_1_1_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:48_1_1_1_1"><span class="lab"><font color="Green" title="E61">A37</font></span></a>, <a class="txt" href="#E6:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E77">A53</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E74">A50</font></span></a>, <a class="txt" href="#E7:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E78">A54</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E73">A49</font></span></a>, <a class="ref" href="euclid_7.html#T11" target="_self" title="EUCLID_7:th.11">Th11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E10:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E81">A62</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><a NAME="E11:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E82">A63</font></span>: 
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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E12:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E83">A64</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>;<br/><a NAME="E13:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E84">A65</font></span>: 
<font color="Maroon" title="c10">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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E72">A48</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E14:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E85">A66</font></span>: 
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="c10">nx</font> holds <br/><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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="48_1_1_1_1_5_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( 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="c10">nx</font> implies <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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/>
<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="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="c10">nx</font> implies <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48_1_1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E86">A67</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i4</font>
 <span class="kw">and </span><br/><a NAME="E2:48_1_1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E87">A68</font></span>: 
<font color="Maroon" title="c14">i4</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">nx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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/>

<span class="lab"><font color="Green" title="E88">A69</font></span>: 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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E86">A67</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
<font color="Maroon" title="c14">i4</font>

;<br/>
<a NAME="E4:48_1_1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E89">A70</font></span>: 
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="48_1_1_1_1_5_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="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/>

<a NAME="E1:48_1_1_1_1_5_1_2_2"/><span class="lab"><font color="Green" title="E90">A71</font></span>: 
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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:48_1_1_1_1_5_1_2_2"/><span class="lab"><font color="Green" title="E91">A72</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">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="E3:48_1_1_1_1_5_1_2_2"/>
( 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="c10">nx</font> implies <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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="48_1_1_1_1_5_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:48_1_1_1_1_5_1_2_2_1"/><span class="lab"><font color="Green" title="E92">A73</font></span>: 
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="c10">nx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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="E2:48_1_1_1_1_5_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A74</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:48_1_1_1_1"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E3:48_1_1_1_1_5_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A75</font></span>: 
<font color="Maroon" title="c8">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="c8">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></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></span>;<br/>
<a NAME="E4:48_1_1_1_1_5_1_2_2_1"/><span class="lab"><font color="Green" title="E95">A76</font></span>: 
<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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E84">A65</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E92">A73</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:48_1_1_1_1_5_1_2_2_1"/><span class="lab"><font color="Green" title="E96">A77</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E28:48_1_1_1_1"><span class="lab"><font color="Green" title="E64">A40</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E93">A74</font></span></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></span>;<br/>
<a NAME="E6:48_1_1_1_1_5_1_2_2_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E7:48_1_1_1_1_5_1_2_2_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E93">A74</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E8:48_1_1_1_1_5_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A78</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_1_1_1_1"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E98">A79</font></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> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E92">A73</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E93">A74</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E95">A76</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E94">A75</font></span></a>, <a class="txt" href="#E5:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E96">A77</font></span></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></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:48_1_1_1_1"><span class="lab"><font color="Green" title="E70">A46</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E97">A78</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
;<br/>
<a NAME="E10:48_1_1_1_1_5_1_2_2_1"/><span class="lab"><font color="Green" title="E99">A80</font></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="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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E93">A74</font></span></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></span>;<br/>
<a NAME="E11:48_1_1_1_1_5_1_2_2_1"/><span class="lab"><font color="Green" title="E100">A81</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<a NAME="E12:48_1_1_1_1_5_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A82</font></span>: 
<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="c10">nx</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E92">A73</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:48_1_1_1_1_5_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A83</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:48_1_1_1_1"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E14:48_1_1_1_1_5_1_2_2_1"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:48_1_1_1_1"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E12:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E101">A82</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E15:48_1_1_1_1_5_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A84</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E5:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E76">A52</font></span></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></span>;<br/>
<a NAME="E16:48_1_1_1_1_5_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E12:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E83">A64</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E102">A83</font></span></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></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1_2_2"><span class="lab"><font color="Green" title="E91">A72</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1_2_2"><span class="lab"><font color="Green" title="E90">A71</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E92">A73</font></span></a>, <a class="txt" href="#E11:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E100">A81</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E102">A83</font></span></a>, <a class="txt" href="#E15:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E103">A84</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E93">A74</font></span></a>, <a class="txt" href="#E9:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E98">A79</font></span></a>, <a class="txt" href="#E10:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E99">A80</font></span></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></span>
;<br/>
<span class="kw">hence </span><a NAME="E18:48_1_1_1_1_5_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1_2_2"><span class="lab"><font color="Green" title="E90">A71</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E102">A83</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:48_1_1_1_1_5_1_2_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:48_1_1_1_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="E6:48_1_1_1_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A85</font></span>: 
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="E7:48_1_1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E91">A86</font></span>: 
<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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E28:48_1_1_1_1"><span class="lab"><font color="Green" title="E64">A40</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E8:48_1_1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E92">A87</font></span>: 
<font color="Maroon" title="c8">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="c8">h2r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E86">A67</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E87">A68</font></span></a>, <a class="txt" href="#E6:48_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E90">A85</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E92">A87</font></span></a>, <a class="txt" href="#E7:48_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E91">A86</font></span></a>, <a class="ref" href="finseq_7.html#T32" title="FINSEQ_7:th.32">FINSEQ_7:32</a></span>
<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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E39:48_1_1_1_1"><span class="lab"><font color="Green" title="E70">A46</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>
;<br/>
<a NAME="E11:48_1_1_1_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A88</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E12:48_1_1_1_1_5_1_2"/>
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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E11:48_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E93">A88</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E89">A70</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E13:48_1_1_1_1_5_1_2"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E87">A68</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E88">A69</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E15:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E86">A89</font></span>: 
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="c10">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#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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="c10">nx</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_5_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( <font color="Maroon" title="c10">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1 &amp; <font color="Maroon" title="c10">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#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">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="c10">nx</font></span>)</span> );<br/>
<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">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#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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="c10">nx</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48_1_1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E87">A90</font></span>: 
<font color="Maroon" title="c10">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font>
 <span class="kw">and </span><br/><a NAME="E2:48_1_1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E88">A91</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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="c10">nx</font></span>)</span></span><br/>

<a NAME="E3:48_1_1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E89">A92</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>;<br/>
<a NAME="E4:48_1_1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E90">A93</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>;<br/>
<a NAME="E5:48_1_1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E91">A94</font></span>: 
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="48_1_1_1_1_5_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="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/>

<span class="kw">assume </span><a NAME="E1:48_1_1_1_1_5_1_3_1"/><span class="lab"><font color="Green" title="E92">A95</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">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="E2:48_1_1_1_1_5_1_3_1"/>
( <font color="Maroon" title="c10">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">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="c10">nx</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_5_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c16">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="c10">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/>
<a NAME="E2:48_1_1_1_1_5_1_3_1_1"/><span class="lab"><font color="Green" title="E93">A96</font></span>: 
<font color="Maroon" title="c10">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<span class="kw">reconsider </span><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/>
<a NAME="E4:48_1_1_1_1_5_1_3_1_1"/><span class="lab"><font color="Green" title="E94">A97</font></span>: 
<font color="Maroon" title="c10">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="c10">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 class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">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="c10">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="kw">reconsider </span><font color="Maroon" title="c19">f1</font> = <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:48_1_1_1_1_5_1_3_1_1"/>
<font color="Maroon" title="c10">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">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>
 <span class="kw">and </span><br/><a NAME="E8:48_1_1_1_1_5_1_3_1_1"/><span class="lab"><font color="Green" title="E95">A98</font></span>: 
<font color="Maroon" title="c10">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">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="c10">nx</font></span>)</span></span><br/>

<a NAME="E9:48_1_1_1_1_5_1_3_1_1"/><span class="lab"><font color="Green" title="E96">A99</font></span>: 
<font color="Maroon" title="c10">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#K4" title="EUCLID_7:func.4">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E89">A92</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E95">A98</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:48_1_1_1_1_5_1_3_1_1"/><span class="lab"><font color="Green" title="E97">A100</font></span>: 
<font color="Maroon" title="c10">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="c10">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 class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<a NAME="E11:48_1_1_1_1_5_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E95">A98</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E12:48_1_1_1_1_5_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A101</font></span>: 
<font color="Maroon" title="c10">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E73">A49</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:48_1_1_1_1_5_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A102</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E14:48_1_1_1_1_5_1_3_1_1"/>
<font color="Maroon" title="c10">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:48_1_1_1_1"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E15:48_1_1_1_1_5_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A103</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E97">A100</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E16:48_1_1_1_1_5_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A104</font></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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">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="c10">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E99">A102</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E17:48_1_1_1_1_5_1_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E100">A103</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E99">A102</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E18:48_1_1_1_1_5_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A105</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E103">A106</font></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">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="c10">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:48_1_1_1_1"><span class="lab"><font color="Green" title="E70">A46</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<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="c10">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E28:48_1_1_1_1"><span class="lab"><font color="Green" title="E64">A40</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E15:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E100">A103</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E99">A102</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<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="c8">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="c10">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E95">A98</font></span></a>, <a class="txt" href="#E15:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E100">A103</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E99">A102</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E94">A97</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E93">A96</font></span></a>, <a class="ref" href="finseq_7.html#T32" title="FINSEQ_7:th.32">FINSEQ_7:32</a></span>
<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="c8">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="c10">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E15:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E100">A103</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E99">A102</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<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="c10">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E18:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E102">A105</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
;<br/>
<a NAME="E20:48_1_1_1_1_5_1_3_1_1"/>
<font color="Maroon" title="c10">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E21:48_1_1_1_1_5_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A107</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">k3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E77">A53</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">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="c10">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E12:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E98">A101</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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="c10">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E100">A103</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E99">A102</font></span></a>, <a class="txt" href="#E19:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E103">A106</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E101">A104</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E105">A108</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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="c10">nx</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">f1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c16">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="c18">f4</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E100">A103</font></span></a>, <a class="txt" href="#E9:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E96">A99</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">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="c16">f3</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c18">f4</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T19" title="RFUNCT_1:th.19">RFUNCT_1:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">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">f4</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c16">f3</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T19" title="RFUNCT_1:th.19">RFUNCT_1:19</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">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="c10">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E104">A107</font></span></a>, <a class="txt" href="#E12:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E98">A101</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>
<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#K4" title="EUCLID_7:func.4">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="c10">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="c10">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="c10">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E90">A93</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1_3_1"><span class="lab"><font color="Green" title="E92">A95</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E95">A98</font></span></a>, <a class="txt" href="#E10:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E97">A100</font></span></a>, <a class="txt" href="#E21:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E104">A107</font></span></a>, <a class="txt" href="#E12:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E98">A101</font></span></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></span>
;<br/>
<span class="kw">hence </span><a NAME="E25:48_1_1_1_1_5_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">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="c10">nx</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:48_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E105">A108</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:48_1_1_1_1_5_1_3_1"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_5_1_3_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_5_1_3_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_1_1_1_5_1_3_2_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">nx</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">nx</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:48_1_1_1_1"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_5_1_3_2_1_1"><span class="kw">case </span></a><a NAME="E1:48_1_1_1_1_5_1_3_2_1_1"/><span class="lab"><font color="Green" title="E92">A109</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">nx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">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="c10">nx</font></span>)</span></span><br/><div class="add"><a NAME="E2:48_1_1_1_1_5_1_3_2_1_1"/><span class="lab"><font color="Green" title="E93">A110</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E89">A92</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E92">A109</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<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
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E37:48_1_1_1_1"><span class="lab"><font color="Green" title="E69">A45</font></span></a>, <a class="txt" href="#E36:48_1_1_1_1"><span class="lab"><font color="Green" title="E68">A44</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E92">A109</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1_1_1"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="txt" href="#E10:48_1_1_1_1"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A111</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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="c10">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="c10">nx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E93">A110</font></span></a>, <a class="ref" href="euclidlp.html#T7" title="EUCLIDLP:th.7">EUCLIDLP:7</a></span>
<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="c10">nx</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T1" title="EUCLID_4:th.1">EUCLID_4:1</a></span>
;<br/><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E92">A109</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>
<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="c10">nx</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:48_1_1_1_1"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E92">A109</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:48_1_1_1_1_5_1_3_2_1_1"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">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="c10">nx</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E94">A111</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_5_1_3_2_1_2"><span class="kw">case </span></a><a NAME="E1:48_1_1_1_1_5_1_3_2_1_2"/><span class="lab"><font color="Green" title="E92">A112</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">nx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">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="c10">nx</font></span>)</span></span><br/><div class="add"><span class="lab"><font color="Green" title="E93">A113</font></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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">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="c10">nx</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:48_1_1_1_1"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="txt" href="#E25:48_1_1_1_1"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:48_1_1_1_1"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E37:48_1_1_1_1"><span class="lab"><font color="Green" title="E69">A45</font></span></a>, <a class="txt" href="#E36:48_1_1_1_1"><span class="lab"><font color="Green" title="E68">A44</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1_1_1"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="txt" href="#E10:48_1_1_1_1"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
;<br/><a NAME="E3:48_1_1_1_1_5_1_3_2_1_2"/><span class="lab"><font color="Green" title="E94">A114</font></span>: 
<font color="Maroon" title="c10">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="c10">nx</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E92">A112</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E4:48_1_1_1_1_5_1_3_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E92">A112</font></span></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/><a NAME="E5:48_1_1_1_1_5_1_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A115</font></span>: 
<font color="Maroon" title="c10">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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:48_1_1_1_1_5_1_3_2_1_2"/>
<font color="Maroon" title="c10">nx</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">nx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E7:48_1_1_1_1_5_1_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A116</font></span>: 
<font color="Maroon" title="c10">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="c10">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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E8:48_1_1_1_1_5_1_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A117</font></span>: 
<font color="Maroon" title="c10">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E74">A50</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E94">A114</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:48_1_1_1_1_5_1_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A118</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E90">A93</font></span></a>, <a class="txt" href="#E5:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E95">A115</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">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="c10">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E94">A114</font></span></a>, <a class="txt" href="#E5:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E95">A115</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E97">A117</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E94">A114</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E93">A113</font></span></a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A119</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E6:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E77">A53</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E74">A50</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E89">A92</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T63" title="RVSUM_1:th.63">RVSUM_1:63</a></span>
;<br/><a NAME="E12:48_1_1_1_1_5_1_3_2_1_2"/><span class="lab"><font color="Green" title="E100">A120</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">nx</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E89">A92</font></span></a>, <a class="txt" href="#E5:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E95">A115</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E97">A117</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">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="c10">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E94">A114</font></span></a>, <a class="txt" href="#E5:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E95">A115</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E97">A117</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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="c10">nx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E85">A66</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E94">A114</font></span></a>, <a class="txt" href="#E5:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E95">A115</font></span></a>, <a class="txt" href="#E7:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E96">A116</font></span></a>, <a class="txt" href="#E9:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E98">A118</font></span></a>, <a class="txt" href="#E12:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E100">A120</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E14:48_1_1_1_1_5_1_3_2_1_2"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">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#K4" title="EUCLID_7:func.4">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="c10">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="c10">nx</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E99">A119</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E7:48_1_1_1_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A121</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E8:48_1_1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E93">A122</font></span>: 
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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E7:48_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E92">A121</font></span></a>, <a class="txt" href="#E5:48_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E91">A94</font></span></a>);<br/></span>
<a NAME="E9:48_1_1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E94">A123</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c10">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="c10">nx</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E87">A90</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E10:48_1_1_1_1_5_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">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="c10">nx</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E87">A90</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:48_1_1_1_1_5_1_3"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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="c10">nx</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E88">A91</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E93">A122</font></span></a>, <a class="txt" href="#E9:48_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E94">A123</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E16:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E87">A124</font></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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">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="c10">nx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:48_1_1_1_1"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="txt" href="#E25:48_1_1_1_1"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><a NAME="E17:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E88">A125</font></span>: 
<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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">h4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_1_1"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E30:48_1_1_1_1"><span class="lab"><font color="Green" title="E65">A41</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E18:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E89">A126</font></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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><a NAME="E19:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E90">A127</font></span>: 
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#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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="48_1_1_1_1_5_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( <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#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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/>
<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48_1_1_1_1_5_1_4"/><span class="lab"><font color="Green" title="E91">A128</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:48_1_1_1_1_5_1_4"/><span class="lab"><font color="Green" title="E92">A129</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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/>

<a NAME="E3:48_1_1_1_1_5_1_4"/><span class="lab"><font color="Green" title="E93">A130</font></span>: 
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="48_1_1_1_1_5_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="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/>

<span class="kw">assume </span><a NAME="E1:48_1_1_1_1_5_1_4_1"/><span class="lab"><font color="Green" title="E94">A131</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">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="E2:48_1_1_1_1_5_1_4_1"/>
( <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#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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="48_1_1_1_1_5_1_4_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_1_1_1_1_5_1_4_1_1"/><span class="lab"><font color="Green" title="E95">A132</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E2:48_1_1_1_1_5_1_4_1_1"/><span class="lab"><font color="Green" title="E96">A133</font></span>: 
<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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<a NAME="E3:48_1_1_1_1_5_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A134</font></span>: 
<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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E95">A132</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:48_1_1_1_1_5_1_4_1_1"/><span class="lab"><font color="Green" title="E98">A135</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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="E5:48_1_1_1_1_5_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A136</font></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="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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></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></span>;<br/>
<a NAME="E6:48_1_1_1_1_5_1_4_1_1"/>
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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E7:48_1_1_1_1_5_1_4_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48_1_1_1_1"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E98">A135</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E8:48_1_1_1_1_5_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A137</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E101">A138</font></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> <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:48_1_1_1_1"><span class="lab"><font color="Green" title="E70">A46</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E28:48_1_1_1_1"><span class="lab"><font color="Green" title="E64">A40</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E98">A135</font></span></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></span>
<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="c8">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E84">A65</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E98">A135</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E97">A134</font></span></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></span>
<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="c8">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_1_1_1_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E98">A135</font></span></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></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E100">A137</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
;<br/>
<a NAME="E10:48_1_1_1_1_5_1_4_1_1"/>
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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E11:48_1_1_1_1_5_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A139</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E95">A132</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E12:48_1_1_1_1_5_1_4_1_1"/><span class="lab"><font color="Green" title="E103">A140</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E98">A135</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E96">A133</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:48_1_1_1_1_5_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A141</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E5:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E76">A52</font></span></a>, <a class="txt" href="#E11:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E102">A139</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E12:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E103">A140</font></span></a>, <a class="txt" href="#E11:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E102">A139</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E12:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E83">A64</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1_4_1"><span class="lab"><font color="Green" title="E94">A131</font></span></a>, <a class="txt" href="#E12:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E103">A140</font></span></a>, <a class="txt" href="#E11:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E102">A139</font></span></a>, <a class="txt" href="#E13:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E104">A141</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E98">A135</font></span></a>, <a class="txt" href="#E9:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E101">A138</font></span></a>, <a class="txt" href="#E5:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E99">A136</font></span></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></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:48_1_1_1_1_5_1_4_1_1"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E12:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E103">A140</font></span></a>, <a class="txt" href="#E11:48_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E102">A139</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:48_1_1_1_1_5_1_4_1"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:48_1_1_1_1_5_1_4"/><span class="lab"><font color="Green" title="E94">A142</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>;<br/>
<a NAME="E5:48_1_1_1_1_5_1_4"/><span class="lab"><font color="Green" title="E95">A143</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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="c10">nx</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E72">A48</font></span></a>, <a class="txt" href="#E15:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E86">A89</font></span></a></span>;<br/>
<a NAME="E6:48_1_1_1_1_5_1_4"/><span class="lab"><font color="Green" title="E96">A144</font></span>: 
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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E7:48_1_1_1_1_5_1_4"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E14:48_1_1_1_1"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E96">A144</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>
<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#K4" title="EUCLID_7:func.4">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="c10">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E94">A142</font></span></a>, <a class="txt" href="#E5:48_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E95">A143</font></span></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></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="c10">nx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#T6" target="_self" title="EUCLID_7:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E88">A125</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E79">A55</font></span></a>, <a class="txt" href="#E4:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E75">A51</font></span></a>, <a class="txt" href="#E18:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E89">A126</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E87">A124</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
;<br/>
<a NAME="E9:48_1_1_1_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A145</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E41:48_1_1_1_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>;<br/>
<a NAME="E10:48_1_1_1_1_5_1_4"/><span class="lab"><font color="Green" title="E98">A146</font></span>: 
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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E9:48_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E97">A145</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E93">A130</font></span></a>);<br/></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E91">A128</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
<font color="Maroon" title="c14">i</font>

;<br/>
<span class="kw">hence </span><a NAME="E12:48_1_1_1_1_5_1_4"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E92">A129</font></span></a>, <a class="txt" href="#E10:48_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E98">A146</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E20:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E91">A147</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</font></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#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</font></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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1"><span class="lab"><font color="Green" title="E46">A22</font></span></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></span>;<br/><a NAME="E21:48_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E92">A148</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E22:48_1_1_1_1_5_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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="E23:48_1_1_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A149</font></span>: 
 <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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">h4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/><a NAME="E24:48_1_1_1_1_5_1"/><span class="kw">then </span>
 <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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">h4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E25:48_1_1_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A150</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E21:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E92">A148</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E26:48_1_1_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E92">A148</font></span></a>, <a class="txt" href="#E23:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E93">A149</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E27:48_1_1_1_1_5_1"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E28:48_1_1_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A151</font></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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><a NAME="E29:48_1_1_1_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#K4" title="EUCLID_7:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>;<br/><a NAME="E30:48_1_1_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A152</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E94">A150</font></span></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></span>;<br/><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</font></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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E7:48_1_1_1_1"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="txt" href="#E9:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E80">A56</font></span></a>, <a class="txt" href="#E30:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E96">A152</font></span></a>, <a class="txt" href="#E20:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E91">A147</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E88">A125</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E79">A55</font></span></a>, <a class="txt" href="#E28:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E95">A151</font></span></a>, <a class="txt" href="#E10:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E81">A62</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E82">A63</font></span></a>, <a class="txt" href="#E25:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E94">A150</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>
;<br/><span class="kw">hence </span><a NAME="E32:48_1_1_1_1_5_1"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E19:48_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E90">A127</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_5_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_5_2"/>
<font color="Maroon" title="c10">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="E2:48_1_1_1_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A153</font></span>: 
<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="c10">nx</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_5_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_1_1_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="c10">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="c10">nx</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E72">A153</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_5_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E73">A154</font></span>: 
<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="c10">nx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="E2:48_1_1_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E74">A155</font></span>: 
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="48_1_1_1_1_5_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48_1_1_1_1_5_2_1_1_1"/><span class="lab"><font color="Green" title="E75">A156</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:48_1_1_1_1_5_2_1_1_1"/><span class="lab"><font color="Green" title="E76">A157</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E3:48_1_1_1_1_5_2_1_1_1"/><span class="lab"><font color="Green" title="E77">A158</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_1_5_2_1_1_1"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_5_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_1_1_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1_5_2_1_1_1"><span class="lab"><font color="Green" title="E77">A158</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_5_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">hence </span><a NAME="E2:48_1_1_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_1_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_2_1_1_1"><span class="lab"><font color="Green" title="E76">A157</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_5_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">hence </span><a NAME="E2:48_1_1_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:48_1_1_1_1"><span class="lab"><font color="Green" title="E61">A37</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E73">A154</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E3:48_1_1_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E75">A159</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E14:48_1_1_1_1"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E4:48_1_1_1_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#K4" title="EUCLID_7:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>;<br/><a NAME="E5:48_1_1_1_1_5_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A160</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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#K4" title="EUCLID_7:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></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></span>;<br/><a NAME="E6:48_1_1_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E77">A161</font></span>: 
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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E7:48_1_1_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E78">A162</font></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"><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="c10">nx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:48_1_1_1_1"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E23:48_1_1_1_1"><span class="lab"><font color="Green" title="E61">A37</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><a NAME="E8:48_1_1_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E79">A163</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</font></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#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</font></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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1"><span class="lab"><font color="Green" title="E46">A22</font></span></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></span>;<br/><a NAME="E9:48_1_1_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E80">A164</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</font></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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E7:48_1_1_1_1"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="txt" href="#E2:48_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E74">A155</font></span></a>, <a class="txt" href="#E5:48_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E76">A160</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E79">A163</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E8:48_1_1_1_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E16:48_1_1_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E9:48_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E80">A164</font></span></a>, <a class="txt" href="#E7:48_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E78">A162</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E77">A161</font></span></a>, <a class="txt" href="#E3:48_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E75">A159</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>
;<br/><span class="kw">hence </span><a NAME="E11:48_1_1_1_1_5_2_1_1"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1_5_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_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="c10">nx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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"><span class="kw">hence </span><a NAME="E2:48_1_1_1_1_5_2_1_2"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_1_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E23:48_1_1_1_1"><span class="lab"><font color="Green" title="E61">A37</font></span></a>, <a class="txt" href="#E25:48_1_1_1_1"><span class="lab"><font color="Green" title="E62">A38</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:48_1_1_1"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E18:48_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 <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span>  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/><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</font></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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="48_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">h2</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">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 <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</font></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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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="E1:48_1_1_2"/><span class="lab"><font color="Green" title="E43">A165</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E2:48_1_1_2"/><span class="lab"><font color="Green" title="E44">A166</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:48_1_1_2"/><span class="lab"><font color="Green" title="E45">A167</font></span>: 
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</font></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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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/>

<div><span class="lab"><font color="Green" title="E46">A168</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_1_1_2_1"/><span class="lab"><font color="Green" title="E47">A169</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:48_1_1_2_1"/><span class="lab"><font color="Green" title="E48">A170</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">h2</font>
 <span class="kw">and </span><br/><a NAME="E4:48_1_1_2_1"/><span class="lab"><font color="Green" title="E49">A171</font></span>: 
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E1:48_1_1_2"><span class="lab"><font color="Green" title="E43">A165</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2"><span class="lab"><font color="Green" title="E44">A166</font></span></a>, <a class="txt" href="#E3:48_1_1_2_1"><span class="lab"><font color="Green" title="E48">A170</font></span></a></span>;<br/><a NAME="E6:48_1_1_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">nx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E3:48_1_1_2_1"><span class="lab"><font color="Green" title="E48">A170</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E7:48_1_1_2_1"/><span class="kw">then </span>
( <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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E3:48_1_1_2_1"><span class="lab"><font color="Green" title="E48">A170</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:48_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_2"><span class="lab"><font color="Green" title="E45">A167</font></span></a>, <a class="txt" href="#E1:48_1_1_2_1"><span class="lab"><font color="Green" title="E47">A169</font></span></a>, <a class="txt" href="#E4:48_1_1_2_1"><span class="lab"><font color="Green" title="E49">A171</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E2:48_1_1_2"><span class="lab"><font color="Green" title="E44">A166</font></span></a>, <a class="txt" href="#E4:48_1_1_2"><span class="lab"><font color="Green" title="E46">A168</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:48_1_1_2"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</font></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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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 class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E19:48_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A172</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/><a NAME="E20:48_1_1"/>
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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E19:48_1_1"><span class="lab"><font color="Green" title="E43">A172</font></span></a>, <a class="txt" href="#E17:48_1_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>);<br/></span><span class="kw">then </span><span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <font color="Maroon" title="c2">F</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="c2">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:48_1_1"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="txt" href="#E15:48_1_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E8:48_1_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c3">F2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:48_1_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def10</a></span>
;<br/><span class="kw">hence </span><a NAME="E22:48_1_1"/>
 <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_2"/><span class="lab"><font color="Green" title="E28">A173</font></span>: 
 <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c2">F</font></span><br/><div class="add"><a NAME="E2:48_1_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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="E3:48_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A174</font></span>: 
 <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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/><a NAME="E4:48_1_2"/>
 <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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2"><span class="lab"><font color="Green" title="E28">A173</font></span></a></span>;<br/><a NAME="E5:48_1_2"/><span class="kw">then </span>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E6:48_1_2"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:48_1_2"><span class="lab"><font color="Green" title="E29">A174</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E7:48_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A175</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E8:48_1_2"/>
 <a href="euclid_7.html#K5" title="EUCLID_7:func.5">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2"><span class="lab"><font color="Green" title="E28">A173</font></span></a>, <a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E9:48_1_2"/>
 <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2"><span class="lab"><font color="Green" title="E28">A173</font></span></a>, <a class="txt" href="#E7:48_1_2"><span class="lab"><font color="Green" title="E30">A175</font></span></a>, <a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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