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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E1:45"/><span class="lab"><font color="Green" title="E23">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="45_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:45_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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_1_1"/><span class="lab"><font color="Green" title="E24">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#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:45_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:45_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A4</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI: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#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E4:45_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A5</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E5:45_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E6:45_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" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/><a NAME="E7:45_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">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="#E1:45"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/><a NAME="E8:45_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">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="#E5:45_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">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="E9:45_1_1"/><span class="lab"><font color="Green" title="E31">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" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>;<br/><a NAME="E10:45_1_1"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="funct_1.html#K1" title="FUNCT_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="E11:45_1_1"/><span class="lab"><font color="Green" title="E33">A11</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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E12:45_1_1"/>
1 <a href="tarski.html#R2" title="TARSKI: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="#E4:45_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E5:45_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E7:45_1_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E13:45_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A12</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#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E14:45_1_1"/><span class="lab"><font color="Green" title="E35">A13</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="xreal_1.html#T233" onmouseover="rs('xreal_1/T233')" onmouseout="rh()">XREAL_1:233</a></span>;<br/><a NAME="E15:45_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A14</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#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="c4">h</font> <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> <font color="Olive" title="b1">i</font>
 ;<br/><a NAME="E16:45_1_1"/><span class="lab"><font color="Green" title="E37">A15</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="#E3:45_1_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E17:45_1_1"/><span class="lab"><font color="Green" title="E38">A16</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E1:45_1_1_1"/><span class="lab"><font color="Green" title="E39">A17</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:45_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#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="funct_1.html#K1" title="FUNCT_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#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"><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:45_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A18</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" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E2:45_1_1_1_1"/><span class="lab"><font color="Green" title="E41">A19</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#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#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="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> <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#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"><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:45_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A20</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="E4:45_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A21</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#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="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> <font color="Olive" title="b1">i</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="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#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"><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/>

<a NAME="E5:45_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A22</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#K1" title="NAT_1:func.1">+</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="#E3:45_1_1_1_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E6:45_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#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/>
<a NAME="E7:45_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A23</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="#E3:45_1_1_1_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:45_1_1_1_1"/><span class="lab"><font color="Green" title="E46">A24</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#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/>
<a NAME="E9:45_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A25</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#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI: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="#E5:45_1_1_1_1"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E10:45_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A26</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#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E11:45_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A27</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="#E4:45_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E12:45_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A28</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" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E51">A29</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#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</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="#E4:45_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
;<br/>
<a NAME="E14:45_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A30</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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E15:45_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A31</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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E16:45_1_1_1_1"/><span class="lab"><font color="Green" title="E54">A32</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="#E14:45_1_1_1_1"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E17:45_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="#E11:45_1_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E11:45_1_1_1_1"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="ref" href="finseq_1.html#D2" onmouseover="rs('finseq_1/D2')" onmouseout="rh()">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#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_1_1_1_1"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E12:45_1_1_1_1"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E19:45_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="#E1:45_1_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E11:45_1_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E20:45_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A33</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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">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">object</a> <span class="kw"> such that </span><br/><a NAME="E22:45_1_1_1_1"/><span class="lab"><font color="Green" title="E56">A34</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="E23:45_1_1_1_1"/><span class="lab"><font color="Green" title="E57">A35</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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_1_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E11:45_1_1_1_1"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E12:45_1_1_1_1"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="txt" href="#E9:45_1_1_1_1"><span class="lab"><font color="Green" title="E47">A25</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_1_1_1_1"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E22:45_1_1_1_1"><span class="lab"><font color="Green" title="E56">A34</font></span></a></span>;<br/>
<a NAME="E25:45_1_1_1_1"/><span class="lab"><font color="Green" title="E58">A36</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="#E11:45_1_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E22:45_1_1_1_1"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<span class="lab"><font color="Green" title="E59">A37</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#T22" onmouseover="rs('finseq_7/T22')" onmouseout="rh()">FINSEQ_7:22</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" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>
;<br/>
<a NAME="E28:45_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A38</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#T18" onmouseover="rs('finseq_7/T18')" onmouseout="rh()">FINSEQ_7:18</a></span>;<br/>
<a NAME="E29:45_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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E61">A39</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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">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="#E16:45_1_1"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">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="#E27:45_1_1_1_1"><span class="lab"><font color="Green" title="E59">A37</font></span></a>, <a class="ref" href="funct_2.html#T1" onmouseover="rs('funct_2/T1')" onmouseout="rh()">FUNCT_2:1</a></span>;<br/>
<a NAME="E32:45_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A40</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" onmouseover="rs('int_5/T39')" onmouseout="rh()">INT_5:39</a></span>;<br/>
<a NAME="E33:45_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="#E27:45_1_1_1_1"><span class="lab"><font color="Green" title="E59">A37</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">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="#E32:45_1_1_1_1"><span class="lab"><font color="Green" title="E62">A40</font></span></a></span>;<br/>
<a NAME="E35:45_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A41</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="#E11:45_1_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E22:45_1_1_1_1"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E36:45_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A42</font></span>: 
<font color="Maroon" title="c13">h4</font> <a href="funct_1.html#K1" title="FUNCT_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="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_1_1_1_1"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="txt" href="#E25:45_1_1_1_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="txt" href="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="euclid_7.html#T10" target="_self" onmouseover="rs('euclid_7/T10')" onmouseout="rh()">Th9</a></span>;<br/>
<a NAME="E37:45_1_1_1_1"/><span class="lab"><font color="Green" title="E65">A43</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="#E4:45_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E38:45_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="E66">A44</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#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</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="#E4:45_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
;<br/>
<a NAME="E40:45_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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E41:45_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A45</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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_5"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:45_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="45_1_1_1_1_5_1"><span class="kw">suppose </span></a><a NAME="E1:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E68">A46</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="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#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"><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E69">A47</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#K1" title="NAT_1:func.1">+</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="#E3:45_1_1_1_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="txt" href="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E70">A48</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="#E11:45_1_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E22:45_1_1_1_1"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="txt" href="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E4:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E71">A49</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="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="#E22:45_1_1_1_1"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="txt" href="#E23:45_1_1_1_1"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>;<br/><a NAME="E5:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E72">A50</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" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>;<br/><a NAME="E6:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E73">A51</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="#E11:45_1_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E22:45_1_1_1_1"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E7:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E74">A52</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#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E8:45_1_1_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A53</font></span>: 
<font color="Maroon" title="c11">h2b</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:45_1_1_1_1"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E6:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E73">A51</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E70">A48</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E69">A47</font></span></a>, <a class="ref" href="euclid_7.html#T10" target="_self" onmouseover="rs('euclid_7/T10')" onmouseout="rh()">Th9</a></span>;<br/><a NAME="E9:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E76">A54</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="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> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="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> <font color="Maroon" title="c14">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45_1_1_1_1_5_1_1"/><span class="lab"><font color="Green" title="E77">A55</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="E2:45_1_1_1_1_5_1_1"/><span class="lab"><font color="Green" title="E78">A56</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">h4</font> <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> <font color="Maroon" title="c14">i</font></span><br/>

<a NAME="E3:45_1_1_1_1_5_1_1"/><span class="lab"><font color="Green" title="E79">A57</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#K1" title="NAT_1:func.1">+</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="#E1:45_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E77">A55</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_5_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:45_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#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> 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#K1" title="NAT_1:func.1">+</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="45_1_1_1_1_5_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_1_1_1_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E80">A58</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#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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">h4</font> <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> <font color="Maroon" title="c14">i</font></span><br/><div class="add"><a NAME="E2:45_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" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E3:45_1_1_1_1_5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A59</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="#E1:45_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E77">A55</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c13">h4</font> <a href="funct_1.html#K1" title="FUNCT_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="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E28:45_1_1_1_1"><span class="lab"><font color="Green" title="E60">A38</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E78">A56</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</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="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E68">A46</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E77">A55</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E78">A56</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E80">A58</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E81">A59</font></span></a>, <a class="ref" href="finseq_7.html#T30" onmouseover="rs('finseq_7/T30')" onmouseout="rh()">FINSEQ_7:30</a></span>
<br/>.= 
<font color="Maroon" title="c7">h2</font> <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="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E78">A56</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E81">A59</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a></span>
<br/>.= 
<font color="Maroon" title="c14">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1_1_1_1"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E78">A56</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E80">A58</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E5:45_1_1_1_1_5_1_1_1_1"/>
<font color="Maroon" title="c13">h4</font> <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> <font color="Maroon" title="c14">i</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="45_1_1_1_1_5_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">h4</font> <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> <font color="Maroon" title="c14">i</font></span><br/><div class="add"><a NAME="E2:45_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#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E79">A57</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:45_1_1_1_1_5_1_1_1_2"/>
<font color="Maroon" title="c13">h4</font> <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> <font color="Maroon" title="c14">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:45_1_1_1_1"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E6:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E73">A51</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E70">A48</font></span></a>, <a class="txt" href="#E7:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E74">A52</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E69">A47</font></span></a>, <a class="ref" href="euclid_7.html#T10" target="_self" onmouseover="rs('euclid_7/T10')" onmouseout="rh()">Th9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E10:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E77">A60</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_1_1_1_1"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a></span>;<br/><a NAME="E11:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E78">A61</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" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E12:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E79">A62</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" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>;<br/><a NAME="E13:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E80">A63</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#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1_1_1"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E68">A46</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E14:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E81">A64</font></span>: 
 for <font color="Olive" title="b1">i4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="xcmplx_0.html#K2" title="XCMPLX_0: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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</span>)</span> );<br/>
<span class="kw">let </span><font color="Maroon" title="c14">i4</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E1:45_1_1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E82">A65</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="E2:45_1_1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E83">A66</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="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="E84">A67</font></span>: 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E82">A65</font></span></a>, <a class="ref" href="xreal_1.html#T233" onmouseover="rs('xreal_1/T233')" onmouseout="rh()">XREAL_1:233</a></span>
<br/>.= 
<font color="Maroon" title="c14">i4</font>

;<br/>
<a NAME="E4:45_1_1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E85">A68</font></span>: 
 for <font color="Olive" title="b1">k3</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E1:45_1_1_1_1_5_1_2_2"/><span class="lab"><font color="Green" title="E86">A69</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" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:45_1_1_1_1_5_1_2_2"/><span class="lab"><font color="Green" title="E87">A70</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E3:45_1_1_1_1_5_1_2_2"/>
( 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_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="E1:45_1_1_1_1_5_1_2_2_1"/><span class="lab"><font color="Green" title="E88">A71</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="E2:45_1_1_1_1_5_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A72</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="#E25:45_1_1_1_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E3:45_1_1_1_1_5_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A73</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h2r</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="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E4:45_1_1_1_1_5_1_2_2_1"/><span class="lab"><font color="Green" title="E91">A74</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#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#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E80">A63</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E88">A71</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:45_1_1_1_1_5_1_2_2_1"/><span class="lab"><font color="Green" title="E92">A75</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">h2b</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="c15">k3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E28:45_1_1_1_1"><span class="lab"><font color="Green" title="E60">A38</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E89">A72</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E6:45_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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E7:45_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#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI: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="#E2:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E89">A72</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E8:45_1_1_1_1_5_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">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#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI: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="#E13:45_1_1_1_1"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E94">A77</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#K1" title="NAT_1:func.1">+</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="funct_1.html#K1" title="FUNCT_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</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="funct_1.html#K1" title="FUNCT_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E88">A71</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E89">A72</font></span></a>, <a class="txt" href="#E4:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E91">A74</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E90">A73</font></span></a>, <a class="txt" href="#E5:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E92">A75</font></span></a>, <a class="ref" href="finseq_7.html#T30" onmouseover="rs('finseq_7/T30')" onmouseout="rh()">FINSEQ_7:30</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:45_1_1_1_1"><span class="lab"><font color="Green" title="E66">A44</font></span></a>, <a class="txt" href="#E13:45_1_1_1_1"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E93">A76</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
;<br/>
<a NAME="E10:45_1_1_1_1_5_1_2_2_1"/><span class="lab"><font color="Green" title="E95">A78</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E89">A72</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E11:45_1_1_1_1_5_1_2_2_1"/><span class="lab"><font color="Green" title="E96">A79</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#K1" title="NAT_1:func.1">+</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#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/>
<a NAME="E12:45_1_1_1_1_5_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A80</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="#E1:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E88">A71</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:45_1_1_1_1_5_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">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>  <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="#E25:45_1_1_1_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E14:45_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="#E25:45_1_1_1_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E12:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E97">A80</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E15:45_1_1_1_1_5_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A82</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="xcmplx_0.html#K2" title="XCMPLX_0: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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E5:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E72">A50</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E16:45_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="xcmplx_0.html#K2" title="XCMPLX_0: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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E12:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E79">A62</font></span></a>, <a class="txt" href="#E13:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E98">A81</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1_2_2"><span class="lab"><font color="Green" title="E87">A70</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1_2_2"><span class="lab"><font color="Green" title="E86">A69</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E88">A71</font></span></a>, <a class="txt" href="#E11:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E96">A79</font></span></a>, <a class="txt" href="#E13:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E98">A81</font></span></a>, <a class="txt" href="#E15:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E99">A82</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E89">A72</font></span></a>, <a class="txt" href="#E9:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E94">A77</font></span></a>, <a class="txt" href="#E10:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E95">A78</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E18:45_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1_2_2"><span class="lab"><font color="Green" title="E86">A69</font></span></a>, <a class="txt" href="#E13:45_1_1_1_1_5_1_2_2_1"><span class="lab"><font color="Green" title="E98">A81</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:45_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:45_1_1_1_1_5_1_2"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">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="E6:45_1_1_1_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A83</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#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E7:45_1_1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E87">A84</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E28:45_1_1_1_1"><span class="lab"><font color="Green" title="E60">A38</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a></span>;<br/>
<a NAME="E8:45_1_1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E88">A85</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E13:45_1_1_1_1"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E82">A65</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E83">A66</font></span></a>, <a class="txt" href="#E6:45_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E86">A83</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E88">A85</font></span></a>, <a class="txt" href="#E7:45_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E87">A84</font></span></a>, <a class="ref" href="finseq_7.html#T30" onmouseover="rs('finseq_7/T30')" onmouseout="rh()">FINSEQ_7:30</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="#E4:45_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E39:45_1_1_1_1"><span class="lab"><font color="Green" title="E66">A44</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>
;<br/>
<a NAME="E11:45_1_1_1_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A86</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E12:45_1_1_1_1_5_1_2"/>
 for <font color="Olive" title="b1">k3</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E11:45_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E89">A86</font></span></a>, <a class="txt" href="#E4:45_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E85">A68</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E13:45_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="#E2:45_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E83">A66</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1_2"><span class="lab"><font color="Green" title="E84">A67</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E15:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E82">A87</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1 &amp; <font color="Maroon" title="c10">nx</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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#K1" title="NAT_1:func.1">+</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="xcmplx_0.html#K2" title="XCMPLX_0: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="xcmplx_0.html#K2" title="XCMPLX_0: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#K1" title="NAT_1:func.1">+</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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="E1:45_1_1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E83">A88</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="E2:45_1_1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E84">A89</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#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="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#K1" title="NAT_1:func.1">+</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="E3:45_1_1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E85">A90</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" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E4:45_1_1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E86">A91</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" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E5:45_1_1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E87">A92</font></span>: 
 for <font color="Olive" title="b1">k3</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E1:45_1_1_1_1_5_1_3_1"/><span class="lab"><font color="Green" title="E88">A93</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:45_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E2:45_1_1_1_1_5_1_3_1_1"/><span class="lab"><font color="Green" title="E89">A94</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="xcmplx_0.html#K2" title="XCMPLX_0: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" onmouseover="rs('nat_1/T11')" onmouseout="rh()">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#K1" title="NAT_1:func.1">+</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="E4:45_1_1_1_1_5_1_3_1_1"/><span class="lab"><font color="Green" title="E90">A95</font></span>: 
<font color="Maroon" title="c10">nx</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</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="xcmplx_0.html#K2" title="XCMPLX_0: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="E7:45_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#K1" title="NAT_1:func.1">+</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="E8:45_1_1_1_1_5_1_3_1_1"/><span class="lab"><font color="Green" title="E91">A96</font></span>: 
<font color="Maroon" title="c10">nx</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="E9:45_1_1_1_1_5_1_3_1_1"/><span class="lab"><font color="Green" title="E92">A97</font></span>: 
<font color="Maroon" title="c10">nx</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="#E5:45_1_1_1_1"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E85">A90</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E91">A96</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:45_1_1_1_1_5_1_3_1_1"/><span class="lab"><font color="Green" title="E93">A98</font></span>: 
<font color="Maroon" title="c10">nx</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/>
<a NAME="E11:45_1_1_1_1_5_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">nx</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E91">A96</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E12:45_1_1_1_1_5_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A99</font></span>: 
<font color="Maroon" title="c10">nx</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E69">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:45_1_1_1_1_5_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A100</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">nx</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E14:45_1_1_1_1_5_1_3_1_1"/>
<font color="Maroon" title="c10">nx</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:45_1_1_1_1"><span class="lab"><font color="Green" title="E63">A41</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E15:45_1_1_1_1_5_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A101</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#K1" title="NAT_1:func.1">+</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="#E10:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E93">A98</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E16:45_1_1_1_1_5_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A102</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E13:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E95">A100</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a></span>;<br/>
<a NAME="E17:45_1_1_1_1_5_1_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">nx</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI: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="#E15:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E96">A101</font></span></a>, <a class="txt" href="#E13:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E95">A100</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E18:45_1_1_1_1_5_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A103</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">nx</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E99">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="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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="funct_1.html#K1" title="FUNCT_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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E39:45_1_1_1_1"><span class="lab"><font color="Green" title="E66">A44</font></span></a>, <a class="txt" href="#E13:45_1_1_1_1"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E28:45_1_1_1_1"><span class="lab"><font color="Green" title="E60">A38</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E15:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E96">A101</font></span></a>, <a class="txt" href="#E13:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E95">A100</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E91">A96</font></span></a>, <a class="txt" href="#E15:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E96">A101</font></span></a>, <a class="txt" href="#E13:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E95">A100</font></span></a>, <a class="txt" href="#E4:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E90">A95</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E89">A94</font></span></a>, <a class="ref" href="finseq_7.html#T30" onmouseover="rs('finseq_7/T30')" onmouseout="rh()">FINSEQ_7:30</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="funct_1.html#K1" title="FUNCT_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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E15:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E96">A101</font></span></a>, <a class="txt" href="#E13:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E95">A100</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E18:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E98">A103</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
;<br/>
<a NAME="E20:45_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="xcmplx_0.html#K2" title="XCMPLX_0: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" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E21:45_1_1_1_1_5_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A105</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E73">A51</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">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#K1" title="NAT_1:func.1">+</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="xcmplx_0.html#K2" title="XCMPLX_0: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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E12:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E94">A99</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</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="xcmplx_0.html#K2" title="XCMPLX_0: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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E15:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E96">A101</font></span></a>, <a class="txt" href="#E13:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E95">A100</font></span></a>, <a class="txt" href="#E19:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E99">A104</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E97">A102</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E101">A106</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="#E15:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E96">A101</font></span></a>, <a class="txt" href="#E9:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E92">A97</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</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#T8" onmouseover="rs('rfunct_1/T8')" onmouseout="rh()">RFUNCT_1:8</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#T8" onmouseover="rs('rfunct_1/T8')" onmouseout="rh()">RFUNCT_1:8</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#K1" title="NAT_1:func.1">+</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="xcmplx_0.html#K2" title="XCMPLX_0: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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E21:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E100">A105</font></span></a>, <a class="txt" href="#E12:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E94">A99</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</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="xcmplx_0.html#K2" title="XCMPLX_0: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#K1" title="NAT_1:func.1">+</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E4:45_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E86">A91</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1_3_1"><span class="lab"><font color="Green" title="E88">A93</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E91">A96</font></span></a>, <a class="txt" href="#E10:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E93">A98</font></span></a>, <a class="txt" href="#E21:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E100">A105</font></span></a>, <a class="txt" href="#E12:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E94">A99</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E25:45_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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="#E23:45_1_1_1_1_5_1_3_1_1"><span class="lab"><font color="Green" title="E101">A106</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:45_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_1_5_1_3_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">nx</font> &amp; <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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</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> ) or ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">nx</font> &amp; <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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</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></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_5_1_3_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:45_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="#E35:45_1_1_1_1"><span class="lab"><font color="Green" title="E63">A41</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_5_1_3_2_1_1"><span class="kw">case </span></a><a NAME="E1:45_1_1_1_1_5_1_3_2_1_1"/><span class="lab"><font color="Green" title="E88">A107</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">nx</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</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="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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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="#E2:45_1_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E85">A90</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E88">A107</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</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" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</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="funct_1.html#K1" title="FUNCT_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E37:45_1_1_1_1"><span class="lab"><font color="Green" title="E65">A43</font></span></a>, <a class="txt" href="#E36:45_1_1_1_1"><span class="lab"><font color="Green" title="E64">A42</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E88">A107</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_1_1_1_1"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E10:45_1_1_1_1"><span class="lab"><font color="Green" title="E48">A26</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_1_1_1_1"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</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="ref" href="euclidlp.html#T2" onmouseover="rs('euclidlp/T2')" onmouseout="rh()">EUCLIDLP:2</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" onmouseover="rs('euclid_4/T1')" onmouseout="rh()">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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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="#E1:45_1_1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E88">A107</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</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="#E25:45_1_1_1_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E88">A107</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:45_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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</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="#E3:45_1_1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E89">A108</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="45_1_1_1_1_5_1_3_2_1_2"><span class="kw">case </span></a><a NAME="E1:45_1_1_1_1_5_1_3_2_1_2"/><span class="lab"><font color="Green" title="E88">A109</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</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="E89">A110</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="#E35:45_1_1_1_1"><span class="lab"><font color="Green" title="E63">A41</font></span></a>, <a class="txt" href="#E25:45_1_1_1_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="txt" href="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</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="funct_1.html#K1" title="FUNCT_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:45_1_1_1_1"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="txt" href="#E37:45_1_1_1_1"><span class="lab"><font color="Green" title="E65">A43</font></span></a>, <a class="txt" href="#E36:45_1_1_1_1"><span class="lab"><font color="Green" title="E64">A42</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_1_1_1_1"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E10:45_1_1_1_1"><span class="lab"><font color="Green" title="E48">A26</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_1_1_1_1"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a></span>
;<br/><a NAME="E3:45_1_1_1_1_5_1_3_2_1_2"/><span class="lab"><font color="Green" title="E90">A111</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E88">A109</font></span></a>, <a class="ref" href="xreal_1.html#T233" onmouseover="rs('xreal_1/T233')" onmouseout="rh()">XREAL_1:233</a></span>;<br/><a NAME="E4:45_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#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E88">A109</font></span></a>, <a class="ref" href="xreal_1.html#T50" onmouseover="rs('xreal_1/T50')" onmouseout="rh()">XREAL_1:50</a></span>;<br/><a NAME="E5:45_1_1_1_1_5_1_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A112</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#K5" title="NUMBERS:func.5">0</a> <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#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E6:45_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#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><a NAME="E7:45_1_1_1_1_5_1_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A113</font></span>: 
<font color="Maroon" title="c10">nx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E8:45_1_1_1_1_5_1_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">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="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="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E70">A48</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E90">A111</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E9:45_1_1_1_1_5_1_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A115</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="#E4:45_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E86">A91</font></span></a>, <a class="txt" href="#E5:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E91">A112</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E90">A111</font></span></a>, <a class="txt" href="#E5:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E91">A112</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E93">A114</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E90">A111</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E89">A110</font></span></a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A116</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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</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#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="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="txt" href="#E6:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E73">A51</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E70">A48</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E85">A90</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</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#T42" onmouseover="rs('rvsum_1/T42')" onmouseout="rh()">RVSUM_1:42</a></span>
;<br/><a NAME="E12:45_1_1_1_1_5_1_3_2_1_2"/><span class="lab"><font color="Green" title="E96">A117</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="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E85">A90</font></span></a>, <a class="txt" href="#E5:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E91">A112</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E93">A114</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E90">A111</font></span></a>, <a class="txt" href="#E5:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E91">A112</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E93">A114</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</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="#E14:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E81">A64</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E90">A111</font></span></a>, <a class="txt" href="#E5:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E91">A112</font></span></a>, <a class="txt" href="#E7:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E92">A113</font></span></a>, <a class="txt" href="#E9:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E94">A115</font></span></a>, <a class="txt" href="#E12:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E96">A117</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E14:45_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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#K1" title="NAT_1:func.1">+</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="#E11:45_1_1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E95">A116</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E7:45_1_1_1_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A118</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E8:45_1_1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E89">A119</font></span>: 
 for <font color="Olive" title="b1">k3</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E7:45_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E88">A118</font></span></a>, <a class="txt" href="#E5:45_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E87">A92</font></span></a>);<br/></span>
<a NAME="E9:45_1_1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E90">A120</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">nx</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E83">A88</font></span></a>, <a class="ref" href="xreal_1.html#T233" onmouseover="rs('xreal_1/T233')" onmouseout="rh()">XREAL_1:233</a></span>;<br/>
<a NAME="E10:45_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#K1" title="NAT_1:func.1">+</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="#E1:45_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E83">A88</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:45_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#K1" title="NAT_1:func.1">+</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="#E2:45_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E84">A89</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E89">A119</font></span></a>, <a class="txt" href="#E9:45_1_1_1_1_5_1_3"><span class="lab"><font color="Green" title="E90">A120</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E16:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E83">A121</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="#E35:45_1_1_1_1"><span class="lab"><font color="Green" title="E63">A41</font></span></a>, <a class="txt" href="#E25:45_1_1_1_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a></span>;<br/><a NAME="E17:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E84">A122</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#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI: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="#E9:45_1_1_1_1"><span class="lab"><font color="Green" title="E47">A25</font></span></a>, <a class="txt" href="#E30:45_1_1_1_1"><span class="lab"><font color="Green" title="E61">A39</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E18:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E85">A123</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_1_1_1_1"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="txt" href="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a></span>;<br/><a NAME="E19:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E86">A124</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</span>)</span> );<br/>
<span class="kw">let </span><font color="Maroon" title="c14">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K1" title="NAT_1:func.1">+</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="E1:45_1_1_1_1_5_1_4"/><span class="lab"><font color="Green" title="E87">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#K1" title="NAT_1:func.1">+</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="E2:45_1_1_1_1_5_1_4"/><span class="lab"><font color="Green" title="E88">A126</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="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="E3:45_1_1_1_1_5_1_4"/><span class="lab"><font color="Green" title="E89">A127</font></span>: 
 for <font color="Olive" title="b1">k3</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E1:45_1_1_1_1_5_1_4_1"/><span class="lab"><font color="Green" title="E90">A128</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k3</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:45_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_1_5_1_4_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:45_1_1_1_1_5_1_4_1_1"/><span class="lab"><font color="Green" title="E91">A129</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E2:45_1_1_1_1_5_1_4_1_1"/><span class="lab"><font color="Green" title="E92">A130</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/>
<a NAME="E3:45_1_1_1_1_5_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A131</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#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"><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E91">A129</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:45_1_1_1_1_5_1_4_1_1"/><span class="lab"><font color="Green" title="E94">A132</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E5:45_1_1_1_1_5_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A133</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E6:45_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E7:45_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI: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="#E15:45_1_1_1_1"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="txt" href="#E4:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E94">A132</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E8:45_1_1_1_1_5_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A134</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI: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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E97">A135</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="funct_1.html#K1" title="FUNCT_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E39:45_1_1_1_1"><span class="lab"><font color="Green" title="E66">A44</font></span></a>, <a class="txt" href="#E13:45_1_1_1_1"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E28:45_1_1_1_1"><span class="lab"><font color="Green" title="E60">A38</font></span></a>, <a class="txt" href="#E4:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E94">A132</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E13:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E80">A63</font></span></a>, <a class="txt" href="#E4:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E94">A132</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E93">A131</font></span></a>, <a class="ref" href="finseq_7.html#T30" onmouseover="rs('finseq_7/T30')" onmouseout="rh()">FINSEQ_7:30</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">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="funct_1.html#K1" title="FUNCT_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E20:45_1_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E4:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E94">A132</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E8:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E96">A134</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
;<br/>
<a NAME="E10:45_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#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E11:45_1_1_1_1_5_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A136</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E91">A129</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E12:45_1_1_1_1_5_1_4_1_1"/><span class="lab"><font color="Green" title="E99">A137</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="#E4:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E94">A132</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E92">A130</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:45_1_1_1_1_5_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A138</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="txt" href="#E5:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E72">A50</font></span></a>, <a class="txt" href="#E11:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E98">A136</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E12:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E99">A137</font></span></a>, <a class="txt" href="#E11:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E98">A136</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E12:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E79">A62</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1_4_1"><span class="lab"><font color="Green" title="E90">A128</font></span></a>, <a class="txt" href="#E12:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E99">A137</font></span></a>, <a class="txt" href="#E11:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E98">A136</font></span></a>, <a class="txt" href="#E13:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E100">A138</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">k3</font></span>)</span> <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="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E4:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E94">A132</font></span></a>, <a class="txt" href="#E9:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E97">A135</font></span></a>, <a class="txt" href="#E5:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E95">A133</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:45_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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="txt" href="#E12:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E99">A137</font></span></a>, <a class="txt" href="#E11:45_1_1_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E98">A136</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:45_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:45_1_1_1_1_5_1_4"/><span class="lab"><font color="Green" title="E90">A139</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" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E5:45_1_1_1_1_5_1_4"/><span class="lab"><font color="Green" title="E91">A140</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#K1" title="NAT_1:func.1">+</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="#E2:45_1_1_1_1"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E68">A46</font></span></a>, <a class="txt" href="#E15:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E82">A87</font></span></a></span>;<br/>
<a NAME="E6:45_1_1_1_1_5_1_4"/><span class="lab"><font color="Green" title="E92">A141</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" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E7:45_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="#E3:45_1_1_1_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="txt" href="#E14:45_1_1_1_1"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E92">A141</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_1_1_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E4:45_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E90">A139</font></span></a>, <a class="txt" href="#E5:45_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E91">A140</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">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#T5" target="_self" onmouseover="rs('euclid_7/T5')" onmouseout="rh()">Th4</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E84">A122</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E75">A53</font></span></a>, <a class="txt" href="#E4:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E71">A49</font></span></a>, <a class="txt" href="#E18:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E85">A123</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E83">A121</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
;<br/>
<a NAME="E9:45_1_1_1_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A142</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_1_1_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1"><span class="lab"><font color="Green" title="E40">A18</font></span></a>, <a class="txt" href="#E41:45_1_1_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E10:45_1_1_1_1_5_1_4"/><span class="lab"><font color="Green" title="E94">A143</font></span>: 
 for <font color="Olive" title="b1">k3</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E9:45_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E93">A142</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E89">A127</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E87">A125</font></span></a>, <a class="ref" href="xreal_1.html#T233" onmouseover="rs('xreal_1/T233')" onmouseout="rh()">XREAL_1:233</a></span>
<br/>.= 
<font color="Maroon" title="c14">i</font>

;<br/>
<span class="kw">hence </span><a NAME="E12:45_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="#E2:45_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E88">A126</font></span></a>, <a class="txt" href="#E10:45_1_1_1_1_5_1_4"><span class="lab"><font color="Green" title="E94">A143</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E20:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E87">A144</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="#E9:45_1_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E21:45_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E88">A145</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="#E4:45_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E22:45_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="E23:45_1_1_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A146</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#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/><a NAME="E24:45_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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E25:45_1_1_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A147</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="#E3:45_1_1_1_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="txt" href="#E21:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E88">A145</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E26:45_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="#E21:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E88">A145</font></span></a>, <a class="txt" href="#E23:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E89">A146</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E27:45_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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E28:45_1_1_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A148</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_1_1_1_1"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a></span>;<br/><a NAME="E29:45_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" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>;<br/><a NAME="E30:45_1_1_1_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A149</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="#E25:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E90">A147</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:45_1_1"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1_1"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="txt" href="#E7:45_1_1_1_1"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="txt" href="#E9:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E76">A54</font></span></a>, <a class="txt" href="#E30:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E92">A149</font></span></a>, <a class="txt" href="#E20:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E87">A144</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E84">A122</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E75">A53</font></span></a>, <a class="txt" href="#E28:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E91">A148</font></span></a>, <a class="txt" href="#E10:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E77">A60</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</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="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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E78">A61</font></span></a>, <a class="txt" href="#E25:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E90">A147</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>
;<br/><span class="kw">hence </span><a NAME="E32:45_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#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"><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_1_1_1_1"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E19:45_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E86">A124</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="45_1_1_1_1_5_2"><span class="kw">suppose </span></a><a NAME="E1:45_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="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#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"><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:45_1_1_1_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A150</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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_5_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:45_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#K1" title="NAT_1:func.1">+</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#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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E68">A150</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_5_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_1_1_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E69">A151</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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">nx</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="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#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"><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:45_1_1_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E70">A152</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="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> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="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> <font color="Maroon" title="c14">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45_1_1_1_1_5_2_1_1_1"/><span class="lab"><font color="Green" title="E71">A153</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="E2:45_1_1_1_1_5_2_1_1_1"/><span class="lab"><font color="Green" title="E72">A154</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">h2</font> <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> <font color="Maroon" title="c14">i</font></span><br/>

<a NAME="E3:45_1_1_1_1_5_2_1_1_1"/><span class="lab"><font color="Green" title="E73">A155</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#K1" title="NAT_1:func.1">+</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="#E1:45_1_1_1_1_5_2_1_1_1"><span class="lab"><font color="Green" title="E71">A153</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_5_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:45_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#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> 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#K1" title="NAT_1:func.1">+</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="#E3:45_1_1_1_1_5_2_1_1_1"><span class="lab"><font color="Green" title="E73">A155</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_5_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_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#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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">h2</font> <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> <font color="Maroon" title="c14">i</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:45_1_1_1_1_5_2_1_1_1_1_1"/>
<font color="Maroon" title="c7">h2</font> <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> <font color="Maroon" title="c14">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1_1_1_1"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_2_1_1_1"><span class="lab"><font color="Green" title="E72">A154</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="45_1_1_1_1_5_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_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#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">h2</font> <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> <font color="Maroon" title="c14">i</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:45_1_1_1_1_5_2_1_1_1_1_2"/>
<font color="Maroon" title="c7">h2</font> <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> <font color="Maroon" title="c14">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:45_1_1_1_1"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E69">A151</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E3:45_1_1_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E71">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>  <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="#E3:45_1_1_1_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="txt" href="#E14:45_1_1_1_1"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E4:45_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" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>;<br/><a NAME="E5:45_1_1_1_1_5_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A157</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="#E3:45_1_1_1_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E6:45_1_1_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E73">A158</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" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E7:45_1_1_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E74">A159</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="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="#E22:45_1_1_1_1"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="txt" href="#E23:45_1_1_1_1"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>;<br/><a NAME="E8:45_1_1_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E75">A160</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="#E9:45_1_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E9:45_1_1_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E76">A161</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_1_1_1_1"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:45_1_1"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1_1"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="txt" href="#E7:45_1_1_1_1"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="txt" href="#E2:45_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E70">A152</font></span></a>, <a class="txt" href="#E5:45_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E72">A157</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E75">A160</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_1_1_1_1"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E8:45_1_1_1_1"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="txt" href="#E16:45_1_1_1_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E1:45_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E69">A151</font></span></a>, <a class="txt" href="#E9:45_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E76">A161</font></span></a>, <a class="txt" href="#E7:45_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E74">A159</font></span></a>, <a class="ref" href="finseq_4.html#T15" onmouseover="rs('finseq_4/T15')" onmouseout="rh()">FINSEQ_4:15</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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E73">A158</font></span></a>, <a class="txt" href="#E3:45_1_1_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E71">A156</font></span></a>, <a class="ref" href="euclid_7.html#D10" target="_self" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>
;<br/><span class="kw">hence </span><a NAME="E11:45_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#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"><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="45_1_1_1_1_5_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_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#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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="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#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"><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:45_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#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"><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#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1_1_1_1"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="txt" href="#E23:45_1_1_1_1"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E25:45_1_1_1_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E3:45_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E18:45_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="funct_1.html#K1" title="FUNCT_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#K5" title="NUMBERS:func.5">0</a> <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"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="c6">h2</font> <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> <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#K5" title="NUMBERS:func.5">0</a> <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="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#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )</span><br/>

<a NAME="E1:45_1_1_2"/><span class="lab"><font color="Green" title="E39">A162</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" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E2:45_1_1_2"/><span class="lab"><font color="Green" title="E40">A163</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="#E4:45_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:45_1_1_2"/><span class="lab"><font color="Green" title="E41">A164</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="c6">h2</font> <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> <font color="Olive" title="b1">i</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="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#K5" title="NUMBERS:func.5">0</a> <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="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#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E42">A165</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:45_1_1_2_1"/><span class="lab"><font color="Green" title="E43">A166</font></span>: 
<font color="Maroon" title="c6">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">object</a> <span class="kw"> such that </span><br/><a NAME="E3:45_1_1_2_1"/><span class="lab"><font color="Green" title="E44">A167</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="E4:45_1_1_2_1"/><span class="lab"><font color="Green" title="E45">A168</font></span>: 
<font color="Maroon" title="c6">h2</font> <a href="funct_1.html#K1" title="FUNCT_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="#E4:45_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E1:45_1_1_2"><span class="lab"><font color="Green" title="E39">A162</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1_2"><span class="lab"><font color="Green" title="E40">A163</font></span></a>, <a class="txt" href="#E3:45_1_1_2_1"><span class="lab"><font color="Green" title="E44">A167</font></span></a></span>;<br/><a NAME="E6:45_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="#E11:45_1_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E3:45_1_1_2_1"><span class="lab"><font color="Green" title="E44">A167</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E7:45_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="#E11:45_1_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E3:45_1_1_2_1"><span class="lab"><font color="Green" title="E44">A167</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:45_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_1_2"><span class="lab"><font color="Green" title="E41">A164</font></span></a>, <a class="txt" href="#E1:45_1_1_2_1"><span class="lab"><font color="Green" title="E43">A166</font></span></a>, <a class="txt" href="#E4:45_1_1_2_1"><span class="lab"><font color="Green" title="E45">A168</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</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="#E4:45_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E2:45_1_1_2"><span class="lab"><font color="Green" title="E40">A163</font></span></a>, <a class="txt" href="#E4:45_1_1_2"><span class="lab"><font color="Green" title="E42">A165</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:45_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#K5" title="NUMBERS:func.5">0</a> <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="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#K5" title="NUMBERS:func.5">0</a> <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="euclid_7.html#D10" target="_self" onmouseover="rs('euclid_7/D10')" onmouseout="rh()">Def10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E19:45_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A169</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 ;<br/><a NAME="E20:45_1_1"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E19:45_1_1"><span class="lab"><font color="Green" title="E39">A169</font></span></a>, <a class="txt" href="#E17:45_1_1"><span class="lab"><font color="Green" title="E38">A16</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="#E14:45_1_1"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E15:45_1_1"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="xreal_1.html#T44" onmouseover="rs('xreal_1/T44')" onmouseout="rh()">XREAL_1:44</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="#E1:45"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E8:45_1_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">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="#E13:45_1_1"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="ref" href="euclid_7.html#D11" target="_self" onmouseover="rs('euclid_7/D11')" onmouseout="rh()">Def11</a></span>
;<br/><span class="kw">hence </span><a NAME="E22:45_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="#E1:45_1_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="euclid_7.html#D11" target="_self" onmouseover="rs('euclid_7/D11')" onmouseout="rh()">Def11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="45_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_1_2"/><span class="lab"><font color="Green" title="E24">A170</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#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:45_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="E3:45_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A171</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#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/><a NAME="E4:45_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="#E1:45_1_2"><span class="lab"><font color="Green" title="E24">A170</font></span></a></span>;<br/><a NAME="E5:45_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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E6:45_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="#E1:45"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E3:45_1_2"><span class="lab"><font color="Green" title="E25">A171</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E7:45_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A172</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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E8:45_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="#E1:45_1_2"><span class="lab"><font color="Green" title="E24">A170</font></span></a>, <a class="ref" href="euclid_7.html#D11" target="_self" onmouseover="rs('euclid_7/D11')" onmouseout="rh()">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E9:45_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="#E1:45_1_2"><span class="lab"><font color="Green" title="E24">A170</font></span></a>, <a class="txt" href="#E7:45_1_2"><span class="lab"><font color="Green" title="E26">A172</font></span></a>, <a class="ref" href="euclid_7.html#D11" target="_self" onmouseover="rs('euclid_7/D11')" onmouseout="rh()">Def11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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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