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<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">f</font> being   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><br/>  for <font color="Olive" title="b2">o</font> being    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">f</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Olive" title="b2">o</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:87"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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">o</font> being    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Olive" title="b1">o</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">o</font> be    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:87_1"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">fo</font> = <font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">S</font> =  <a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span>;<br/>
<a NAME="E2:87_1"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</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> <span class="p2">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span></span>)</span></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="E3:87_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">xx</font> = <font color="Maroon" title="c5">x</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:87_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span></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"> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<a NAME="E3:87_1_1"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#D8" target="_self" onmouseover="rs('flexary1/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<a NAME="E4:87_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</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> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:87_1_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:87_1_1_1"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E4:87_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E5:87_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A6</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/>
<a NAME="E6:87_1_1_1"/>
<font color="Maroon" title="c1">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="txt" href="#E1:87_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:87_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:87_1_1_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E3:87_1_1_1"><span class="lab"><font color="Green" title="E50">A5</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="E5:87_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>
 
;<br/>
<span class="kw">hence </span><a NAME="E6:87_1_1"/>
<span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T72" onmouseover="rs('rvsum_1/T72')" onmouseout="rh()">RVSUM_1:72</a>, <a class="txt" href="#E3:87_1_1"><span class="lab"><font color="Green" title="E49">A4</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="E4:87_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A7</font></span>: 
 <a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span> <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> <span class="p2">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&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="#E2:87_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a></span>;<br/>
<a NAME="E5:87_1"/>
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:87_1"/>
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T72" onmouseover="rs('rvsum_1/T72')" onmouseout="rh()">RVSUM_1:72</a>, <a class="txt" href="#E4:87_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="rvsum_1.html#T81" onmouseover="rs('rvsum_1/T81')" onmouseout="rh()">RVSUM_1:81</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="E2:87"/><span class="lab"><font color="Green" title="E47">A8</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">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</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="87_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">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">S<sub>1</sub></font>[<font color="Maroon" title="c1">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_2"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">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">S<sub>1</sub></font>[<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">i1</font> = <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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">o</font> being    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Olive" title="b1">o</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">o</font> be    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E2:87_2"/><span class="lab"><font color="Green" title="E49">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</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">  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">fo</font> = <font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font>;<br/>
<a NAME="E3:87_2"/><span class="lab"><font color="Green" title="E50">A11</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</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="E4:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A12</font></span>: 
<font color="Maroon" title="c1">i</font> <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="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87_2"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E5:87_2"/>
 <a href="matrix_0.html#K9" title="MATRIX_0:func.9">Values</a> <font color="Maroon" title="c4">o</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="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#D7" target="_self" onmouseover="rs('flexary1/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:87_2"/><span class="lab"><font color="Green" title="E52">A13</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">o</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <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="#E3:87_2"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="txt" href="#E2:87_2"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="ref" href="flexary1.html#T1" target="_self" onmouseover="rs('flexary1/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c7">y</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">ox</font> = <font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">rox</font> =  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>;<br/>
<a NAME="E9:87_2"/><span class="lab"><font color="Green" title="E53">A14</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_2"><span class="lab"><font color="Green" title="E52">A13</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/>
<a NAME="E10:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A15</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <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="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">C</font> =  <a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/>
<a NAME="E11:87_2"/><span class="lab"><font color="Green" title="E55">A16</font></span>: 
<font color="Maroon" title="c1">i</font> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_2"><span class="lab"><font color="Green" title="E52">A13</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/>
<a NAME="E12:87_2"/><span class="lab"><font color="Green" title="E56">A17</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<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="E13:87_2"/><span class="lab"><font color="Green" title="E57">A18</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T116" onmouseover="rs('zfmisc_1/T116')" onmouseout="rh()">ZFMISC_1:116</a>, <a class="txt" href="#E11:87_2"><span class="lab"><font color="Green" title="E55">A16</font></span></a></span>;<br/>
<a NAME="E14:87_2"/><span class="lab"><font color="Green" title="E58">A19</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T38" onmouseover="rs('finseq_1/T38')" onmouseout="rh()">FINSEQ_1:38</a></span>;<br/>
<a NAME="E15:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A20</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:87_2"><span class="lab"><font color="Green" title="E57">A18</font></span></a>, <a class="txt" href="#E12:87_2"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="ref" href="finseq_1.html#T31" onmouseover="rs('finseq_1/T31')" onmouseout="rh()">FINSEQ_1:31</a></span>;<br/>
<a NAME="E16:87_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> 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="xboole_1.html#T79" onmouseover="rs('xboole_1/T79')" onmouseout="rh()">XBOOLE_1:79</a>, <a class="ref" href="finseq_3.html#T91" onmouseover="rs('finseq_3/T91')" onmouseout="rh()">FINSEQ_3:91</a>, <a class="txt" href="#E12:87_2"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="txt" href="#E14:87_2"><span class="lab"><font color="Green" title="E58">A19</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">P</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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E18:87_2"/><span class="lab"><font color="Green" title="E60">A21</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:87_2"><span class="lab"><font color="Green" title="E59">A20</font></span></a>, <a class="ref" href="rfinseq.html#T26" onmouseover="rs('rfinseq/T26')" onmouseout="rh()">RFINSEQ:26</a>, <a class="ref" href="rfinseq.html#T4" onmouseover="rs('rfinseq/T4')" onmouseout="rh()">RFINSEQ:4</a></span>;<br/>
<a NAME="E19:87_2"/><span class="lab"><font color="Green" title="E61">A22</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:87_2"><span class="lab"><font color="Green" title="E56">A17</font></span></a></span>;<br/>
<a NAME="E20:87_2"/><span class="lab"><font color="Green" title="E62">A23</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:87_2"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="txt" href="#E9:87_2"><span class="lab"><font color="Green" title="E53">A14</font></span></a>, <a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a>, <a class="ref" href="xboole_1.html#T33" onmouseover="rs('xboole_1/T33')" onmouseout="rh()">XBOOLE_1:33</a></span>;<br/>
<a NAME="E21:87_2"/><span class="lab"><font color="Green" title="E63">A24</font></span>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:87_2"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="txt" href="#E10:87_2"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="ref" href="xboole_1.html#T33" onmouseover="rs('xboole_1/T33')" onmouseout="rh()">XBOOLE_1:33</a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>;<br/>
<a NAME="E22:87_2"/><span class="lab"><font color="Green" title="E64">A25</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:87_2"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="txt" href="#E12:87_2"><span class="lab"><font color="Green" title="E56">A17</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E65">A26</font></span>: <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:87_2"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T116" onmouseover="rs('zfmisc_1/T116')" onmouseout="rh()">ZFMISC_1:116</a>, <a class="txt" href="#E11:87_2"><span class="lab"><font color="Green" title="E55">A16</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T53" onmouseover="rs('xboole_1/T53')" onmouseout="rh()">XBOOLE_1:53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T24" onmouseover="rs('xboole_1/T24')" onmouseout="rh()">XBOOLE_1:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:87_2"><span class="lab"><font color="Green" title="E64">A25</font></span></a>, <a class="ref" href="xboole_1.html#T45" onmouseover="rs('xboole_1/T45')" onmouseout="rh()">XBOOLE_1:45</a>, <a class="txt" href="#E21:87_2"><span class="lab"><font color="Green" title="E63">A24</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E24:87_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">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"><font color="Maroon" title="c1">i</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="#E2:87_2"><span class="lab"><font color="Green" title="E49">A10</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="E25:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A27</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T10" onmouseover="rs('finseq_1/T10')" onmouseout="rh()">FINSEQ_1:10</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">fi</font> = <font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font>;<br/>
<a NAME="E26:87_2"/><span class="lab"><font color="Green" title="E67">A28</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="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</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>, <a class="txt" href="#E2:87_2"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/>
<a NAME="E27:87_2"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:87_2"><span class="lab"><font color="Green" title="E61">A22</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">oC</font> = <font color="Maroon" title="c4">o</font> <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>) as    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:87_2"><span class="lab"><font color="Green" title="E66">A27</font></span></a>, <a class="txt" href="#E7:87_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="ref" href="flexary1.html#T37" target="_self" onmouseover="rs('flexary1/T37')" onmouseout="rh()">Th37</a>, <a class="txt" href="#E23:87_2"><span class="lab"><font color="Green" title="E65">A26</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">FO</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font>;<br/>
<a NAME="E29:87_2"/><span class="lab"><font color="Green" title="E68">A29</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">oC</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="c4">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T30" onmouseover="rs('funct_7/T30')" onmouseout="rh()">FUNCT_7:30</a></span>;<br/>
<a NAME="E30:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A30</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">oC</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="c4">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/>
<a NAME="E31:87_2"/><span class="lab"><font color="Green" title="E70">A31</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</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="c15">oC</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">FOx</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">H</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E33:87_2"/><span class="lab"><font color="Green" title="E71">A32</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</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"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c18">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T80" onmouseover="rs('finseq_1/T80')" onmouseout="rh()">FINSEQ_1:80</a></span>;<br/>
<a NAME="E34:87_2"/><span class="lab"><font color="Green" title="E72">A33</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c8">x</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="c4">o</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E35:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A34</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a>, <a class="txt" href="#E30:87_2"><span class="lab"><font color="Green" title="E69">A30</font></span></a>, <a class="txt" href="#E31:87_2"><span class="lab"><font color="Green" title="E70">A31</font></span></a></span>;<br/>
<a NAME="E36:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A35</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></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> <font color="Maroon" title="c8">x</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="E37:87_2"/><span class="lab"><font color="Green" title="E75">A36</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:87_2"><span class="lab"><font color="Green" title="E72">A33</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">x1</font> = <font color="Maroon" title="c8">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:87_2"><span class="lab"><font color="Green" title="E72">A33</font></span></a></span>;<br/>
<a NAME="E39:87_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x1</font> <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="#E34:87_2"><span class="lab"><font color="Green" title="E72">A33</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a>, <a class="txt" href="#E30:87_2"><span class="lab"><font color="Green" title="E69">A30</font></span></a>, <a class="txt" href="#E31:87_2"><span class="lab"><font color="Green" title="E70">A31</font></span></a></span>;<br/>
<a NAME="E40:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A37</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><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="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T55" onmouseover="rs('finseq_3/T55')" onmouseout="rh()">FINSEQ_3:55</a></span>;<br/>
<a NAME="E41:87_2"/><span class="lab"><font color="Green" title="E77">A38</font></span>: 
<font color="Maroon" title="c19">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">x1</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="E42:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A39</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c19">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T82" onmouseover="rs('finseq_1/T82')" onmouseout="rh()">FINSEQ_1:82</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">H</font> = <font color="Maroon" title="c18">H</font> as   <a href="pre_poly.html#V1" title="PRE_POLY:attr.1">FinSequence-yielding</a>   <a href="valued_2.html#V7" title="VALUED_2:attr.7">complex-functions-valued</a>  <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="#E33:87_2"><span class="lab"><font color="Green" title="E71">A32</font></span></a>, <a class="ref" href="flexary1.html#T43" target="_self" onmouseover="rs('flexary1/T43')" onmouseout="rh()">Th43</a>, <a class="ref" href="flexary1.html#T44" target="_self" onmouseover="rs('flexary1/T44')" onmouseout="rh()">Th44</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">FF</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><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="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>, <font color="Maroon" title="c22">FOx1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c19">x1</font> as   <a href="pre_poly.html#V1" title="PRE_POLY:attr.1">FinSequence-yielding</a>   <a href="valued_2.html#V7" title="VALUED_2:attr.7">complex-functions-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<a NAME="E45:87_2"/>
 <a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">FOx1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c21">FF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c22">FOx1</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c21">FF</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T46" target="_self" onmouseover="rs('flexary1/T46')" onmouseout="rh()">Th46</a></span>;<br/>
<a NAME="E46:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A40</font></span>: 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">FOx1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c21">FF</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p2">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c22">FOx1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p2">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c21">FF</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_2.html#T32" onmouseover="rs('rvsum_2/T32')" onmouseout="rh()">RVSUM_2:32</a></span>;<br/>
<a NAME="E47:87_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">FOx1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c21">FF</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c20">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:87_2"><span class="lab"><font color="Green" title="E78">A39</font></span></a>, <a class="txt" href="#E40:87_2"><span class="lab"><font color="Green" title="E76">A37</font></span></a>, <a class="txt" href="#E36:87_2"><span class="lab"><font color="Green" title="E74">A35</font></span></a>, <a class="txt" href="#E37:87_2"><span class="lab"><font color="Green" title="E75">A36</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a>, <a class="txt" href="#E33:87_2"><span class="lab"><font color="Green" title="E71">A32</font></span></a></span>;<br/>
<a NAME="E48:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A41</font></span>: 
 <a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">FOx1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c21">FF</font></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c20">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T46" target="_self" onmouseover="rs('flexary1/T46')" onmouseout="rh()">Th46</a></span>;<br/>
<a NAME="E49:87_2"/><span class="lab"><font color="Green" title="E81">A42</font></span>: 
 <a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c21">FF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T45" target="_self" onmouseover="rs('flexary1/T45')" onmouseout="rh()">Th45</a></span>;<br/>
<a NAME="E50:87_2"/><span class="lab"><font color="Green" title="E82">A43</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="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</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="c4">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">fox</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">h</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E52:87_2"/><span class="lab"><font color="Green" title="E83">A44</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c24">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T80" onmouseover="rs('finseq_1/T80')" onmouseout="rh()">FINSEQ_1:80</a></span>;<br/>
<a NAME="E53:87_2"/><span class="lab"><font color="Green" title="E84">A45</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:87_2"><span class="lab"><font color="Green" title="E72">A33</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a>, <a class="txt" href="#E50:87_2"><span class="lab"><font color="Green" title="E82">A43</font></span></a></span>;<br/>
<a NAME="E54:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A46</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></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> <font color="Maroon" title="c8">x</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="E55:87_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x1</font> <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="#E34:87_2"><span class="lab"><font color="Green" title="E72">A33</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a>, <a class="txt" href="#E50:87_2"><span class="lab"><font color="Green" title="E82">A43</font></span></a></span>;<br/>
<a NAME="E56:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A47</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</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"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c19">x1</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><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="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T55" onmouseover="rs('finseq_3/T55')" onmouseout="rh()">FINSEQ_3:55</a></span>;<br/>
<a NAME="E57:87_2"/><span class="lab"><font color="Green" title="E87">A48</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c19">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:87_2"><span class="lab"><font color="Green" title="E77">A38</font></span></a>, <a class="ref" href="finseq_1.html#T82" onmouseover="rs('finseq_1/T82')" onmouseout="rh()">FINSEQ_1:82</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c25">h</font> = <font color="Maroon" title="c24">h</font> as   <a href="pre_poly.html#V1" title="PRE_POLY:attr.1">FinSequence-yielding</a>   <a href="valued_2.html#V7" title="VALUED_2:attr.7">complex-functions-valued</a>  <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="#E52:87_2"><span class="lab"><font color="Green" title="E83">A44</font></span></a>, <a class="ref" href="flexary1.html#T43" target="_self" onmouseover="rs('flexary1/T43')" onmouseout="rh()">Th43</a>, <a class="ref" href="flexary1.html#T44" target="_self" onmouseover="rs('flexary1/T44')" onmouseout="rh()">Th44</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">ff</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>, <font color="Maroon" title="c27">fox1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c19">x1</font> as   <a href="pre_poly.html#V1" title="PRE_POLY:attr.1">FinSequence-yielding</a>   <a href="valued_2.html#V7" title="VALUED_2:attr.7">complex-functions-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<a NAME="E60:87_2"/>
 <a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">fox1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">ff</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c27">fox1</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c26">ff</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T46" target="_self" onmouseover="rs('flexary1/T46')" onmouseout="rh()">Th46</a></span>;<br/>
<a NAME="E61:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A49</font></span>: 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">fox1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">ff</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p2">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c27">fox1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p2">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c26">ff</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_2.html#T32" onmouseover="rs('rvsum_2/T32')" onmouseout="rh()">RVSUM_2:32</a></span>;<br/>
<a NAME="E62:87_2"/>
<font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">fox1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">ff</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c25">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:87_2"><span class="lab"><font color="Green" title="E83">A44</font></span></a>, <a class="txt" href="#E56:87_2"><span class="lab"><font color="Green" title="E86">A47</font></span></a>, <a class="txt" href="#E57:87_2"><span class="lab"><font color="Green" title="E87">A48</font></span></a>, <a class="txt" href="#E54:87_2"><span class="lab"><font color="Green" title="E85">A46</font></span></a>, <a class="txt" href="#E37:87_2"><span class="lab"><font color="Green" title="E75">A36</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>;<br/>
<a NAME="E63:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A50</font></span>: 
 <a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">fox1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c26">ff</font></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c25">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T46" target="_self" onmouseover="rs('flexary1/T46')" onmouseout="rh()">Th46</a></span>;<br/>
<a NAME="E64:87_2"/><span class="lab"><font color="Green" title="E90">A51</font></span>: 
 <a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c26">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T45" target="_self" onmouseover="rs('flexary1/T45')" onmouseout="rh()">Th45</a></span>;<br/>
<a NAME="E65:87_2"/><span class="lab"><font color="Green" title="E91">A52</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">fox1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">FOx1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:87_2"><span class="lab"><font color="Green" title="E77">A38</font></span></a>, <a class="txt" href="#E53:87_2"><span class="lab"><font color="Green" title="E84">A45</font></span></a>, <a class="txt" href="#E35:87_2"><span class="lab"><font color="Green" title="E73">A34</font></span></a>, <a class="txt" href="#E57:87_2"><span class="lab"><font color="Green" title="E87">A48</font></span></a>, <a class="txt" href="#E42:87_2"><span class="lab"><font color="Green" title="E78">A39</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/>
<a NAME="E66:87_2"/>
 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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">x1</font> holds <br/><font color="Maroon" title="c27">fox1</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="Maroon" title="c22">FOx1</font> <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="87_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">j</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="c28">j</font> &amp; <font color="Maroon" title="c28">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">x1</font> implies <font color="Maroon" title="c27">fox1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">FOx1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_2_2"/><span class="lab"><font color="Green" title="E92">A53</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">j</font> &amp; <font color="Maroon" title="c28">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">x1</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="c27">fox1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">FOx1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font></span><br/>

<a NAME="E2:87_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A54</font></span>: 
<font color="Maroon" title="c28">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:87_2"><span class="lab"><font color="Green" title="E77">A38</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="E3:87_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A55</font></span>: 
<font color="Maroon" title="c28">j</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="c4">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:87_2"><span class="lab"><font color="Green" title="E72">A33</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="E4:87_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A56</font></span>: 
<font color="Maroon" title="c28">j</font> <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="c4">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_2"><span class="lab"><font color="Green" title="E92">A53</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E5:87_2_2"/><span class="lab"><font color="Green" title="E96">A57</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">oC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_2_2"><span class="lab"><font color="Green" title="E94">A55</font></span></a>, <a class="txt" href="#E1:87_2_2"><span class="lab"><font color="Green" title="E92">A53</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="txt" href="#E29:87_2"><span class="lab"><font color="Green" title="E68">A29</font></span></a>, <a class="ref" href="fomodel2.html#D6" onmouseover="rs('fomodel2/D6')" onmouseout="rh()">FOMODEL2:def 6</a></span>;<br/>
<a NAME="E6:87_2_2"/>
<font color="Maroon" title="c28">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_2"><span class="lab"><font color="Green" title="E92">A53</font></span></a></span>;<br/>
<a NAME="E7:87_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A58</font></span>: 
( <font color="Maroon" title="c27">fox1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> &amp; <font color="Maroon" title="c22">FOx1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E8:87_2_2"/>
( <font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">o</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">o</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87_2_2"><span class="lab"><font color="Green" title="E95">A56</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/>
<a NAME="E9:87_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A59</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" onmouseover="rs('finseq_1/D11')" onmouseout="rh()">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E10:87_2_2"/>
 not <font color="Maroon" title="c1">i</font> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:87_2_2_1"/>
<font color="Maroon" title="c1">i</font> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:87_2_2_1"/><span class="lab"><font color="Green" title="E99">A60</font></span>: 
( <font color="Maroon" title="c29">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</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="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E4:87_2_2_1"/>
<font color="Maroon" title="c4">o</font> <a href="flexary1.html#K5" title="FLEXARY1:func.5">_</a> (<font color="Maroon" title="c28">j</font>,<font color="Maroon" title="c29">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</font> <a href="flexary1.html#K5" title="FLEXARY1:func.5">_</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_2_2_1"><span class="lab"><font color="Green" title="E99">A60</font></span></a>, <a class="txt" href="#E7:87_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:87_2_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_2_2_1"><span class="lab"><font color="Green" title="E99">A60</font></span></a>, <a class="txt" href="#E4:87_2_2"><span class="lab"><font color="Green" title="E95">A56</font></span></a>, <a class="txt" href="#E7:87_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="ref" href="flexary1.html#D6" target="_self" onmouseover="rs('flexary1/D6')" onmouseout="rh()">Def6</a>, <a class="txt" href="#E2:87_2_2"><span class="lab"><font color="Green" title="E93">A54</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="E11:87_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A61</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:87_2"><span class="lab"><font color="Green" title="E66">A27</font></span></a>, <a class="txt" href="#E9:87_2_2"><span class="lab"><font color="Green" title="E98">A59</font></span></a>, <a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a>, <a class="ref" href="zfmisc_1.html#T34" onmouseover="rs('zfmisc_1/T34')" onmouseout="rh()">ZFMISC_1:34</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T65" onmouseover="rs('relat_1/T65')" onmouseout="rh()">RELAT_1:65</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" onmouseover="rs('relat_1/T36')" onmouseout="rh()">RELAT_1:36</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:87_2_2"><span class="lab"><font color="Green" title="E99">A61</font></span></a>, <a class="ref" href="relat_1.html#T53" onmouseover="rs('relat_1/T53')" onmouseout="rh()">RELAT_1:53</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:87_2_2"/>
<font color="Maroon" title="c27">fox1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">FOx1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:87_2_2"><span class="lab"><font color="Green" title="E96">A57</font></span></a>, <a class="txt" href="#E2:87_2_2"><span class="lab"><font color="Green" title="E93">A54</font></span></a>, <a class="ref" href="funct_7.html#T32" onmouseover="rs('funct_7/T32')" onmouseout="rh()">FUNCT_7:32</a>, <a class="txt" href="#E7:87_2_2"><span class="lab"><font color="Green" title="E97">A58</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="E67:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A62</font></span>: 
<font color="Maroon" title="c27">fox1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">FOx1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:87_2"><span class="lab"><font color="Green" title="E91">A52</font></span></a></span>;<br/>
<a NAME="E68:87_2"/><span class="lab"><font color="Green" title="E93">A63</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_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="c20">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:87_2"><span class="lab"><font color="Green" title="E71">A32</font></span></a>, <a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<a NAME="E69:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A64</font></span>: 
<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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_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="c20">H</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_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="c25">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:87_2"><span class="lab"><font color="Green" title="E83">A44</font></span></a>, <a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a>, <a class="txt" href="#E50:87_2"><span class="lab"><font color="Green" title="E82">A43</font></span></a>, <a class="txt" href="#E31:87_2"><span class="lab"><font color="Green" title="E70">A31</font></span></a>, <a class="txt" href="#E30:87_2"><span class="lab"><font color="Green" title="E69">A30</font></span></a></span>;<br/>
<a NAME="E70:87_2"/>
 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#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="c20">H</font> holds <br/><font color="Maroon" title="c20">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="Maroon" title="c25">h</font> <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="87_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">j</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="c28">j</font> &amp; <font color="Maroon" title="c28">j</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="c20">H</font> implies <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c29">jx</font> = <font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font>;<br/>
<span class="kw">assume </span><a NAME="E1:87_2_3"/><span class="lab"><font color="Green" title="E95">A65</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">j</font> &amp; <font color="Maroon" title="c28">j</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="c20">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"> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font></span><br/>

<a NAME="E2:87_2_3"/><span class="kw">then </span>
( <font color="Maroon" title="c28">j</font> <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="c20">H</font> &amp; <font color="Maroon" title="c28">j</font> <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="c25">h</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:87_2"><span class="lab"><font color="Green" title="E94">A64</font></span></a>, <a class="txt" href="#E35:87_2"><span class="lab"><font color="Green" title="E73">A34</font></span></a>, <a class="txt" href="#E53:87_2"><span class="lab"><font color="Green" title="E84">A45</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E3:87_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A66</font></span>: 
( <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</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="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span> &amp; <font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</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="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:87_2"><span class="lab"><font color="Green" title="E73">A34</font></span></a>, <a class="txt" href="#E53:87_2"><span class="lab"><font color="Green" title="E84">A45</font></span></a>, <a class="txt" href="#E33:87_2"><span class="lab"><font color="Green" title="E71">A32</font></span></a>, <a class="txt" href="#E52:87_2"><span class="lab"><font color="Green" title="E83">A44</font></span></a>, <a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/>
<a NAME="E4:87_2_3"/>
<font color="Maroon" title="c28">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="#E1:87_2_3"><span class="lab"><font color="Green" title="E95">A65</font></span></a></span>;<br/>
<a NAME="E5:87_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A67</font></span>: 
<font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">x</font>
 
;<br/>
<a NAME="E6:87_2_3"/>
<font color="Maroon" title="c28">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</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="E7:87_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_3"><span class="lab"><font color="Green" title="E95">A65</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="E8:87_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A68</font></span>: 
<font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font> <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="c4">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_3"><span class="lab"><font color="Green" title="E95">A65</font></span></a>, <a class="txt" href="#E68:87_2"><span class="lab"><font color="Green" title="E93">A63</font></span></a>, <a class="txt" href="#E35:87_2"><span class="lab"><font color="Green" title="E73">A34</font></span></a>, <a class="txt" href="#E31:87_2"><span class="lab"><font color="Green" title="E70">A31</font></span></a>, <a class="txt" href="#E29:87_2"><span class="lab"><font color="Green" title="E68">A29</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E9:87_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A69</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</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="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</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="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">oC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:87_2"><span class="lab"><font color="Green" title="E68">A29</font></span></a>, <a class="ref" href="fomodel2.html#D6" onmouseover="rs('fomodel2/D6')" onmouseout="rh()">FOMODEL2:def 6</a></span>;<br/>
<a NAME="E10:87_2_3"/>
( <font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">o</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">o</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87_2_3"><span class="lab"><font color="Green" title="E98">A68</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/>
<a NAME="E11:87_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A70</font></span>: 
<font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" onmouseover="rs('finseq_1/D11')" onmouseout="rh()">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E12:87_2_3"/>
 not <font color="Maroon" title="c1">i</font> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_2_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:87_2_3_1"/>
<font color="Maroon" title="c1">i</font> <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span></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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c30">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:87_2_3_1"/><span class="lab"><font color="Green" title="E101">A71</font></span>: 
( <font color="Maroon" title="c30">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</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="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E4:87_2_3_1"/>
<font color="Maroon" title="c4">o</font> <a href="flexary1.html#K5" title="FLEXARY1:func.5">_</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span>,<font color="Maroon" title="c30">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</font> <a href="flexary1.html#K5" title="FLEXARY1:func.5">_</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_2_3_1"><span class="lab"><font color="Green" title="E101">A71</font></span></a>, <a class="txt" href="#E7:87_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a></span>;<br/>
<a NAME="E5:87_2_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_2_3_1"><span class="lab"><font color="Green" title="E101">A71</font></span></a>, <a class="txt" href="#E8:87_2_3"><span class="lab"><font color="Green" title="E98">A68</font></span></a>, <a class="txt" href="#E7:87_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="ref" href="flexary1.html#D6" target="_self" onmouseover="rs('flexary1/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E6:87_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E7:87_2_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_3"><span class="lab"><font color="Green" title="E95">A65</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="E13:87_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A72</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:87_2"><span class="lab"><font color="Green" title="E66">A27</font></span></a>, <a class="txt" href="#E11:87_2_3"><span class="lab"><font color="Green" title="E100">A70</font></span></a>, <a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a>, <a class="ref" href="zfmisc_1.html#T34" onmouseover="rs('zfmisc_1/T34')" onmouseout="rh()">ZFMISC_1:34</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T65" onmouseover="rs('relat_1/T65')" onmouseout="rh()">RELAT_1:65</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" onmouseover="rs('relat_1/T36')" onmouseout="rh()">RELAT_1:36</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:87_2_3"><span class="lab"><font color="Green" title="E101">A72</font></span></a>, <a class="ref" href="relat_1.html#T53" onmouseover="rs('relat_1/T53')" onmouseout="rh()">RELAT_1:53</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:87_2_3"/>
<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87_2_3"><span class="lab"><font color="Green" title="E99">A69</font></span></a>, <a class="txt" href="#E5:87_2_3"><span class="lab"><font color="Green" title="E97">A67</font></span></a>, <a class="ref" href="funct_7.html#T32" onmouseover="rs('funct_7/T32')" onmouseout="rh()">FUNCT_7:32</a>, <a class="txt" href="#E3:87_2_3"><span class="lab"><font color="Green" title="E96">A66</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="E71:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A73</font></span>: 
<font color="Maroon" title="c20">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:87_2"><span class="lab"><font color="Green" title="E94">A64</font></span></a>, <a class="txt" href="#E35:87_2"><span class="lab"><font color="Green" title="E73">A34</font></span></a>, <a class="txt" href="#E53:87_2"><span class="lab"><font color="Green" title="E84">A45</font></span></a></span>;<br/>
<a NAME="E72:87_2"/><span class="lab"><font color="Green" title="E96">A74</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">oC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="txt" href="#E29:87_2"><span class="lab"><font color="Green" title="E68">A29</font></span></a>, <a class="ref" href="fomodel2.html#D6" onmouseover="rs('fomodel2/D6')" onmouseout="rh()">FOMODEL2:def 6</a></span>;<br/>
<a NAME="E73:87_2"/><span class="lab"><font color="Green" title="E97">A75</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="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:87_2"><span class="lab"><font color="Green" title="E54">A15</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="E74:87_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">g</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c29">PP</font> = <font color="Maroon" title="c13">P</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="c28">g</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E73:87_2"><span class="lab"><font color="Green" title="E97">A75</font></span></a></span>;<br/>
<a NAME="E77:87_2"/><span class="lab"><font color="Green" title="E98">A76</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="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></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="E78:87_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 
<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="E79:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A77</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c13">P</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c13">P</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c28">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E73:87_2"><span class="lab"><font color="Green" title="E97">A75</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a>, <a class="ref" href="relat_1.html#T28" onmouseover="rs('relat_1/T28')" onmouseout="rh()">RELAT_1:28</a></span>;<br/>
<a NAME="E80:87_2"/><span class="kw">then </span>
<font color="Maroon" title="c28">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c29">PP</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="#E77:87_2"><span class="lab"><font color="Green" title="E98">A76</font></span></a>, <a class="ref" href="funct_2.html#T52" onmouseover="rs('funct_2/T52')" onmouseout="rh()">FUNCT_2:52</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="c30">G</font> = <font color="Maroon" title="c28">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c29">PP</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a>, <a class="txt" href="#E79:87_2"><span class="lab"><font color="Green" title="E99">A77</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E100">A78</font></span>:  <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c28">g</font> = 
<a href="binop_2.html#K3" title="BINOP_2:func.3">addcomplex</a> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <font color="Maroon" title="c28">g</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#D11" onmouseover="rs('rvsum_1/D11')" onmouseout="rh()">RVSUM_1:def 11</a></span>
<br/>.= 
<a href="binop_2.html#K3" title="BINOP_2:func.3">addcomplex</a> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">"**"</a> <font color="Maroon" title="c30">G</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T7" onmouseover="rs('finsop_1/T7')" onmouseout="rh()">FINSOP_1:7</a></span>
<br/>.= 
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <font color="Maroon" title="c30">G</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#D11" onmouseover="rs('rvsum_1/D11')" onmouseout="rh()">RVSUM_1:def 11</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c31">F</font> = <font color="Maroon" title="c3">f</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="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T1" onmouseover="rs('funct_2/T1')" onmouseout="rh()">FUNCT_2:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c32">I1</font> = <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_2"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="txt" href="#E2:87_2"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c33">C1</font> =  <a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:87_2"><span class="lab"><font color="Green" title="E64">A25</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="E86:87_2"/><span class="lab"><font color="Green" title="E101">A79</font></span>: 
(  not  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  not  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87_2"><span class="lab"><font color="Green" title="E51">A12</font></span></a>, <a class="ref" href="relat_1.html#T42" onmouseover="rs('relat_1/T42')" onmouseout="rh()">RELAT_1:42</a></span>;<br/>
<a NAME="E87:87_2"/>
<font color="Maroon" title="c30">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">P</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" onmouseover="rs('relat_1/T36')" onmouseout="rh()">RELAT_1:36</a></span>;<br/>
<a NAME="E88:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A80</font></span>: 
<font color="Maroon" title="c30">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">F</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c33">C1</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:87_2"><span class="lab"><font color="Green" title="E60">A21</font></span></a>, <a class="txt" href="#E86:87_2"><span class="lab"><font color="Green" title="E101">A79</font></span></a>, <a class="txt" href="#E4:87_2"><span class="lab"><font color="Green" title="E51">A12</font></span></a>, <a class="ref" href="finseqop.html#T8" onmouseover="rs('finseqop/T8')" onmouseout="rh()">FINSEQOP:8</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T65" onmouseover="rs('relat_1/T65')" onmouseout="rh()">RELAT_1:65</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" onmouseover="rs('relat_1/T36')" onmouseout="rh()">RELAT_1:36</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:87_2"><span class="lab"><font color="Green" title="E62">A23</font></span></a>, <a class="txt" href="#E25:87_2"><span class="lab"><font color="Green" title="E66">A27</font></span></a>, <a class="ref" href="relat_1.html#T53" onmouseover="rs('relat_1/T53')" onmouseout="rh()">RELAT_1:53</a></span>
;<br/>
<a NAME="E90:87_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E72:87_2"><span class="lab"><font color="Green" title="E96">A74</font></span></a>, <a class="txt" href="#E7:87_2"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="ref" href="funct_7.html#T31" onmouseover="rs('funct_7/T31')" onmouseout="rh()">FUNCT_7:31</a></span>;<br/>
<a NAME="E91:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A81</font></span>: 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">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="txt" href="#E88:87_2"><span class="lab"><font color="Green" title="E102">A80</font></span></a>, <a class="ref" href="rvsum_2.html#T31" onmouseover="rs('rvsum_2/T31')" onmouseout="rh()">RVSUM_2:31</a>, <a class="txt" href="#E82:87_2"><span class="lab"><font color="Green" title="E100">A78</font></span></a>, <a class="txt" href="#E72:87_2"><span class="lab"><font color="Green" title="E96">A74</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E104">A82</font></span>:  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> = 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="txt" href="#E26:87_2"><span class="lab"><font color="Green" title="E67">A28</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p3">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c22">FOx1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p3">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c21">FF</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p2">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c20">H</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:87_2"><span class="lab"><font color="Green" title="E80">A41</font></span></a>, <a class="txt" href="#E46:87_2"><span class="lab"><font color="Green" title="E79">A40</font></span></a>, <a class="ref" href="rvsum_2.html#T32" onmouseover="rs('rvsum_2/T32')" onmouseout="rh()">RVSUM_2:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p3">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c27">fox1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p2">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c25">h</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:87_2"><span class="lab"><font color="Green" title="E92">A62</font></span></a>, <a class="txt" href="#E71:87_2"><span class="lab"><font color="Green" title="E95">A73</font></span></a>, <a class="txt" href="#E49:87_2"><span class="lab"><font color="Green" title="E81">A42</font></span></a>, <a class="ref" href="rvsum_2.html#T30" onmouseover="rs('rvsum_2/T30')" onmouseout="rh()">RVSUM_2:30</a></span>
;<br/>
<span class="lab"><font color="Green" title="E105">A83</font></span>:  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p3">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c27">fox1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p3">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c26">ff</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p2">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c25">h</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E61:87_2"><span class="lab"><font color="Green" title="E88">A49</font></span></a>, <a class="txt" href="#E63:87_2"><span class="lab"><font color="Green" title="E89">A50</font></span></a>, <a class="ref" href="rvsum_2.html#T32" onmouseover="rs('rvsum_2/T32')" onmouseout="rh()">RVSUM_2:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p3">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c27">fox1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p2">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c25">h</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:87_2"><span class="lab"><font color="Green" title="E90">A51</font></span></a>, <a class="ref" href="rvsum_2.html#T30" onmouseover="rs('rvsum_2/T30')" onmouseout="rh()">RVSUM_2:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p4">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c27">fox1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c15">oC</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p3">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <font color="Maroon" title="c25">h</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">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="txt" href="#E91:87_2"><span class="lab"><font color="Green" title="E103">A81</font></span></a></span>
;<br/>
<a NAME="E94:87_2"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T55" onmouseover="rs('finseq_3/T55')" onmouseout="rh()">FINSEQ_3:55</a>, <a class="txt" href="#E2:87_2"><span class="lab"><font color="Green" title="E49">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E95:87_2"/>
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c4">o</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_2.html#T31" onmouseover="rs('rvsum_2/T31')" onmouseout="rh()">RVSUM_2:31</a>, <a class="txt" href="#E93:87_2"><span class="lab"><font color="Green" title="E105">A83</font></span></a>, <a class="txt" href="#E92:87_2"><span class="lab"><font color="Green" title="E104">A82</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="E3:87"/><span class="lab"><font color="Green" title="E48">A84</font></span>: 
 for <font color="Olive" title="b1">i</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">i</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="#E1:87"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E47">A8</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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">o</font> being    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> holds   <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Olive" title="b1">o</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">o</font> be    <a href="flexary1.html#M1" title="FLEXARY1:mode.1">DoubleReorganization</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">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">  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span></span>)</span></span><br/>



<a NAME="E4:87"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E48">A84</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:87"/>
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="flexary1.html#K6" title="FLEXARY1:func.6">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="flexary1.html#NK6" title="FLEXARY1:NK.6">*.</a> <font color="Maroon" title="c2">o</font></span>)</span></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/>


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