<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">b</font> being  <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c1">A</font>  st  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Olive" title="b2">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> holds <br/> ex <font color="Olive" title="b3">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Olive" title="b1">S</font></span>)</span> &amp;  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b3">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c1">A</font>  st  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Olive" title="b1">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> holds <br/> ex <font color="Olive" title="b2">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> &amp;  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b2">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">b</font> be   <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> implies  ex <font color="Olive" title="b1">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> &amp;  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b1">f</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:26"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> &amp;  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b1">f</font> )</span><br/>

<a NAME="E2:26"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">cs</font> =  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span>;<br/>
<a NAME="E3:26"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">cS</font> =  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span>;<br/>
<a NAME="E4:26"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Th5</a></span>;<br/>
<a NAME="E5:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:26"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Th5</a></span>;<br/>
<a NAME="E7:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A5</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</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="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E9:26"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c7">g</font>
 <span class="kw">and </span><br/><a NAME="E10:26"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#D3" target="_self" title="UPROOTS:def.3">Def3</a></span>;<br/>
<a NAME="E11:26"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E12:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A9</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E13:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E14:26"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E15:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A12</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/>
<a NAME="E17:26"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></span>;<br/>
<a NAME="E18:26"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">f</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> &amp;  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">f</font> )</span><br/>

<span class="kw">thus </span><a NAME="E20:26"/>
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">f</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font> or  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">f</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">dd</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">j</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="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> ;<br/><div><span class="lab"><font color="Green" title="E29">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:26_1_1_1"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
( ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font> &amp; not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> ) or ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:26_1_1_1"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<font color="Maroon" title="c12">j</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="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:26_1_1_1"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E2:26_1_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c12">j</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a></span>;<br/><a NAME="E7:26_1_1_1"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E5:26_1_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><a NAME="E8:26_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E2:26_1_1_1"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>;<br/><a NAME="E9:26_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">j</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="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E4:26_1_1_1"><span class="lab"><font color="Green" title="E31">A16</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:26_1_1_1"/>
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">j</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="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">reconsider </span><font color="Maroon" title="c11">gr</font> = <font color="Maroon" title="c7">g</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><a NAME="E4:26_1_1"/><span class="lab"><font color="Green" title="E30">A18</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">j</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="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </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="c9">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">j</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="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span>  or <font color="Maroon" title="c12">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="c9">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">j</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="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">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="c9">f</font></span><br/>

<a NAME="E2:26_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</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="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:26_1_1_2"/>
<font color="Maroon" title="c12">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="c9">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">dd</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">j</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="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span>  as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">d</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:26_1_1"/><span class="lab"><font color="Green" title="E31">A19</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">d</font>
 <span class="kw">and </span><br/><a NAME="E8:26_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="fsm_1.html#T1" title="FSM_1:th.1">FSM_1:1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">h</font> = <font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c15">F</font> = <font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>;<br/><a NAME="E9:26_1_1"/>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c12">dd</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">dd</font> &amp; <font color="Maroon" title="c12">dd</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">cdd</font> =  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c12">dd</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c17">cdi</font> = <font color="Maroon" title="c16">cdd</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">cdi</font> = <font color="Maroon" title="c16">cdd</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c12">dd</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">dd</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T7" target="_self" title="UPROOTS:th.7">Th7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">cdi</font> = <font color="Maroon" title="c18">cdi</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c12">dd</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> $1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c20">z</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E14:26_1_1"/><span class="lab"><font color="Green" title="E32">A20</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">dd</font>
 <span class="kw">and </span><br/><a NAME="E15:26_1_1"/><span class="lab"><font color="Green" title="E33">A21</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">dd</font> holds <br/><font color="Maroon" title="c20">z</font> <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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></span><span class="kw">set </span><font color="Maroon" title="c21">p</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c20">z</font>;<br/><a NAME="E16:26_1_1"/><span class="lab"><font color="Green" title="E34">A22</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">cdi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">dd</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c22">cSr</font> = <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1: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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1: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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font></span>)</span></span>)</span> implies <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font> ) &amp; ( <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1: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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font></span>)</span></span>)</span> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1: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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font></span>)</span></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:26_1_1_3_1"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1: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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:26_1_1_3_1"/><span class="lab"><font color="Green" title="E35">A23</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1: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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:26_1_1_3_1"/><span class="lab"><font color="Green" title="E36">A24</font></span>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1: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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E5:26_1_1_3_1"/><span class="lab"><font color="Green" title="E37">A25</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1: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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1_3_1"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">j</font> = <font color="Maroon" title="c24">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1_3_1"><span class="lab"><font color="Green" title="E35">A23</font></span></a></span>;<br/><a NAME="E7:26_1_1_3_1"/><span class="lab"><font color="Green" title="E38">A26</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K4" title="FINSOP_1:func.4">findom</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c12">dd</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1_3_1"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></span>;<br/><a NAME="E8:26_1_1_3_1"/><span class="kw">then </span>
not <font color="Maroon" title="c25">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">dd</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E9:26_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A27</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_1_3_1"><span class="lab"><font color="Green" title="E38">A26</font></span></a></span>;<br/><a NAME="E10:26_1_1_3_1"/>
<font color="Maroon" title="c24">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="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1_3_1"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></span>;<br/><a NAME="E11:26_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26_1_1_3_1"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><span class="kw">hence </span><a NAME="E12:26_1_1_3_1"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_3_1"><span class="lab"><font color="Green" title="E36">A24</font></span></a>, <a class="txt" href="#E5:26_1_1_3_1"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:26_1_1_3"/><span class="lab"><font color="Green" title="E35">A28</font></span>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1: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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font></span>)</span></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:26_1_1_3"/><span class="lab"><font color="Green" title="E36">A29</font></span>: 
<font color="Maroon" title="c24">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="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:26_1_1_3"/><span class="lab"><font color="Green" title="E37">A30</font></span>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_1_1_3_2"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">dd</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:26_1_1_3_2"/><span class="lab"><font color="Green" title="E38">A31</font></span>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">and </span><br/><a NAME="E4:26_1_1_3_2"/><span class="lab"><font color="Green" title="E39">A32</font></span>: 
( <font color="Maroon" title="c25">j</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="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><a NAME="E5:26_1_1_3_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E4:26_1_1_3_2"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><span class="kw">hence </span><a NAME="E6:26_1_1_3_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1_3"><span class="lab"><font color="Green" title="E35">A28</font></span></a>, <a class="txt" href="#E5:26_1_1_3"><span class="lab"><font color="Green" title="E37">A30</font></span></a>, <a class="txt" href="#E3:26_1_1_3_2"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E7:26_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E4:26_1_1_3"><span class="lab"><font color="Green" title="E36">A29</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E8:26_1_1_3"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1: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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_3"><span class="lab"><font color="Green" title="E36">A29</font></span></a>, <a class="txt" href="#E5:26_1_1_3"><span class="lab"><font color="Green" title="E37">A30</font></span></a>, <a class="ref" href="funct_1.html#T73" title="FUNCT_1:th.73">FUNCT_1:73</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E18:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A33</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1: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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>;<br/><a NAME="E19:26_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finsop_1.html#K4" title="FINSOP_1:func.4">findom</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c12">dd</font></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="c9">f</font></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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><a NAME="E20:26_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1: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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1: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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font></span>)</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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E21:26_1_1"/><span class="kw">then </span>
 <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26_1_1"><span class="lab"><font color="Green" title="E35">A33</font></span></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></span>;<br/><a NAME="E22:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A34</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></span>;<br/><a NAME="E23:26_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</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="c9">f</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">dd</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">dd</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></span>;<br/><a NAME="E24:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A35</font></span>: 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">dd</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E22:26_1_1"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></span>;<br/><div><span class="lab"><font color="Green" title="E38">A36</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_4"><span class="kw">now </span></a><div class="add"><a NAME="E1:26_1_1_4"/><span class="lab"><font color="Green" title="E39">A37</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/><span class="kw">assume </span><a NAME="E2:26_1_1_4"/>
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:26_1_1_4"/><span class="lab"><font color="Green" title="E40">A38</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E5:26_1_1_4"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_4"><span class="lab"><font color="Green" title="E40">A38</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:26_1_1_4"/><span class="lab"><font color="Green" title="E41">A39</font></span>: 
<font color="Maroon" title="c24">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x</font>
 <span class="kw">and </span><br/><a NAME="E8:26_1_1_4"/>
<font color="Maroon" title="c24">j</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="c9">f</font>
 <span class="kw">and </span><br/><a NAME="E9:26_1_1_4"/><span class="lab"><font color="Green" title="E42">A40</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a></span>;<br/><a NAME="E10:26_1_1_4"/><span class="lab"><font color="Green" title="E43">A41</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_4"><span class="lab"><font color="Green" title="E40">A38</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E11:26_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c24">j</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="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_1_4"><span class="lab"><font color="Green" title="E41">A39</font></span></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>;<br/><a NAME="E12:26_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><a NAME="E13:26_1_1_4"/><span class="kw">then </span>
not <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26_1_1_4"><span class="lab"><font color="Green" title="E42">A40</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E14:26_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_1_1_4"><span class="lab"><font color="Green" title="E43">A41</font></span></a>, <a class="txt" href="#E7:26_1_1_4"><span class="lab"><font color="Green" title="E41">A39</font></span></a>, <a class="txt" href="#E1:26_1_1_4"><span class="lab"><font color="Green" title="E39">A37</font></span></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E7:26_1_1"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>
;<br/><a NAME="E27:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A42</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="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font></span>)</span> implies <font color="Maroon" title="c23">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c23">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font></span>)</span> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:26_1_1_6_1"/><span class="lab"><font color="Green" title="E40">A43</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_6_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_1_6_1_1"/>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> or <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_6_1"><span class="lab"><font color="Green" title="E40">A43</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_6_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_6_1_1_1"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:26_1_1_6_1_1_1"/>
<font color="Maroon" title="c23">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E27:26_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_6_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_6_1_1_2"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:26_1_1_6_1_1_2"/>
<font color="Maroon" title="c23">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E27:26_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:26_1_1_6"/><span class="lab"><font color="Green" title="E40">A44</font></span>: 
<font color="Maroon" title="c23">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">i</font> = <font color="Maroon" title="c23">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_6_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_1_6_2"/>
( <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_6_2_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_6_2_1"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:26_1_1_6_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E27:26_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E2:26_1_1_6"><span class="lab"><font color="Green" title="E40">A44</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:26_1_1_6_2_1"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_6_2_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_6_2_2"/><span class="lab"><font color="Green" title="E41">A45</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:26_1_1_6_2_2"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E27:26_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="txt" href="#E2:26_1_1_6"><span class="lab"><font color="Green" title="E40">A44</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><a NAME="E3:26_1_1_6_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_6_2_2"><span class="lab"><font color="Green" title="E41">A45</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>;<br/><a NAME="E4:26_1_1_6_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/><a NAME="E5:26_1_1_6_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E27:26_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="txt" href="#E2:26_1_1_6"><span class="lab"><font color="Green" title="E40">A44</font></span></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E6:26_1_1_6_2_2"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E29:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A46</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</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="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>;<br/><a NAME="E30:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A47</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c20">z</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="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>;<br/><a NAME="E31:26_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">cdd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">dd</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Th5</a></span>;<br/><a NAME="E32:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A48</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">cdd</font> <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="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_1"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E24:26_1_1"><span class="lab"><font color="Green" title="E37">A35</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E33:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A49</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c19">cdi</font> <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="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/><a NAME="E34:26_1_1"/><span class="lab"><font color="Green" title="E44">A50</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">z</font> or <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_1_1_7"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:26_1_1_7"/><span class="lab"><font color="Green" title="E45">A51</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font>
 <span class="kw">and </span><br/><a NAME="E4:26_1_1_7"/><span class="lab"><font color="Green" title="E46">A52</font></span>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:26_1_1_7"/><span class="lab"><font color="Green" title="E47">A53</font></span>: 
( <font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">cdi</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">x</font> &amp; <font color="Maroon" title="c19">cdi</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">d</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26_1_1"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E33:26_1_1"><span class="lab"><font color="Green" title="E43">A49</font></span></a>, <a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E3:26_1_1_7"><span class="lab"><font color="Green" title="E45">A51</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E6:26_1_1_7"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c24">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E7:26_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A54</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E8:26_1_1_7"/>
<font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c24">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_1_7"><span class="lab"><font color="Green" title="E47">A53</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E9:26_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A55</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E10:26_1_1_7"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E15:26_1_1"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="txt" href="#E3:26_1_1_7"><span class="lab"><font color="Green" title="E45">A51</font></span></a>, <a class="txt" href="#E4:26_1_1_7"><span class="lab"><font color="Green" title="E46">A52</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:26_1_1_7"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_1"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E7:26_1_1_7"><span class="lab"><font color="Green" title="E48">A54</font></span></a>, <a class="txt" href="#E9:26_1_1_7"><span class="lab"><font color="Green" title="E49">A55</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E45">A56</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_8"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c20">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></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="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:26_1_1_8"/><span class="lab"><font color="Green" title="E46">A57</font></span>: 
<font color="Maroon" title="c23">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c20">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></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="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_8_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_1_8_1"/>
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> or <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:26_1_1"><span class="lab"><font color="Green" title="E40">A46</font></span></a>, <a class="txt" href="#E1:26_1_1_8"><span class="lab"><font color="Green" title="E46">A57</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_8_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_8_1_1"/><span class="lab"><font color="Green" title="E47">A58</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c20">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></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="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:26_1_1_8_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E3:26_1_1_8_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T25" title="FUNCT_1:th.25">FUNCT_1:25</a></span>;<br/><a NAME="E4:26_1_1_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A59</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">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="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/><a NAME="E5:26_1_1_8_1_1"/>
not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:26_1_1"><span class="lab"><font color="Green" title="E38">A36</font></span></a>, <a class="txt" href="#E1:26_1_1_8_1_1"><span class="lab"><font color="Green" title="E47">A58</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E6:26_1_1_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A60</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c20">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">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"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>;<br/><a NAME="E7:26_1_1_8_1_1"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</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> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E8:26_1_1_8_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c20">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E27:26_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="txt" href="#E6:26_1_1_8_1_1"><span class="lab"><font color="Green" title="E49">A60</font></span></a>, <a class="txt" href="#E4:26_1_1_8_1_1"><span class="lab"><font color="Green" title="E48">A59</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_8_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_8_1_2"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c20">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></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="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:26_1_1_8_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c20">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> &amp; <font color="Maroon" title="c20">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:26_1_1_8_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c20">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:26_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="txt" href="#E34:26_1_1"><span class="lab"><font color="Green" title="E44">A50</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E36:26_1_1"/><span class="lab"><font color="Green" title="E46">A61</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c20">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">p</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c20">z</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> <span class="p2">(<span class="default"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>,<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="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:26_1_1"><span class="lab"><font color="Green" title="E41">A47</font></span></a>, <a class="txt" href="#E35:26_1_1"><span class="lab"><font color="Green" title="E45">A56</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/><a NAME="E38:26_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/><a NAME="E39:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A62</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="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E40:26_1_1"/><span class="lab"><font color="Green" title="E48">A63</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c16">cdd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">dd</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_9"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c23">p</font> implies <font color="Maroon" title="c24">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c24">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c23">p</font> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c23">p</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:26_1_1_9_1"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c23">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:26_1_1_9_1"/><span class="lab"><font color="Green" title="E49">A64</font></span>: 
<font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c23">p</font>
 <span class="kw">and </span><br/><a NAME="E4:26_1_1_9_1"/><span class="lab"><font color="Green" title="E50">A65</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_9_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_1_9_1_1"/>
( <font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> or <font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1_9_1"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_9_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_9_1_1_1"/><span class="lab"><font color="Green" title="E51">A66</font></span>: 
<font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:26_1_1_9_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>;<br/><a NAME="E3:26_1_1_9_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E4:26_1_1_9_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A67</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">a</font></span>)</span> <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="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/><a NAME="E5:26_1_1_9_1_1_1"/>
not <font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:26_1_1"><span class="lab"><font color="Green" title="E38">A36</font></span></a>, <a class="txt" href="#E1:26_1_1_9_1_1_1"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A68</font></span>: <font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_9_1_1_1"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
;<br/><a NAME="E7:26_1_1_9_1_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E27:26_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E8:26_1_1_9_1_1_1"/>
<font color="Maroon" title="c24">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_9_1"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E6:26_1_1_9_1_1_1"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E4:26_1_1_9_1_1_1"><span class="lab"><font color="Green" title="E52">A67</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_9_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_9_1_1_2"/>
<font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:26_1_1_9_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c20">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">z</font> &amp; <font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:26_1_1_9_1_1_2"/>
<font color="Maroon" title="c24">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:26_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="txt" href="#E34:26_1_1"><span class="lab"><font color="Green" title="E44">A50</font></span></a>, <a class="txt" href="#E4:26_1_1_9_1"><span class="lab"><font color="Green" title="E50">A65</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:26_1_1_9"/><span class="lab"><font color="Green" title="E49">A69</font></span>: 
<font color="Maroon" title="c24">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"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c23">p</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">j</font> = <font color="Maroon" title="c24">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_9_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_1_9_2"/>
( <font color="Maroon" title="c25">j</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="c11">gr</font> or  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</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="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">gr</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1_9"><span class="lab"><font color="Green" title="E49">A69</font></span></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_9_2_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_9_2_1"/><span class="lab"><font color="Green" title="E50">A70</font></span>: 
<font color="Maroon" title="c25">j</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="c11">gr</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c23">p</font></span><br/><div class="add"><a NAME="E2:26_1_1_9_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A71</font></span>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E3:26_1_1_9_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A72</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_9_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_1_1_9_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:26_1_1_9_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A73</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span> &amp; <font color="Olive" title="b1">k</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="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E3:26_1_1_9_2_1"><span class="lab"><font color="Green" title="E52">A72</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E2:26_1_1_9_2_1"><span class="lab"><font color="Green" title="E51">A71</font></span></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:26_1_1_9_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1_9_2_1"><span class="lab"><font color="Green" title="E51">A71</font></span></a>, <a class="txt" href="#E2:26_1_1_9_2_1_1"><span class="lab"><font color="Green" title="E53">A73</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E3:26_1_1_9_2_1"><span class="lab"><font color="Green" title="E52">A72</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E2:26_1_1_9_2_1"><span class="lab"><font color="Green" title="E51">A71</font></span></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></span>
<br/>.= 
<font color="Maroon" title="c25">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E1:26_1_1_9_2_1"><span class="lab"><font color="Green" title="E50">A70</font></span></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:26_1_1_9_2_1"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c23">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:26_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="txt" href="#E30:26_1_1"><span class="lab"><font color="Green" title="E41">A47</font></span></a>, <a class="txt" href="#E3:26_1_1_9_2_1"><span class="lab"><font color="Green" title="E52">A72</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_9_2_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_9_2_2"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</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="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">gr</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c23">p</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:26_1_1_9_2_2"/><span class="lab"><font color="Green" title="E50">A74</font></span>: 
<font color="Maroon" title="c26">n</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="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:26_1_1_9_2_2"/><span class="lab"><font color="Green" title="E51">A75</font></span>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">gr</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">n</font>
 ;<br/><a NAME="E5:26_1_1_9_2_2"/><span class="lab"><font color="Green" title="E52">A76</font></span>: 
<font color="Maroon" title="c16">cdd</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">dd</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:26_1_1"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E32:26_1_1"><span class="lab"><font color="Green" title="E42">A48</font></span></a>, <a class="txt" href="#E40:26_1_1"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E3:26_1_1_9_2_2"><span class="lab"><font color="Green" title="E50">A74</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">cdd</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">n</font></span>)</span> = 
<font color="Maroon" title="c20">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">cdd</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:26_1_1"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E32:26_1_1"><span class="lab"><font color="Green" title="E42">A48</font></span></a>, <a class="txt" href="#E40:26_1_1"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E3:26_1_1_9_2_2"><span class="lab"><font color="Green" title="E50">A74</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">cdd</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:26_1_1"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E32:26_1_1"><span class="lab"><font color="Green" title="E42">A48</font></span></a>, <a class="txt" href="#E40:26_1_1"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E15:26_1_1"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="txt" href="#E3:26_1_1_9_2_2"><span class="lab"><font color="Green" title="E50">A74</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">cdi</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">cdd</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:26_1_1"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E32:26_1_1"><span class="lab"><font color="Green" title="E42">A48</font></span></a>, <a class="txt" href="#E40:26_1_1"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E16:26_1_1"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E3:26_1_1_9_2_2"><span class="lab"><font color="Green" title="E50">A74</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:26_1_1"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E32:26_1_1"><span class="lab"><font color="Green" title="E42">A48</font></span></a>, <a class="txt" href="#E3:26_1_1_9_2_2"><span class="lab"><font color="Green" title="E50">A74</font></span></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></span>
<br/>.= 
<font color="Maroon" title="c25">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E4:26_1_1_9_2_2"><span class="lab"><font color="Green" title="E51">A75</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:26_1_1_9_2_2"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c23">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E27:26_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E30:26_1_1"><span class="lab"><font color="Green" title="E41">A47</font></span></a>, <a class="txt" href="#E5:26_1_1_9_2_2"><span class="lab"><font color="Green" title="E52">A76</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E42:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A77</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c23">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="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>;<br/><a NAME="E43:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A78</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">p</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="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:26_1_1"><span class="lab"><font color="Green" title="E41">A47</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_10"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">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="c9">f</font> implies <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></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="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:26_1_1_10"/><span class="lab"><font color="Green" title="E51">A79</font></span>: 
<font color="Maroon" title="c24">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="c9">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></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="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_10_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_1_10_1"/>
( <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_10_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_10_1_1"/><span class="lab"><font color="Green" title="E52">A80</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></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="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c25">cdix</font> = <font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c24">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">px</font> = <font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">j</font> = <font color="Maroon" title="c24">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_10"><span class="lab"><font color="Green" title="E51">A79</font></span></a></span>;<br/><a NAME="E5:26_1_1_10_1_1"/><span class="lab"><font color="Green" title="E53">A81</font></span>: 
<font color="Maroon" title="c27">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E1:26_1_1_10"><span class="lab"><font color="Green" title="E51">A79</font></span></a>, <a class="txt" href="#E1:26_1_1_10_1_1"><span class="lab"><font color="Green" title="E52">A80</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A82</font></span>: <font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">x</font> = 
<font color="Maroon" title="c20">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E15:26_1_1"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="txt" href="#E5:26_1_1_10_1_1"><span class="lab"><font color="Green" title="E53">A81</font></span></a></span>
;<br/><a NAME="E7:26_1_1_10_1_1"/><span class="lab"><font color="Green" title="E55">A83</font></span>: 
<font color="Maroon" title="c19">cdi</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">dd</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E5:26_1_1_10_1_1"><span class="lab"><font color="Green" title="E53">A81</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><a NAME="E8:26_1_1_10_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">cdi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">dd</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E9:26_1_1_10_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A84</font></span>: 
<font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">cdi</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E5:26_1_1_10_1_1"><span class="lab"><font color="Green" title="E53">A81</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">cdix</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_1"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E24:26_1_1"><span class="lab"><font color="Green" title="E37">A35</font></span></a>, <a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E1:26_1_1_10_1_1"><span class="lab"><font color="Green" title="E52">A80</font></span></a>, <a class="txt" href="#E5:26_1_1_10_1_1"><span class="lab"><font color="Green" title="E53">A81</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">px</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E7:26_1_1"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E39:26_1_1"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E24:26_1_1"><span class="lab"><font color="Green" title="E37">A35</font></span></a>, <a class="txt" href="#E6:26_1_1_10_1_1"><span class="lab"><font color="Green" title="E54">A82</font></span></a>, <a class="txt" href="#E9:26_1_1_10_1_1"><span class="lab"><font color="Green" title="E56">A84</font></span></a>, <a class="txt" href="#E7:26_1_1_10_1_1"><span class="lab"><font color="Green" title="E55">A83</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E27:26_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="txt" href="#E43:26_1_1"><span class="lab"><font color="Green" title="E50">A78</font></span></a>, <a class="txt" href="#E1:26_1_1_10"><span class="lab"><font color="Green" title="E51">A79</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_10_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_10_1_2"/><span class="lab"><font color="Green" title="E52">A85</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></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="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c25">px</font> = <font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E3:26_1_1_10_1_2"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_10"><span class="lab"><font color="Green" title="E51">A79</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/><a NAME="E4:26_1_1_10_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_10_1_2"><span class="lab"><font color="Green" title="E52">A85</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>;<br/><a NAME="E5:26_1_1_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A86</font></span>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/><a NAME="E6:26_1_1_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A87</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span>)</span> <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="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_10_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_1_1_10_1_2_1"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">dd</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:26_1_1_10_1_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x</font> &amp; <font color="Olive" title="b1">j</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="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><span class="kw">hence </span><a NAME="E3:26_1_1_10_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_10_1_2"><span class="lab"><font color="Green" title="E52">A85</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A88</font></span>: <font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E1:26_1_1_10"><span class="lab"><font color="Green" title="E51">A79</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
;<br/><span class="kw">thus </span><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> = 
<font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_10"><span class="lab"><font color="Green" title="E51">A79</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">px</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_1_10_1_2"><span class="lab"><font color="Green" title="E53">A86</font></span></a>, <a class="txt" href="#E8:26_1_1_10_1_2"><span class="lab"><font color="Green" title="E55">A88</font></span></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></span>
<br/>.= 
<font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">px</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E6:26_1_1_10_1_2"><span class="lab"><font color="Green" title="E54">A87</font></span></a>, <a class="txt" href="#E8:26_1_1_10_1_2"><span class="lab"><font color="Green" title="E55">A88</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">px</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E6:26_1_1_10_1_2"><span class="lab"><font color="Green" title="E54">A87</font></span></a>, <a class="txt" href="#E8:26_1_1_10_1_2"><span class="lab"><font color="Green" title="E55">A88</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E27:26_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="txt" href="#E43:26_1_1"><span class="lab"><font color="Green" title="E50">A78</font></span></a>, <a class="txt" href="#E1:26_1_1_10"><span class="lab"><font color="Green" title="E51">A79</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E45:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A89</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E27:26_1_1"><span class="lab"><font color="Green" title="E39">A42</font></span></a>, <a class="txt" href="#E43:26_1_1"><span class="lab"><font color="Green" title="E50">A78</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/><a NAME="E46:26_1_1"/><span class="lab"><font color="Green" title="E52">A90</font></span>: 
<font color="Maroon" title="c23">p</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">a</font>, <font color="Maroon" title="c25">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c23">p</font> or  not <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c23">p</font> or  not <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">c</font> or <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26_1_1_11"/><span class="lab"><font color="Green" title="E53">A91</font></span>: 
( <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c23">p</font> &amp; <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c23">p</font> )
 <span class="kw">and </span><br/><a NAME="E2:26_1_1_11"/><span class="lab"><font color="Green" title="E54">A92</font></span>: 
<font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_11_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_1_11_1"/>
( ( <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> ) or ( <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font> ) or ( <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font> &amp; <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> ) or ( <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font> &amp; <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:26_1_1"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E1:26_1_1_11"><span class="lab"><font color="Green" title="E53">A91</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_11_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_11_1_1"/><span class="lab"><font color="Green" title="E55">A93</font></span>: 
( <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c</font></span><br/><div class="add"><a NAME="E2:26_1_1_11_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>;<br/><a NAME="E3:26_1_1_11_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/><a NAME="E4:26_1_1_11_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A94</font></span>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></span>;<br/><a NAME="E5:26_1_1_11_1_1"/>
<font color="Maroon" title="c24">a</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="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_11_1_1"><span class="lab"><font color="Green" title="E55">A93</font></span></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>;<br/><a NAME="E6:26_1_1_11_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A95</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></span>;<br/><a NAME="E7:26_1_1_11_1_1"/>
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_11_1_1"><span class="lab"><font color="Green" title="E55">A93</font></span></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>;<br/><a NAME="E8:26_1_1_11_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A96</font></span>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/><a NAME="E9:26_1_1_11_1_1"/>
not <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:26_1_1"><span class="lab"><font color="Green" title="E38">A36</font></span></a>, <a class="txt" href="#E1:26_1_1_11_1_1"><span class="lab"><font color="Green" title="E55">A93</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A97</font></span>: <font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">c</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_11_1_1"><span class="lab"><font color="Green" title="E55">A93</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
;<br/><a NAME="E11:26_1_1_11_1_1"/><span class="lab"><font color="Green" title="E60">A98</font></span>: 
<font color="Maroon" title="c25">c</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="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_11_1_1"><span class="lab"><font color="Green" title="E55">A93</font></span></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>;<br/><a NAME="E12:26_1_1_11_1_1"/>
not <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:26_1_1"><span class="lab"><font color="Green" title="E38">A36</font></span></a>, <a class="txt" href="#E1:26_1_1_11_1_1"><span class="lab"><font color="Green" title="E55">A93</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_11_1_1"><span class="lab"><font color="Green" title="E55">A93</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
;<br/><a NAME="E14:26_1_1_11_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1_11"><span class="lab"><font color="Green" title="E54">A92</font></span></a>, <a class="txt" href="#E10:26_1_1_11_1_1"><span class="lab"><font color="Green" title="E59">A97</font></span></a>, <a class="txt" href="#E4:26_1_1_11_1_1"><span class="lab"><font color="Green" title="E56">A94</font></span></a>, <a class="txt" href="#E8:26_1_1_11_1_1"><span class="lab"><font color="Green" title="E58">A96</font></span></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></span>;<br/><span class="kw">hence </span><a NAME="E15:26_1_1_11_1_1"/>
<font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_1_1_11_1_1"><span class="lab"><font color="Green" title="E57">A95</font></span></a>, <a class="txt" href="#E11:26_1_1_11_1_1"><span class="lab"><font color="Green" title="E60">A98</font></span></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_11_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_11_1_2"/><span class="lab"><font color="Green" title="E55">A99</font></span>: 
( <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c</font></span><br/><div class="add"><a NAME="E2:26_1_1_11_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>;<br/><a NAME="E3:26_1_1_11_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E4:26_1_1_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A100</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font></span>)</span> <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="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/><a NAME="E5:26_1_1_11_1_2"/>
not <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:26_1_1"><span class="lab"><font color="Green" title="E38">A36</font></span></a>, <a class="txt" href="#E1:26_1_1_11_1_2"><span class="lab"><font color="Green" title="E55">A99</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A101</font></span>: <font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_11_1_2"><span class="lab"><font color="Green" title="E55">A99</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
;<br/><font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">c</font> = 
<font color="Maroon" title="c20">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_11_1_2"><span class="lab"><font color="Green" title="E55">A99</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c25">c</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E15:26_1_1"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="txt" href="#E1:26_1_1_11_1_2"><span class="lab"><font color="Green" title="E55">A99</font></span></a></span>
;<br/><a NAME="E8:26_1_1_11_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c25">c</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E2:26_1_1_11"><span class="lab"><font color="Green" title="E54">A92</font></span></a>, <a class="txt" href="#E6:26_1_1_11_1_2"><span class="lab"><font color="Green" title="E57">A101</font></span></a>, <a class="txt" href="#E4:26_1_1_11_1_2"><span class="lab"><font color="Green" title="E56">A100</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E9:26_1_1_11_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c25">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E10:26_1_1_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A102</font></span>: 
<font color="Maroon" title="c19">cdi</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26_1_1"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E1:26_1_1_11_1_2"><span class="lab"><font color="Green" title="E55">A99</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E11:26_1_1_11_1_2"/>
<font color="Maroon" title="c19">cdi</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c19">cdi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26_1_1"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E1:26_1_1_11_1_2"><span class="lab"><font color="Green" title="E55">A99</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E12:26_1_1_11_1_2"/>
<font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:26_1_1"><span class="lab"><font color="Green" title="E43">A49</font></span></a>, <a class="txt" href="#E10:26_1_1_11_1_2"><span class="lab"><font color="Green" title="E58">A102</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_11_1_3"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_11_1_3"/><span class="lab"><font color="Green" title="E55">A103</font></span>: 
( <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font> &amp; <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c</font></span><br/><div class="add"><a NAME="E2:26_1_1_11_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></span>;<br/><a NAME="E3:26_1_1_11_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">c</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E4:26_1_1_11_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A104</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">c</font></span>)</span> <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="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/><a NAME="E5:26_1_1_11_1_3"/>
not <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:26_1_1"><span class="lab"><font color="Green" title="E38">A36</font></span></a>, <a class="txt" href="#E1:26_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A103</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A105</font></span>: <font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">c</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A103</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
;<br/><font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">a</font> = 
<font color="Maroon" title="c20">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A103</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c24">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E15:26_1_1"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="txt" href="#E1:26_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A103</font></span></a></span>
;<br/><a NAME="E8:26_1_1_11_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c24">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E2:26_1_1_11"><span class="lab"><font color="Green" title="E54">A92</font></span></a>, <a class="txt" href="#E6:26_1_1_11_1_3"><span class="lab"><font color="Green" title="E57">A105</font></span></a>, <a class="txt" href="#E4:26_1_1_11_1_3"><span class="lab"><font color="Green" title="E56">A104</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E9:26_1_1_11_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E10:26_1_1_11_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A106</font></span>: 
<font color="Maroon" title="c19">cdi</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26_1_1"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E1:26_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A103</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E11:26_1_1_11_1_3"/>
<font color="Maroon" title="c19">cdi</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c19">cdi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26_1_1"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E1:26_1_1_11_1_3"><span class="lab"><font color="Green" title="E55">A103</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E12:26_1_1_11_1_3"/>
<font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:26_1_1"><span class="lab"><font color="Green" title="E43">A49</font></span></a>, <a class="txt" href="#E10:26_1_1_11_1_3"><span class="lab"><font color="Green" title="E58">A106</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_11_1_4"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_11_1_4"/><span class="lab"><font color="Green" title="E55">A107</font></span>: 
( <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font> &amp; <font color="Maroon" title="c25">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c</font></span><br/><div class="add"><a NAME="E2:26_1_1_11_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A108</font></span>: 
( <font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">cdi</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">a</font> &amp; <font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c25">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">cdi</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26_1_1"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><span class="lab"><font color="Green" title="E57">A109</font></span>: <font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c25">c</font> = 
<font color="Maroon" title="c20">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_11_1_4"><span class="lab"><font color="Green" title="E55">A107</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c25">c</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E15:26_1_1"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="txt" href="#E1:26_1_1_11_1_4"><span class="lab"><font color="Green" title="E55">A107</font></span></a></span>
;<br/><font color="Maroon" title="c23">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">a</font> = 
<font color="Maroon" title="c20">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_11_1_4"><span class="lab"><font color="Green" title="E55">A107</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">cdi</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c24">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E15:26_1_1"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="txt" href="#E1:26_1_1_11_1_4"><span class="lab"><font color="Green" title="E55">A107</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E5:26_1_1_11_1_4"/>
<font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26_1_1"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E14:26_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E2:26_1_1_11"><span class="lab"><font color="Green" title="E54">A92</font></span></a>, <a class="txt" href="#E1:26_1_1_11_1_4"><span class="lab"><font color="Green" title="E55">A107</font></span></a>, <a class="txt" href="#E3:26_1_1_11_1_4"><span class="lab"><font color="Green" title="E57">A109</font></span></a>, <a class="txt" href="#E2:26_1_1_11_1_4"><span class="lab"><font color="Green" title="E56">A108</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">reconsider </span><font color="Maroon" title="c24">cadd</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">dd</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E48:26_1_1"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T111" title="RVSUM_1:th.111">RVSUM_1:111</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A110</font></span>:  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c11">gr</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c11">gr</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>

<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></span>
;<br/><a NAME="E50:26_1_1"/>
<font color="Maroon" title="c23">p</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:26_1_1"><span class="lab"><font color="Green" title="E49">A77</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E51:26_1_1"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E46:26_1_1"><span class="lab"><font color="Green" title="E52">A90</font></span></a>, <a class="txt" href="#E45:26_1_1"><span class="lab"><font color="Green" title="E51">A89</font></span></a>, <a class="txt" href="#E49:26_1_1"><span class="lab"><font color="Green" title="E53">A110</font></span></a>, <a class="ref" href="finsop_1.html#T8" title="FINSOP_1:th.8">FINSOP_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">f</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:26_1_2"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="pre_poly.html#T8" title="PRE_POLY:th.8">PRE_POLY:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
