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<b>let </b><font color="Maroon" title="c1">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><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>;<br/><b>let </b><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>;<br/>





<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E19">A1</font></i>: 
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">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>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">cS</font> =  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font>;<br/>
<b>set </b><font color="Maroon" title="c5">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="E2:23"/>
 <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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="uproots.html#D1" target="_self" title="UPROOTS:def.1">Def1</a></i>;<br/>
<a NAME="E3:23"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:23"/><i><font color="Green" title="E21">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">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>
 
<b>by </b><i><a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Th5</a></i>;<br/>
<a NAME="E5:23"/><i><font color="Green" title="E22">A4</font></i>: 
 <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="uproots.html#T4" target="_self" title="UPROOTS:th.4">Th4</a></i>;<br/>
<a NAME="E6:23"/><i><font color="Green" title="E23">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:23"/><b>then </b><i><font color="Green" title="E24">A6</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E3:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="uproots.html#E4:23"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<b>then reconsider </b><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> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E9:23"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">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>
 
<b>by </b><i><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></i>;<br/>
<a NAME="E10:23"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">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>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">f</font>
;<br/>

<b>thus </b><a NAME="E12:23"/>
<font color="Maroon" title="c7">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>
 ;<br/>

<b>set </b><font color="Maroon" title="c8">cs</font> =  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span>;<br/>
<a NAME="E13:23"/><i><font color="Green" title="E25">A7</font></i>: 
 <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="uproots.html#T4" target="_self" title="UPROOTS:th.4">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">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> <b> such that </b><br/><a NAME="E15:23"/><i><font color="Green" title="E26">A8</font></i>: 
 <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">g</font>
 <b>and </b><br/><a NAME="E16:23"/><i><font color="Green" title="E27">A9</font></i>: 
<font color="Maroon" title="c9">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="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D3" target="_self" title="UPROOTS:def.3">Def3</a></i>;<br/>
<a NAME="E17:23"/>
 <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="polynom1.html#K11" title="POLYNOM1:func.11">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#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="uproots.html#D1" target="_self" title="UPROOTS:def.1">Def1</a></i>;<br/>
<a NAME="E18:23"/><b>then </b><i><font color="Green" title="E28">A10</font></i>: 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">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="polynom1.html#K11" title="POLYNOM1:func.11">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#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E19:23"/><i><font color="Green" title="E29">A11</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">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="polynom1.html#K11" title="POLYNOM1:func.11">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="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Th5</a></i>;<br/>
<a NAME="E20:23"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="uproots.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E21:23"/><b>then </b><i><font color="Green" title="E30">A12</font></i>: 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c9">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#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E16:23"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="uproots.html#E18:23"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="uproots.html#E19:23"><i><font color="Green" title="E29">A11</font></i></a>, <a class="txt" href="uproots.html#E6:23"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E22:23"/><b>then </b><i><font color="Green" title="E31">A13</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E23:23"/><i><font color="Green" title="E32">A14</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="uproots.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="card_1.html#T80" title="CARD_1:th.80">CARD_1:80</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1"/>
(  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font> or  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font> )
 </span><b>by </b><i><a class="txt" href="uproots.html#E23:23"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>suppose </b></a><a NAME="E1:23_1_1"/><i><font color="Green" title="E33">A15</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c10">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> <b> such that </b><br/><a NAME="E3:23_1_1"/><i><font color="Green" title="E34">A16</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">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#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">d</font>
 <b>and </b><br/><a NAME="E4:23_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">d</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T1" title="FSM_1:th.1">FSM_1:1</a></i>;<br/><b>set </b><font color="Maroon" title="c11">h</font> = <font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0;<br/><a NAME="E5:23_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="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">d</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>;<br/><a NAME="E6:23_1_1"/><b>then </b><i><font color="Green" title="E35">A17</font></i>: 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</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="c10">d</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">gr</font> = <font color="Maroon" title="c9">g</font>, <font color="Maroon" title="c13">fr</font> = <font color="Maroon" title="c7">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>  ;<br/><b>set </b><font color="Maroon" title="c14">F</font> = <font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span>;<br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</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="c9">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font>
<b>by </b><i><a class="txt" href="uproots.html#E22:23"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="uproots.html#E3:23_1_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>
;<br/><a NAME="E9:23_1_1"/><b>then </b><i><font color="Green" title="E36">A18</font></i>: 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon" title="c15">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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) </span>}</span> ;<br/><div><i><font color="Green" title="E37">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c16">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:23_1_1_2"/><i><font color="Green" title="E38">A20</font></i>: 
( ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font> &amp; not <font color="Maroon" title="c16">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="c16">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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font> ) )
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:23_1_1_2"/><i><font color="Green" title="E39">A21</font></i>: 
<font color="Maroon" title="c17">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">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>
 <b>and </b><br/><a NAME="E5:23_1_1_2"/><i><font color="Green" title="E40">A22</font></i>: 
<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="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="uproots.html#E4:23"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="uproots.html#E2:23_1_1_2"><i><font color="Green" title="E38">A20</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c17">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>  <b>by </b><i><a class="txt" href="uproots.html#E4:23_1_1_2"><i><font color="Green" title="E39">A21</font></i></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/><a NAME="E7:23_1_1_2"/>
<font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">x</font>
 <b>by </b><i><a class="txt" href="uproots.html#E4:23_1_1_2"><i><font color="Green" title="E39">A21</font></i></a>, <a class="txt" href="uproots.html#E5:23_1_1_2"><i><font color="Green" title="E40">A22</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E8:23_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="uproots.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="uproots.html#E2:23_1_1_2"><i><font color="Green" title="E38">A20</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/><a NAME="E9:23_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="uproots.html#E4:23_1_1_2"><i><font color="Green" title="E39">A21</font></i></a>, <a class="txt" href="uproots.html#E3:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E10:23_1_1_2"/>
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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) </span>}</span>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/></div><b>end;</b></div><a NAME="E11:23_1_1"/><i><font color="Green" title="E38">A23</font></i>: 
<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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_1_3"/>
<font color="Maroon" title="c16">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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) </span>}</span> 
 ;<br/>

<a NAME="E2:23_1_1_3"/><b>then </b>
 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="c16">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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
;<br/>
<b>hence </b><a NAME="E3:23_1_1_3"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c16">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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) </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>  <b>by </b><i><a class="txt" href="uproots.html#E10:23_1_1"><i><font color="Green" title="E37">A19</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><a NAME="E13:23_1_1"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c16">dd</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">dd</font> &amp; <font color="Maroon" title="c16">dd</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Th5</a>, <a class="txt" href="uproots.html#E11:23_1_1"><i><font color="Green" title="E38">A23</font></i></a>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">cdd</font> =  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c16">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>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>set </b><font color="Maroon" title="c18">cdi</font> = <font color="Maroon" title="c17">cdd</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ;<br/><b>reconsider </b><font color="Maroon" title="c19">cdi</font> = <font color="Maroon" title="c17">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="c16">dd</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c16">dd</font></span>)</span> <b>by </b><i><a class="ref" href="uproots.html#T7" target="_self" title="UPROOTS:th.7">Th7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">cadd</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c16">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>  <b>by </b><i><a class="ref" href="graph_5.html#T5" title="GRAPH_5:th.5">GRAPH_5:5</a></i>;<br/><a NAME="E17:23_1_1"/><i><font color="Green" title="E39">A24</font></i>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c20">cadd</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c16">dd</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c21">cdi</font> = <font color="Maroon" title="c19">cdi</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c16">dd</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/><b>set </b><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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">dd</font></span>)</span>;<br/><a NAME="E19:23_1_1"/><i><font color="Green" title="E40">A25</font></i>: 
<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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">dd</font></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="uproots.html#E5:23"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></i>;<br/><a NAME="E20:23_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">dd</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">dd</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><a NAME="E21:23_1_1"/><b>then </b><i><font color="Green" title="E41">A26</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">dd</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">dd</font>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:23_1_1_4_1"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">dd</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c24">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_4_1"/><i><font color="Green" title="E42">A27</font></i>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">dd</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:23_1_1_4_1"/><i><font color="Green" title="E43">A28</font></i>: 
<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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">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>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:23_1_1_4_1"/><i><font color="Green" title="E44">A29</font></i>: 
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">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> &amp; <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#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">dd</font> )
 <b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1_4_1"><i><font color="Green" title="E42">A27</font></i></a>, <a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/><b>reconsider </b><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>  <b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1_4_1"><i><font color="Green" title="E42">A27</font></i></a></i>;<br/><a NAME="E7:23_1_1_4_1"/><i><font color="Green" title="E45">A30</font></i>: 
<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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">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>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1_4_1"><i><font color="Green" title="E42">A27</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E8:23_1_1_4_1"/>
not <font color="Maroon" title="c25">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">dd</font>
 <b>by </b><i><a class="txt" href="uproots.html#E5:23_1_1_4_1"><i><font color="Green" title="E44">A29</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E9:23_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="uproots.html#E5:23_1_1_4_1"><i><font color="Green" title="E44">A29</font></i></a></i>;<br/><a NAME="E10:23_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> 0
 <b>by </b><i><a class="txt" href="uproots.html#E5:23_1_1_4_1"><i><font color="Green" title="E44">A29</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E11:23_1_1_4_1"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="uproots.html#E4:23_1_1_4_1"><i><font color="Green" title="E43">A28</font></i></a>, <a class="txt" href="uproots.html#E7:23_1_1_4_1"><i><font color="Green" title="E45">A30</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:23_1_1_4"/><i><font color="Green" title="E42">A31</font></i>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c24">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:23_1_1_4"/><i><font color="Green" title="E43">A32</font></i>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">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>
 <b>and </b><br/><a NAME="E5:23_1_1_4"/><i><font color="Green" title="E44">A33</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="uproots.html#E4:23"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_4_2"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">dd</font>
 ;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:23_1_1_4_2"/><i><font color="Green" title="E45">A34</font></i>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x</font>
 <b>and </b><br/><a NAME="E4:23_1_1_4_2"/><i><font color="Green" title="E46">A35</font></i>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font>
 <b>and </b><br/><a NAME="E5:23_1_1_4_2"/><i><font color="Green" title="E47">A36</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E6:23_1_1_4_2"/>
0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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>
 <b>by </b><i><a class="txt" href="uproots.html#E5:23_1_1_4_2"><i><font color="Green" title="E47">A36</font></i></a>, <a class="txt" href="uproots.html#E4:23_1_1_4_2"><i><font color="Green" title="E46">A35</font></i></a>, <a class="txt" href="uproots.html#E3:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E2:23_1_1_4"><i><font color="Green" title="E42">A31</font></i></a>, <a class="txt" href="uproots.html#E3:23_1_1_4_2"><i><font color="Green" title="E45">A34</font></i></a>, <a class="txt" href="uproots.html#E5:23_1_1_4"><i><font color="Green" title="E44">A33</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div><a NAME="E7:23_1_1_4"/><b>then </b>
<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#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">dd</font>
 <b>by </b><i><a class="txt" href="uproots.html#E4:23_1_1_4"><i><font color="Green" title="E43">A32</font></i></a>, <a class="txt" href="uproots.html#E3:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E8:23_1_1_4"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">dd</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E4:23_1_1_4"><i><font color="Green" title="E43">A32</font></i></a>, <a class="txt" href="uproots.html#E5:23_1_1_4"><i><font color="Green" title="E44">A33</font></i></a>, <a class="ref" href="funct_1.html#T73" title="FUNCT_1:th.73">FUNCT_1:73</a></i>;<br/></div><b>end;</b></div><a NAME="E23:23_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">dd</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E24:23_1_1"/><b>then </b>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font>,<span class="p1">(<span class="default"><a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">dd</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="uproots.html#E19:23_1_1"><i><font color="Green" title="E40">A25</font></i></a>, <a class="txt" href="uproots.html#E21:23_1_1"><i><font color="Green" title="E41">A26</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/><a NAME="E25:23_1_1"/><b>then </b><i><font color="Green" title="E42">A37</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">dd</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T81" title="CARD_1:th.81">CARD_1:81</a></i>;<br/><a NAME="E26:23_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">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#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c16">dd</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E11:23_1_1"><i><font color="Green" title="E38">A23</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>;<br/><a NAME="E27:23_1_1"/><b>then </b><i><font color="Green" title="E43">A38</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c16">dd</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="uproots.html#E7:23"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="uproots.html#E25:23_1_1"><i><font color="Green" title="E42">A37</font></i></a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>;<br/><a NAME="E28:23_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">cdd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c16">dd</font>
 <b>by </b><i><a class="ref" href="uproots.html#D1" target="_self" title="UPROOTS:def.1">Def1</a></i>;<br/><a NAME="E29:23_1_1"/><b>then </b><i><font color="Green" title="E44">A39</font></i>: 
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c17">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="c10">d</font>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="txt" href="uproots.html#E27:23_1_1"><i><font color="Green" title="E43">A38</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E30:23_1_1"/><i><font color="Green" title="E45">A40</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c17">cdd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">dd</font>
 <b>by </b><i><a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Th5</a></i>;<br/><a NAME="E31:23_1_1"/><i><font color="Green" title="E46">A41</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">cdi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">dd</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E32:23_1_1"/><i><font color="Green" title="E47">A42</font></i>: 
<font color="Maroon" title="c17">cdd</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="uproots.html#T4" target="_self" title="UPROOTS:th.4">Th4</a></i>;<br/><a NAME="E33:23_1_1"/><b>then </b><i><font color="Green" title="E48">A43</font></i>: 
<font color="Maroon" title="c21">cdi</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#T62" title="FUNCT_1:th.62">FUNCT_1:62</a></i>;<br/><a NAME="E34:23_1_1"/><i><font color="Green" title="E49">A44</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c21">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="c10">d</font>
 <b>by </b><i><a class="txt" href="uproots.html#E32:23_1_1"><i><font color="Green" title="E47">A42</font></i></a>, <a class="txt" href="uproots.html#E29:23_1_1"><i><font color="Green" title="E44">A39</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <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="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c23">z</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E36:23_1_1"/><i><font color="Green" title="E50">A45</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">dd</font>
 <b>and </b><br/><a NAME="E37:23_1_1"/><i><font color="Green" title="E51">A46</font></i>: 
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="c16">dd</font> holds <br/><font color="Maroon" title="c23">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>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E38:23_1_1"/><i><font color="Green" title="E52">A47</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c23">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_1_5"/>
<font color="Maroon" title="c24">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="c23">z</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c25">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_5"/><i><font color="Green" title="E53">A48</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font>
 <b>and </b><br/><a NAME="E4:23_1_1_5"/><i><font color="Green" title="E54">A49</font></i>: 
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:23_1_1_5"/><i><font color="Green" title="E55">A50</font></i>: 
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c25">x</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1_5"><i><font color="Green" title="E53">A48</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="txt" href="uproots.html#E37:23_1_1"><i><font color="Green" title="E51">A46</font></i></a>, <a class="txt" href="uproots.html#E4:23_1_1_5"><i><font color="Green" title="E54">A49</font></i></a></i>;<br/>
<a NAME="E6:23_1_1_5"/><i><font color="Green" title="E56">A51</font></i>: 
<font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">cdi</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c25">x</font>
 
<b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1_5"><i><font color="Green" title="E53">A48</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="txt" href="uproots.html#E31:23_1_1"><i><font color="Green" title="E46">A41</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E7:23_1_1_5"/>
<font color="Maroon" title="c21">cdi</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c25">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="c10">d</font>
 
<b>by </b><i><a class="txt" href="uproots.html#E34:23_1_1"><i><font color="Green" title="E49">A44</font></i></a>, <a class="txt" href="uproots.html#E3:23_1_1_5"><i><font color="Green" title="E53">A48</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="txt" href="uproots.html#E31:23_1_1"><i><font color="Green" title="E46">A41</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E8:23_1_1_5"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c25">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">d</font> )
 
<b>by </b><i><a class="txt" href="uproots.html#E6:23_1_1_5"><i><font color="Green" title="E56">A51</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:23_1_1_5"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c25">x</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c25">x</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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="c10">d</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E10:23_1_1_5"/>
<font color="Maroon" title="c24">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="txt" href="uproots.html#E5:23_1_1_5"><i><font color="Green" title="E55">A50</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon" title="c24">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c23">z</font>;<br/><a NAME="E39:23_1_1"/><i><font color="Green" title="E53">A52</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c23">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#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>;<br/><div><i><font color="Green" title="E54">A53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_6"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">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> 
 ;<br/><b>then consider </b><font color="Maroon" title="c25">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_6"/><i><font color="Green" title="E55">A54</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:23_1_1_6"/><i><font color="Green" title="E56">A55</font></i>: 
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font> )
 <b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1_6"><i><font color="Green" title="E55">A54</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c26">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> <b> such that </b><br/><a NAME="E6:23_1_1_6"/><i><font color="Green" title="E57">A56</font></i>: 
<font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font>
 <b>and </b><br/><a NAME="E7:23_1_1_6"/>
<font color="Maroon" title="c26">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font>
 <b>and </b><br/><a NAME="E8:23_1_1_6"/><i><font color="Green" title="E58">A57</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a></i>;<br/><a NAME="E9:23_1_1_6"/><i><font color="Green" title="E59">A58</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_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> <span class="p3">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="uproots.html#E19:23"><i><font color="Green" title="E29">A11</font></i></a>, <a class="txt" href="uproots.html#E13:23"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><a NAME="E10:23_1_1_6"/>
<font color="Maroon" title="c26">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">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>
 <b>by </b><i><a class="txt" href="uproots.html#E4:23_1_1_6"><i><font color="Green" title="E56">A55</font></i></a>, <a class="txt" href="uproots.html#E6:23_1_1_6"><i><font color="Green" title="E57">A56</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E11:23_1_1_6"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c26">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E12:23_1_1_6"/><b>then </b>
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="c26">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="uproots.html#E8:23_1_1_6"><i><font color="Green" title="E58">A57</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/><b>hence </b><a NAME="E13:23_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E6:23_1_1_6"><i><font color="Green" title="E57">A56</font></i></a>, <a class="txt" href="uproots.html#E4:23_1_1_6"><i><font color="Green" title="E56">A55</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1_6"><i><font color="Green" title="E59">A58</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:23_1_1_7_1"/><i><font color="Green" title="E55">A59</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_7_1_1"/>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font> )
 </span><b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_7_1"><i><font color="Green" title="E55">A59</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_7_1_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_1_7_1_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_7_1_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_1_7_1_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E11:23_1_1"><i><font color="Green" title="E38">A23</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:23_1_1_7"/><i><font color="Green" title="E55">A60</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c26">i</font> = <font color="Maroon" title="c25">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_7_2"/>
( <font color="Maroon" title="c7">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c7">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_2_1"><b>suppose </b></a><a NAME="E1:23_1_1_7_2_1"/>
<font color="Maroon" title="c7">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:23_1_1_7_2_1"/><b>then </b>
<font color="Maroon" title="c26">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font>
 <b>by </b><i><a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="txt" href="uproots.html#E2:23_1_1_7"><i><font color="Green" title="E55">A60</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_7_2_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_2_2"><b>suppose </b></a><a NAME="E1:23_1_1_7_2_2"/><i><font color="Green" title="E56">A61</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:23_1_1_7_2_2"/>
<font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c26">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="uproots.html#E2:23_1_1_7"><i><font color="Green" title="E55">A60</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E3:23_1_1_7_2_2"/><b>then </b>
<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="c26">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_7_2_2"><i><font color="Green" title="E56">A61</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/><a NAME="E4:23_1_1_7_2_2"/><b>then </b>
<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="c26">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_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> <span class="p3">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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>
 <b>by </b><i><a class="txt" href="uproots.html#E13:23"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="uproots.html#E19:23"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><a NAME="E5:23_1_1_7_2_2"/><b>then </b>
<font color="Maroon" title="c26">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="uproots.html#E2:23_1_1_7"><i><font color="Green" title="E55">A60</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><b>hence </b><a NAME="E6:23_1_1_7_2_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E42:23_1_1"/><b>then </b><i><font color="Green" title="E55">A62</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E43:23_1_1"/><b>then </b><i><font color="Green" title="E56">A63</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c23">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1_1_8"/><i><font color="Green" title="E57">A64</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_8_1"/>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font> )
 </span><b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_8"><i><font color="Green" title="E57">A64</font></i></a>, <a class="txt" href="uproots.html#E42:23_1_1"><i><font color="Green" title="E55">A62</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_8_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_8_1_1"/><i><font color="Green" title="E58">A65</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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>
 ;<br/><div class="add"><a NAME="E2:23_1_1_8_1_1"/><b>then </b>
not <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font>
 <b>by </b><i><a class="txt" href="uproots.html#E40:23_1_1"><i><font color="Green" title="E54">A53</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:23_1_1_8_1_1"/><b>then </b><i><font color="Green" title="E59">A66</font></i>: 
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c23">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">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="polynom1.html#K11" title="POLYNOM1:func.11">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">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/><a NAME="E4:23_1_1_8_1_1"/>
<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="polynom1.html#K11" title="POLYNOM1:func.11">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">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="polynom1.html#K11" title="POLYNOM1:func.11">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>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_8_1_1"><i><font color="Green" title="E58">A65</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E5:23_1_1_8_1_1"/><b>then </b>
<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="polynom1.html#K11" title="POLYNOM1:func.11">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">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="polynom1.html#K11" title="POLYNOM1:func.11">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>
 <b>by </b><i><a class="ref" href="funct_1.html#T25" title="FUNCT_1:th.25">FUNCT_1:25</a></i>;<br/><a NAME="E6:23_1_1_8_1_1"/><b>then </b><i><font color="Green" title="E60">A67</font></i>: 
<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="polynom1.html#K11" title="POLYNOM1:func.11">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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">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="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E13:23"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><a NAME="E7:23_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#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E23:23"><i><font color="Green" title="E32">A14</font></i></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/><b>hence </b><a NAME="E8:23_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="polynom1.html#K11" title="POLYNOM1:func.11">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="c23">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E18:23"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="uproots.html#E3:23_1_1_8_1_1"><i><font color="Green" title="E59">A66</font></i></a>, <a class="txt" href="uproots.html#E6:23_1_1_8_1_1"><i><font color="Green" title="E60">A67</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_8_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_8_1_2"/><i><font color="Green" title="E58">A68</font></i>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_8_1_2"/><b>then </b><i><font color="Green" title="E59">A69</font></i>: 
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c23">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><a NAME="E3:23_1_1_8_1_2"/>
<font color="Maroon" title="c23">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">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="c23">z</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_8_1_2"><i><font color="Green" title="E58">A68</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E4:23_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="polynom1.html#K11" title="POLYNOM1:func.11">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="c23">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E2:23_1_1_8_1_2"><i><font color="Green" title="E59">A69</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="uproots.html#E38:23_1_1"><i><font color="Green" title="E52">A47</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c25">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c23">z</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>, <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span> <b>by </b><i><a class="txt" href="uproots.html#E43:23_1_1"><i><font color="Green" title="E56">A63</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/><a NAME="E46:23_1_1"/><i><font color="Green" title="E57">A70</font></i>: 
<font color="Maroon" title="c25">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="23_1_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">a</font>, <font color="Maroon" title="c27">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:23_1_1_9"/><i><font color="Green" title="E58">A71</font></i>: 
<font color="Maroon" title="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c25">p</font>
 <b>and </b><br/><a NAME="E2:23_1_1_9"/><i><font color="Green" title="E59">A72</font></i>: 
<font color="Maroon" title="c27">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c25">p</font>
 <b>and </b><br/><a NAME="E3:23_1_1_9"/><i><font color="Green" title="E60">A73</font></i>: 
<font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">c</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_9_1"/>
( ( <font color="Maroon" title="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font> ) or ( <font color="Maroon" title="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font> &amp; <font color="Maroon" title="c27">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font> &amp; <font color="Maroon" title="c27">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font> ) )
 </span><b>by </b><i><a class="txt" href="uproots.html#E39:23_1_1"><i><font color="Green" title="E53">A52</font></i></a>, <a class="txt" href="uproots.html#E1:23_1_1_9"><i><font color="Green" title="E58">A71</font></i></a>, <a class="txt" href="uproots.html#E2:23_1_1_9"><i><font color="Green" title="E59">A72</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_9_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_9_1_1"/><i><font color="Green" title="E61">A74</font></i>: 
( <font color="Maroon" title="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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> )
 ;<br/><div class="add"><a NAME="E2:23_1_1_9_1_1"/><b>then </b>
not <font color="Maroon" title="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font>
 <b>by </b><i><a class="txt" href="uproots.html#E40:23_1_1"><i><font color="Green" title="E54">A53</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E62">A75</font></i>: <font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c26">a</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c26">a</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_1"><i><font color="Green" title="E61">A74</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><a NAME="E4:23_1_1_9_1_1"/>
not <font color="Maroon" title="c27">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_1"><i><font color="Green" title="E61">A74</font></i></a>, <a class="txt" href="uproots.html#E40:23_1_1"><i><font color="Green" title="E54">A53</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E63">A76</font></i>: <font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">c</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">c</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_1"><i><font color="Green" title="E61">A74</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><a NAME="E6:23_1_1_9_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="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_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> <span class="p3">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_1"><i><font color="Green" title="E61">A74</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E7:23_1_1_9_1_1"/><b>then </b>
<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="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">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="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E13:23"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><a NAME="E8:23_1_1_9_1_1"/><b>then </b><i><font color="Green" title="E64">A77</font></i>: 
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c26">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="c26">a</font>
 <b>by </b><i><a class="txt" href="uproots.html#E13:23"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>;<br/><a NAME="E9:23_1_1_9_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="c27">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_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> <span class="p3">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_1"><i><font color="Green" title="E61">A74</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E10:23_1_1_9_1_1"/><b>then </b>
<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="c27">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">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="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E13:23"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><a NAME="E11:23_1_1_9_1_1"/><b>then </b><i><font color="Green" title="E65">A78</font></i>: 
<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="c26">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="c27">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E13:23"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="uproots.html#E3:23_1_1_9"><i><font color="Green" title="E60">A73</font></i></a>, <a class="txt" href="uproots.html#E3:23_1_1_9_1_1"><i><font color="Green" title="E62">A75</font></i></a>, <a class="txt" href="uproots.html#E5:23_1_1_9_1_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="uproots.html#E8:23_1_1_9_1_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>;<br/><a NAME="E12:23_1_1_9_1_1"/>
<font color="Maroon" title="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">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>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_1"><i><font color="Green" title="E61">A74</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E13:23_1_1_9_1_1"/><b>then </b><i><font color="Green" title="E66">A79</font></i>: 
<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="c26">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a</font>
 <b>by </b><i><a class="txt" href="uproots.html#E5:23"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>;<br/><a NAME="E14:23_1_1_9_1_1"/>
<font color="Maroon" title="c27">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">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>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_1"><i><font color="Green" title="E61">A74</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><b>hence </b><a NAME="E15:23_1_1_9_1_1"/>
<font color="Maroon" title="c26">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">c</font>
 <b>by </b><i><a class="txt" href="uproots.html#E5:23"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="uproots.html#E11:23_1_1_9_1_1"><i><font color="Green" title="E65">A78</font></i></a>, <a class="txt" href="uproots.html#E13:23_1_1_9_1_1"><i><font color="Green" title="E66">A79</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_9_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_9_1_2"/><i><font color="Green" title="E61">A80</font></i>: 
( <font color="Maroon" title="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font> )
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E62">A81</font></i>: <font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">c</font> = 
<font color="Maroon" title="c23">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">c</font>
<b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_2"><i><font color="Green" title="E61">A80</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="txt" href="uproots.html#E37:23_1_1"><i><font color="Green" title="E51">A46</font></i></a></i>
;<br/><a NAME="E3:23_1_1_9_1_2"/>
not <font color="Maroon" title="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_2"><i><font color="Green" title="E61">A80</font></i></a>, <a class="txt" href="uproots.html#E40:23_1_1"><i><font color="Green" title="E54">A53</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E63">A82</font></i>: <font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c26">a</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c26">a</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_2"><i><font color="Green" title="E61">A80</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><a NAME="E5:23_1_1_9_1_2"/>
<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="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_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> <span class="p3">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_2"><i><font color="Green" title="E61">A80</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E6:23_1_1_9_1_2"/><b>then </b>
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c26">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="polynom1.html#K11" title="POLYNOM1:func.11">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>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E7:23_1_1_9_1_2"/><b>then </b><i><font color="Green" title="E64">A83</font></i>: 
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c26">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">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="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E13:23"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c28">ccc</font> = <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="polynom1.html#K11" title="POLYNOM1:func.11">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="c26">a</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/><a NAME="E9:23_1_1_9_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1_9"><i><font color="Green" title="E60">A73</font></i></a>, <a class="txt" href="uproots.html#E2:23_1_1_9_1_2"><i><font color="Green" title="E62">A81</font></i></a>, <a class="txt" href="uproots.html#E4:23_1_1_9_1_2"><i><font color="Green" title="E63">A82</font></i></a>, <a class="txt" href="uproots.html#E7:23_1_1_9_1_2"><i><font color="Green" title="E64">A83</font></i></a>, <a class="txt" href="uproots.html#E18:23"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:23_1_1_9_1_2"/><b>then </b>
<font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E11:23_1_1_9_1_2"/><b>then </b>
<font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E12:23_1_1_9_1_2"/><b>then </b><i><font color="Green" title="E65">A84</font></i>: 
<font color="Maroon" title="c21">cdi</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c27">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_2"><i><font color="Green" title="E61">A80</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="txt" href="uproots.html#E31:23_1_1"><i><font color="Green" title="E46">A41</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E13:23_1_1_9_1_2"/>
<font color="Maroon" title="c21">cdi</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c27">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="c21">cdi</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_2"><i><font color="Green" title="E61">A80</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="txt" href="uproots.html#E31:23_1_1"><i><font color="Green" title="E46">A41</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E14:23_1_1_9_1_2"/>
<font color="Maroon" title="c26">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">c</font>
 <b>by </b><i><a class="txt" href="uproots.html#E34:23_1_1"><i><font color="Green" title="E49">A44</font></i></a>, <a class="txt" href="uproots.html#E12:23_1_1_9_1_2"><i><font color="Green" title="E65">A84</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_9_1_3"><b>suppose </b></a><a NAME="E1:23_1_1_9_1_3"/><i><font color="Green" title="E61">A85</font></i>: 
( <font color="Maroon" title="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font> &amp; <font color="Maroon" title="c27">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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> )
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E62">A86</font></i>: <font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font> = 
<font color="Maroon" title="c23">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font>
<b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">a</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_3"><i><font color="Green" title="E61">A85</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="txt" href="uproots.html#E37:23_1_1"><i><font color="Green" title="E51">A46</font></i></a></i>
;<br/><a NAME="E3:23_1_1_9_1_3"/>
not <font color="Maroon" title="c27">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_3"><i><font color="Green" title="E61">A85</font></i></a>, <a class="txt" href="uproots.html#E40:23_1_1"><i><font color="Green" title="E54">A53</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E63">A87</font></i>: <font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">c</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">c</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_3"><i><font color="Green" title="E61">A85</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><a NAME="E5:23_1_1_9_1_3"/>
<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="c27">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_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> <span class="p3">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_3"><i><font color="Green" title="E61">A85</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E6:23_1_1_9_1_3"/><b>then </b>
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">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="polynom1.html#K11" title="POLYNOM1:func.11">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>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E7:23_1_1_9_1_3"/><b>then </b><i><font color="Green" title="E64">A88</font></i>: 
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">c</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">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="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E13:23"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c28">ccc</font> = <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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">c</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/><a NAME="E9:23_1_1_9_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">a</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1_9"><i><font color="Green" title="E60">A73</font></i></a>, <a class="txt" href="uproots.html#E2:23_1_1_9_1_3"><i><font color="Green" title="E62">A86</font></i></a>, <a class="txt" href="uproots.html#E4:23_1_1_9_1_3"><i><font color="Green" title="E63">A87</font></i></a>, <a class="txt" href="uproots.html#E7:23_1_1_9_1_3"><i><font color="Green" title="E64">A88</font></i></a>, <a class="txt" href="uproots.html#E18:23"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:23_1_1_9_1_3"/><b>then </b>
<font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E11:23_1_1_9_1_3"/><b>then </b>
<font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E12:23_1_1_9_1_3"/><b>then </b><i><font color="Green" title="E65">A89</font></i>: 
<font color="Maroon" title="c21">cdi</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c26">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_3"><i><font color="Green" title="E61">A85</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="txt" href="uproots.html#E31:23_1_1"><i><font color="Green" title="E46">A41</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E13:23_1_1_9_1_3"/>
<font color="Maroon" title="c21">cdi</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c26">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="c21">cdi</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_3"><i><font color="Green" title="E61">A85</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="txt" href="uproots.html#E31:23_1_1"><i><font color="Green" title="E46">A41</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E14:23_1_1_9_1_3"/>
<font color="Maroon" title="c26">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">c</font>
 <b>by </b><i><a class="txt" href="uproots.html#E34:23_1_1"><i><font color="Green" title="E49">A44</font></i></a>, <a class="txt" href="uproots.html#E12:23_1_1_9_1_3"><i><font color="Green" title="E65">A89</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_9_1_4"><b>suppose </b></a><a NAME="E1:23_1_1_9_1_4"/><i><font color="Green" title="E61">A90</font></i>: 
( <font color="Maroon" title="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font> &amp; <font color="Maroon" title="c27">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font> )
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E62">A91</font></i>: <font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font> = 
<font color="Maroon" title="c23">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font>
<b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">a</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_4"><i><font color="Green" title="E61">A90</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="txt" href="uproots.html#E37:23_1_1"><i><font color="Green" title="E51">A46</font></i></a></i>
;<br/><i><font color="Green" title="E63">A92</font></i>: <font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">c</font> = 
<font color="Maroon" title="c23">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">c</font>
<b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_4"><i><font color="Green" title="E61">A90</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">c</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_4"><i><font color="Green" title="E61">A90</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="txt" href="uproots.html#E37:23_1_1"><i><font color="Green" title="E51">A46</font></i></a></i>
;<br/><a NAME="E4:23_1_1_9_1_4"/>
( <font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">cdi</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c26">a</font> &amp; <font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c27">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">cdi</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c27">c</font> )
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_9_1_4"><i><font color="Green" title="E61">A90</font></i></a>, <a class="txt" href="uproots.html#E31:23_1_1"><i><font color="Green" title="E46">A41</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E5:23_1_1_9_1_4"/>
<font color="Maroon" title="c26">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">c</font>
 <b>by </b><i><a class="txt" href="uproots.html#E31:23_1_1"><i><font color="Green" title="E46">A41</font></i></a>, <a class="txt" href="uproots.html#E33:23_1_1"><i><font color="Green" title="E48">A43</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="txt" href="uproots.html#E3:23_1_1_9"><i><font color="Green" title="E60">A73</font></i></a>, <a class="txt" href="uproots.html#E1:23_1_1_9_1_4"><i><font color="Green" title="E61">A90</font></i></a>, <a class="txt" href="uproots.html#E2:23_1_1_9_1_4"><i><font color="Green" title="E62">A91</font></i></a>, <a class="txt" href="uproots.html#E3:23_1_1_9_1_4"><i><font color="Green" title="E63">A92</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><div><i><font color="Green" title="E58">A93</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:23_1_1_10_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c25">p</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c27">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_10_1"/><i><font color="Green" title="E59">A94</font></i>: 
<font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c25">p</font>
 <b>and </b><br/><a NAME="E4:23_1_1_10_1"/><i><font color="Green" title="E60">A95</font></i>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_10_1_1"/>
( <font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font> )
 </span><b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1_10_1"><i><font color="Green" title="E59">A94</font></i></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_10_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_10_1_1_1"/><i><font color="Green" title="E61">A96</font></i>: 
<font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="polynom1.html#K11" title="POLYNOM1:func.11">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>
 ;<br/><div class="add"><a NAME="E2:23_1_1_10_1_1_1"/><b>then </b>
not <font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font>
 <b>by </b><i><a class="txt" href="uproots.html#E40:23_1_1"><i><font color="Green" title="E54">A53</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E62">A97</font></i>: <font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">a</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">a</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_10_1_1_1"><i><font color="Green" title="E61">A96</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><a NAME="E4:23_1_1_10_1_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="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_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> <span class="p3">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_10_1_1_1"><i><font color="Green" title="E61">A96</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E5:23_1_1_10_1_1_1"/><b>then </b>
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">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="polynom1.html#K11" title="POLYNOM1:func.11">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>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E6:23_1_1_10_1_1_1"/><b>then </b><i><font color="Green" title="E63">A98</font></i>: 
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">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="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E13:23"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><a NAME="E7:23_1_1_10_1_1_1"/>
 <a href="finsop_1.html#K5" title="FINSOP_1:func.5">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="polynom1.html#K11" title="POLYNOM1:func.11">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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E23:23"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="uproots.html#E18:23"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/><b>hence </b><a NAME="E8:23_1_1_10_1_1_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E6:23_1_1_10_1_1_1"><i><font color="Green" title="E63">A98</font></i></a>, <a class="txt" href="uproots.html#E3:23_1_1_10_1_1_1"><i><font color="Green" title="E62">A97</font></i></a>, <a class="txt" href="uproots.html#E4:23_1_1_10_1"><i><font color="Green" title="E60">A95</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_10_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_10_1_1_2"/><i><font color="Green" title="E61">A99</font></i>: 
<font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_10_1_1_2"/><b>then </b><i><font color="Green" title="E62">A100</font></i>: 
<font color="Maroon" title="c23">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">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="c23">z</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E3:23_1_1_10_1_1_2"/>
<font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">a</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_10_1_1_2"><i><font color="Green" title="E61">A99</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_10_1_1_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E4:23_1_1_10_1"><i><font color="Green" title="E60">A95</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="uproots.html#E38:23_1_1"><i><font color="Green" title="E52">A47</font></i></a>, <a class="txt" href="uproots.html#E2:23_1_1_10_1_1_2"><i><font color="Green" title="E62">A100</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:23_1_1_10"/><i><font color="Green" title="E59">A101</font></i>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c27">j</font> = <font color="Maroon" title="c26">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_10_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_10_2"/>
( <font color="Maroon" title="c27">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c12">gr</font> or  ex <font color="Olive" title="b1">n</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">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span> &amp; <font color="Maroon" title="c27">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="c12">gr</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">n</font> ) )
 </span><b>by </b><i><a class="txt" href="uproots.html#E2:23_1_1_10"><i><font color="Green" title="E59">A101</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_10_2_1"><b>suppose </b></a><a NAME="E1:23_1_1_10_2_1"/><i><font color="Green" title="E60">A102</font></i>: 
<font color="Maroon" title="c27">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c12">gr</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_10_2_1"/><b>then </b><i><font color="Green" title="E61">A103</font></i>: 
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="uproots.html#E18:23"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="uproots.html#E19:23"><i><font color="Green" title="E29">A11</font></i></a>, <a class="txt" href="uproots.html#E21:23"><i><font color="Green" title="E30">A12</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E3:23_1_1_10_2_1"/><b>then </b><i><font color="Green" title="E62">A104</font></i>: 
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="uproots.html#E3:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="uproots.html#E4:23"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="uproots.html#E5:23"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_10_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_10_2_1_1"/><i><font color="Green" title="E63">A105</font></i>: 
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">j</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font>
 ;<br/><i><font color="Green" title="E64">A106</font></i>: <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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">j</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1_10_2_1"><i><font color="Green" title="E62">A104</font></i></a>, <a class="txt" href="uproots.html#E3:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">j</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="uproots.html#E4:23"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="uproots.html#E5:23"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="uproots.html#E2:23_1_1_10_2_1"><i><font color="Green" title="E61">A103</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>
;<br/><a NAME="E3:23_1_1_10_2_1_1"/>
 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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">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="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_10_2_1_1"><i><font color="Green" title="E63">A105</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_10_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E2:23_1_1_10_2_1"><i><font color="Green" title="E61">A103</font></i></a>, <a class="txt" href="uproots.html#E2:23_1_1_10_2_1_1"><i><font color="Green" title="E64">A106</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div><b>then </b><font color="Maroon" title="c25">p</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"><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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">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="polynom1.html#K11" title="POLYNOM1:func.11">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1_10_2_1"><i><font color="Green" title="E62">A104</font></i></a>, <a class="txt" href="uproots.html#E3:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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="polynom1.html#K11" title="POLYNOM1:func.11">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">j</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="uproots.html#E4:23"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="uproots.html#E5:23"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="uproots.html#E2:23_1_1_10_2_1"><i><font color="Green" title="E61">A103</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>
<br/>.= 
<font color="Maroon" title="c27">j</font>
<b>by </b><i><a class="txt" href="uproots.html#E18:23"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="uproots.html#E21:23"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="uproots.html#E1:23_1_1_10_2_1"><i><font color="Green" title="E60">A102</font></i></a>, <a class="txt" href="uproots.html#E13:23"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>
;<br/><b>hence </b><a NAME="E6:23_1_1_10_2_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c25">p</font>
 <b>by </b><i><a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="uproots.html#E43:23_1_1"><i><font color="Green" title="E56">A63</font></i></a>, <a class="txt" href="uproots.html#E3:23_1_1_10_2_1"><i><font color="Green" title="E62">A104</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_10_2_2"><b>suppose </b></a><a NAME="E1:23_1_1_10_2_2"/>
 ex <font color="Olive" title="b1">n</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">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span> &amp; <font color="Maroon" title="c27">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="c12">gr</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">n</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c28">n</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> <b> such that </b><br/><a NAME="E3:23_1_1_10_2_2"/><i><font color="Green" title="E60">A107</font></i>: 
<font color="Maroon" title="c28">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span>
 <b>and </b><br/><a NAME="E4:23_1_1_10_2_2"/><i><font color="Green" title="E61">A108</font></i>: 
<font color="Maroon" title="c27">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="c12">gr</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c28">n</font>
 ;<br/><a NAME="E5:23_1_1_10_2_2"/><i><font color="Green" title="E62">A109</font></i>: 
<font color="Maroon" title="c17">cdd</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c28">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">dd</font>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1_10_2_2"><i><font color="Green" title="E60">A107</font></i></a>, <a class="txt" href="uproots.html#E29:23_1_1"><i><font color="Green" title="E44">A39</font></i></a>, <a class="txt" href="uproots.html#E30:23_1_1"><i><font color="Green" title="E45">A40</font></i></a>, <a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>then </b><font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">cdd</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c28">n</font></span>)</span> = 
<font color="Maroon" title="c23">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">cdd</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c28">n</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">cdd</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c28">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E5:23_1_1_10_2_2"><i><font color="Green" title="E62">A109</font></i></a>, <a class="txt" href="uproots.html#E37:23_1_1"><i><font color="Green" title="E51">A46</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">cdi</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">cdd</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c28">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E5:23_1_1_10_2_2"><i><font color="Green" title="E62">A109</font></i></a>, <a class="txt" href="uproots.html#E31:23_1_1"><i><font color="Green" title="E46">A41</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c28">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#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1_10_2_2"><i><font color="Green" title="E60">A107</font></i></a>, <a class="txt" href="uproots.html#E29:23_1_1"><i><font color="Green" title="E44">A39</font></i></a>, <a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="uproots.html#E32:23_1_1"><i><font color="Green" title="E47">A42</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>
<br/>.= 
<font color="Maroon" title="c27">j</font>
<b>by </b><i><a class="txt" href="uproots.html#E21:23"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="uproots.html#E4:23_1_1_10_2_2"><i><font color="Green" title="E61">A108</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><b>hence </b><a NAME="E7:23_1_1_10_2_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c25">p</font>
 <b>by </b><i><a class="txt" href="uproots.html#E7:23"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="uproots.html#E11:23_1_1"><i><font color="Green" title="E38">A23</font></i></a>, <a class="txt" href="uproots.html#E43:23_1_1"><i><font color="Green" title="E56">A63</font></i></a>, <a class="txt" href="uproots.html#E5:23_1_1_10_2_2"><i><font color="Green" title="E62">A109</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E48:23_1_1"/><b>then </b><i><font color="Green" title="E59">A110</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c25">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E49:23_1_1"/><b>then </b>
<font color="Maroon" title="c25">p</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><a NAME="E50:23_1_1"/><b>then </b><i><font color="Green" title="E60">A111</font></i>: 
<font color="Maroon" title="c25">p</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E46:23_1_1"><i><font color="Green" title="E57">A70</font></i></a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/><a NAME="E51:23_1_1"/><i><font color="Green" title="E61">A112</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c25">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E43:23_1_1"><i><font color="Green" title="E56">A63</font></i></a>, <a class="txt" href="uproots.html#E47:23_1_1"><i><font color="Green" title="E58">A93</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1_1_11"/><i><font color="Green" title="E62">A113</font></i>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_11_1"/>
( <font color="Maroon" title="c7">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c7">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_11_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_11_1_1"/><i><font color="Green" title="E63">A114</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c27">j</font> = <font color="Maroon" title="c26">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_11"><i><font color="Green" title="E62">A113</font></i></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/><a NAME="E3:23_1_1_11_1_1"/><i><font color="Green" title="E64">A115</font></i>: 
<font color="Maroon" title="c27">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">z</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_11"><i><font color="Green" title="E62">A113</font></i></a>, <a class="txt" href="uproots.html#E1:23_1_1_11_1_1"><i><font color="Green" title="E63">A114</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E65">A116</font></i>: <font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> = 
<font color="Maroon" title="c23">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">x</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1_11_1_1"><i><font color="Green" title="E64">A115</font></i></a>, <a class="txt" href="uproots.html#E37:23_1_1"><i><font color="Green" title="E51">A46</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a></i>
;<br/><a NAME="E5:23_1_1_11_1_1"/>
<font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E7:23"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="uproots.html#E43:23_1_1"><i><font color="Green" title="E56">A63</font></i></a>, <a class="txt" href="uproots.html#E48:23_1_1"><i><font color="Green" title="E59">A110</font></i></a>, <a class="txt" href="uproots.html#E1:23_1_1_11"><i><font color="Green" title="E62">A113</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c28">px</font> = <font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/><a NAME="E7:23_1_1_11_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">cdi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">dd</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E8:23_1_1_11_1_1"/><b>then </b><i><font color="Green" title="E66">A117</font></i>: 
<font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">cdi</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c26">x</font>
 <b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1_11_1_1"><i><font color="Green" title="E64">A115</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E9:23_1_1_11_1_1"/><i><font color="Green" title="E67">A118</font></i>: 
<font color="Maroon" title="c21">cdi</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c26">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c16">dd</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E17:23_1_1"><i><font color="Green" title="E39">A24</font></i></a>, <a class="txt" href="uproots.html#E3:23_1_1_11_1_1"><i><font color="Green" title="E64">A115</font></i></a>, <a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">cdix</font> = <font color="Maroon" title="c21">cdi</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c26">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/><b>thus </b><font color="Maroon" title="c7">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c26">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c29">cdix</font>
<b>by </b><i><a class="txt" href="uproots.html#E3:23_1_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="txt" href="uproots.html#E27:23_1_1"><i><font color="Green" title="E43">A38</font></i></a>, <a class="txt" href="uproots.html#E1:23_1_1_11_1_1"><i><font color="Green" title="E63">A114</font></i></a>, <a class="txt" href="uproots.html#E8:23_1_1_11_1_1"><i><font color="Green" title="E66">A117</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1_11_1_1"><i><font color="Green" title="E67">A118</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c28">px</font>
<b>by </b><i><a class="txt" href="uproots.html#E22:23"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="uproots.html#E3:23_1_1"><i><font color="Green" title="E34">A16</font></i></a>, <a class="txt" href="uproots.html#E6:23_1_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="uproots.html#E27:23_1_1"><i><font color="Green" title="E43">A38</font></i></a>, <a class="txt" href="uproots.html#E4:23_1_1_11_1_1"><i><font color="Green" title="E65">A116</font></i></a>, <a class="txt" href="uproots.html#E8:23_1_1_11_1_1"><i><font color="Green" title="E66">A117</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1_11_1_1"><i><font color="Green" title="E67">A118</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c25">p</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="txt" href="uproots.html#E7:23"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="uproots.html#E51:23_1_1"><i><font color="Green" title="E61">A112</font></i></a>, <a class="txt" href="uproots.html#E1:23_1_1_11"><i><font color="Green" title="E62">A113</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_11_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_11_1_2"/><i><font color="Green" title="E63">A119</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><i><font color="Green" title="E64">A120</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_11_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_11_1_2_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">dd</font>
 ;<br/><a NAME="E2:23_1_1_11_1_2_1"/><b>then </b>
 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="c26">x</font> &amp; <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><b>hence </b><a NAME="E3:23_1_1_11_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_11_1_2"><i><font color="Green" title="E63">A119</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:23_1_1_11_1_2"/>
<font color="Maroon" title="c7">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c26">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_11"><i><font color="Green" title="E62">A113</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E4:23_1_1_11_1_2"/><b>then </b>
<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="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_11_1_2"><i><font color="Green" title="E63">A119</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/><a NAME="E5:23_1_1_11_1_2"/><b>then </b><i><font color="Green" title="E65">A121</font></i>: 
<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="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">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="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Th5</a></i>;<br/><a NAME="E6:23_1_1_11_1_2"/><b>then </b><i><font color="Green" title="E66">A122</font></i>: 
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c26">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">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="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E13:23"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/><i><font color="Green" title="E67">A123</font></i>: <font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">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="polynom1.html#K11" title="POLYNOM1:func.11">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="c26">x</font>
<b>by </b><i><a class="txt" href="uproots.html#E36:23_1_1"><i><font color="Green" title="E50">A45</font></i></a>, <a class="txt" href="uproots.html#E2:23_1_1_11_1_2"><i><font color="Green" title="E64">A120</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<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="polynom1.html#K11" title="POLYNOM1:func.11">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="c26">x</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_11"><i><font color="Green" title="E62">A113</font></i></a>, <a class="txt" href="uproots.html#E3:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="uproots.html#E7:23"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><b>then reconsider </b><font color="Maroon" title="c27">px</font> = <font color="Maroon" title="c25">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="uproots.html#E6:23_1_1_11_1_2"><i><font color="Green" title="E66">A122</font></i></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/><b>thus </b><font color="Maroon" title="c7">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c26">x</font> = 
<font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c26">x</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:23_1_1_11"><i><font color="Green" title="E62">A113</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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="c27">px</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E13:23"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="uproots.html#E5:23_1_1_11_1_2"><i><font color="Green" title="E65">A121</font></i></a>, <a class="txt" href="uproots.html#E7:23_1_1_11_1_2"><i><font color="Green" title="E67">A123</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>
<br/>.= 
<font color="Maroon" title="c9">g</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c27">px</font>
<b>by </b><i><a class="txt" href="uproots.html#E16:23"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="uproots.html#E6:23_1_1_11_1_2"><i><font color="Green" title="E66">A122</font></i></a>, <a class="txt" href="uproots.html#E7:23_1_1_11_1_2"><i><font color="Green" title="E67">A123</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c27">px</font>
<b>by </b><i><a class="txt" href="uproots.html#E18:23"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="uproots.html#E21:23"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="uproots.html#E6:23_1_1_11_1_2"><i><font color="Green" title="E66">A122</font></i></a>, <a class="txt" href="uproots.html#E7:23_1_1_11_1_2"><i><font color="Green" title="E67">A123</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c25">p</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="txt" href="uproots.html#E7:23"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="uproots.html#E51:23_1_1"><i><font color="Green" title="E61">A112</font></i></a>, <a class="txt" href="uproots.html#E1:23_1_1_11"><i><font color="Green" title="E62">A113</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E53:23_1_1"/><b>then </b><i><font color="Green" title="E62">A124</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c25">p</font>
 <b>by </b><i><a class="txt" href="uproots.html#E7:23"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="uproots.html#E9:23_1_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="txt" href="uproots.html#E51:23_1_1"><i><font color="Green" title="E61">A112</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><a NAME="E54:23_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="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="rvsum_1.html#T111" title="RVSUM_1:th.111">RVSUM_1:111</a></i>;<br/><b>then </b> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c12">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="c12">gr</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</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="c12">gr</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>
;<br/><b>hence </b><a NAME="E56:23_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="c7">f</font>
 <b>by </b><i><a class="txt" href="uproots.html#E15:23"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="uproots.html#E50:23_1_1"><i><font color="Green" title="E60">A111</font></i></a>, <a class="txt" href="uproots.html#E53:23_1_1"><i><font color="Green" title="E62">A124</font></i></a>, <a class="ref" href="finsop_1.html#T8" title="FINSOP_1:th.8">FINSOP_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2"><b>suppose </b></a><a NAME="E1:23_1_2"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_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="c7">f</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="uproots.html#E15:23"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="uproots.html#E16:23"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="triang_1.html#T3" title="TRIANG_1:th.3">TRIANG_1:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
