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<b>set </b><font color="Maroon" title="c2">f</font> = <font color="Maroon" title="c1">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<a NAME="E1:83_1_1"/><i><font color="Green" title="E42">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c1">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-yielding</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-support</a> )</span><br/>

<b>thus </b><a NAME="E3:83_1_1"/>
<font color="Maroon" title="c3">f</font> is <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-yielding</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> is <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-support</a></span><br/>

<a NAME="E4:83_1_1"/>
 <a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:83_1_1_1"/>
not  <a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:83_1_1_1"/><i><font color="Green" title="E43">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <font color="Maroon" title="c3">f</font>
 <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="E4:83_1_1_1"/>
 <a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41" target="_self" title="POLYNOM1:th.41">Th41</a></i>;<br/>
<a NAME="E5:83_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="polynom1.html#E1:83_1_1"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="polynom1.html#E3:83_1_1_1"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E6:83_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom1.html#E3:83_1_1_1"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="polynom1.html#D7" target="_self" title="POLYNOM1:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:83_1_1"/>
<font color="Maroon" title="c3">f</font> is <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-support</a>
 <b>by </b><i><a class="ref" href="polynom1.html#D8" target="_self" title="POLYNOM1:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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