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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">b</font>, <font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c4">b2</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="E1:24"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c4">b2</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">S</font> =  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c2">b</font>;<br/>
<b>set </b><font color="Maroon" title="c6">SS</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c4">b2</font></span>)</span>;<br/>
<a NAME="E2:24"/><i><font color="Green" title="E21">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b2</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="E3:24"/><b>then </b>
(  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c4">b2</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="ref" href="polynom1.html#T41" title="POLYNOM1:th.41">POLYNOM1:41</a></i>;<br/>
<a NAME="E4:24"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">SS</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c4">b2</font></span>)</span> as  <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/>
<a NAME="E6:24"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c2">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">SS</font>
 
<b>by </b><i><a class="txt" href="uproots.html#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="polynom1.html#T91" title="POLYNOM1:th.91">POLYNOM1:91</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E8:24"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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="c7">SS</font></span>)</span>
 <b>and </b><br/><a NAME="E9:24"/><i><font color="Green" title="E23">A4</font></i>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c2">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="c8">f</font>
 <b>by </b><i><a class="ref" href="uproots.html#T16" target="_self" title="UPROOTS:th.16">Th16</a></i>;<br/>
<a NAME="E10:24"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">SS</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">f1r</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="E12:24"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c9">f1r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</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="c7">SS</font></span>)</span>
 <b>and </b><br/><a NAME="E13:24"/><i><font color="Green" title="E25">A6</font></i>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b1</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">f1r</font>
 <b>by </b><i><a class="ref" href="uproots.html#T16" target="_self" title="UPROOTS:th.16">Th16</a></i>;<br/>
<a NAME="E14:24"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c4">b2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">SS</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">f2r</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="E16:24"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c10">f2r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</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="c7">SS</font></span>)</span>
 <b>and </b><br/><a NAME="E17:24"/><i><font color="Green" title="E27">A8</font></i>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c4">b2</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="c10">f2r</font>
 <b>by </b><i><a class="ref" href="uproots.html#T16" target="_self" title="UPROOTS:th.16">Th16</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">cS</font> =  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c7">SS</font>;<br/>
<a NAME="E18:24"/>
 <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="c7">SS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">SS</font>
 
<b>by </b><i><a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Th5</a></i>;<br/>
<a NAME="E19:24"/><b>then </b><i><font color="Green" title="E28">A9</font></i>: 
(  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c8">f</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"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c7">SS</font></span>)</span> &amp;  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c9">f1r</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"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c7">SS</font></span>)</span> &amp;  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c10">f2r</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"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c7">SS</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="uproots.html#E2:24"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="uproots.html#E8:24"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="uproots.html#E12:24"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="uproots.html#E16:24"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E20:24"/><b>then </b><i><font color="Green" title="E29">A10</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f1r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f2r</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f1r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">k</font> be    <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>assume </b><a NAME="E1:24_1"/><i><font color="Green" title="E30">A11</font></i>: 
<font color="Maroon" title="c12">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="c9">f1r</font>
 ;<br/><i><font color="Green" title="E31">A12</font></i>: <font color="Maroon" title="c9">f1r</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">k</font> = 
<font color="Maroon" title="c9">f1r</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
<b>by </b><i><a class="txt" href="uproots.html#E1:24_1"><i><font color="Green" title="E30">A11</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="c3">b1</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="c7">SS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:24_1"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="uproots.html#E12:24"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><i><font color="Green" title="E32">A13</font></i>: <font color="Maroon" title="c10">f2r</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">k</font> = 
<font color="Maroon" title="c10">f2r</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
<b>by </b><i><a class="txt" href="uproots.html#E1:24_1"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="uproots.html#E19:24"><i><font color="Green" title="E28">A9</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="c4">b2</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="c7">SS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:24_1"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="uproots.html#E16:24"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="uproots.html#E19:24"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><a NAME="E4:24_1"/>
<font color="Maroon" title="c8">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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="c7">SS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E1:24_1"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="uproots.html#E8:24"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="uproots.html#E19:24"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><b>hence </b><a NAME="E5:24_1"/>
<font color="Maroon" title="c8">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1r</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2r</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E2:24_1"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="uproots.html#E3:24_1"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="uproots.html#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E22:24"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E9:24"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="uproots.html#E13:24"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="uproots.html#E17:24"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="uproots.html#E20:24"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="integra1.html#T23" title="INTEGRA1:th.23">INTEGRA1:23</a></i>;<br/>


</div>
