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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c1">n</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c2">T</font> );<br/>
<a NAME="E1:40_1_1"/><i><font color="Green" title="E22">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</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> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c4">P</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font>] holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
;<br/>
<a NAME="E2:40_1_1"/><i><font color="Green" title="E23">A2</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> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c4">P</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c4">P</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<b>assume </b><a NAME="E1:40_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c4">P</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<b>then consider </b><font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">p2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1_1"/><i><font color="Green" title="E24">A3</font></i>: 
( <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">p1</font> = <font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c9">p2</font> = <font color="Maroon" title="c7">p2</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="txt" href="groeb_2.html#E3:40_1_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>take </b>
 <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">T</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>, <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">T</font>]</span><br/>

<b>take </b>
<font color="Maroon" title="c8">p1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p2</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p2</font> &amp; <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Olive" title="b1">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp;  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c8">p1</font>,<font color="Olive" title="b1">p2</font>,<font color="Maroon" title="c2">T</font> )</span><br/>

<b>take </b>
<font color="Maroon" title="c9">p2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> &amp; <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c9">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp;  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">T</font> )</span><br/>

<b>thus </b><a NAME="E5:40_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p1</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:40_1_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> &amp; <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c9">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp;  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">T</font> )</span><br/>

<b>thus </b><a NAME="E6:40_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:40_1_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c9">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp;  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">T</font> )</span><br/>

<b>thus </b><a NAME="E7:40_1_1_1"/>
( <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c9">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp;  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:40_1_1_1"><i><font color="Green" title="E24">A3</font></i></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>consider </b><font color="Maroon" title="c5">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:40_1_1"/><i><font color="Green" title="E24">A4</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c4">P</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp; ( 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> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c4">P</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="groeb_2.html#E2:40_1_1"><i><font color="Green" title="E23">A2</font></i></a>);<br/></i>
<div><i><font color="Green" title="E25">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">v</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="c5">f</font> implies <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_2.html#K4" title="GROEB_2:func.4">S-Poly</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> )</span><br/><b>assume </b><a NAME="E1:40_1_1_2"/>
<font color="Maroon" title="c6">v</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="c5">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="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_2.html#K4" title="GROEB_2:func.4">S-Poly</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span><br/><b>then consider </b><font color="Maroon" title="c7">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1_2"/><i><font color="Green" title="E26">A6</font></i>: 
( <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">u</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/><b>consider </b><font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c9">p2</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E5:40_1_1_2"/><i><font color="Green" title="E27">A7</font></i>: 
( <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c7">u</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> &amp; <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c9">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E4:40_1_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="groeb_2.html#E3:40_1_1_2"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/><b>thus </b><a NAME="E6:40_1_1_2"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_2.html#K4" title="GROEB_2:func.4">S-Poly</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:40_1_1_2"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="groeb_2.html#E5:40_1_1_2"><i><font color="Green" title="E27">A7</font></i></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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_2.html#K4" title="GROEB_2:func.4">S-Poly</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c6">v</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="c5">f</font> )</span><br/><b>assume </b><a NAME="E1:40_1_1_3"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_2.html#K4" title="GROEB_2:func.4">S-Poly</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">v</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="c5">f</font></span><br/><b>then consider </b><font color="Maroon" title="c7">p1</font>, <font color="Maroon" title="c8">p2</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E3:40_1_1_3"/><i><font color="Green" title="E26">A8</font></i>: 
( <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c7">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c8">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> )
 ;<br/><a NAME="E4:40_1_1_3"/><i><font color="Green" title="E27">A9</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E4:40_1_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="groeb_2.html#E3:40_1_1_3"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">p1'</font>, <font color="Maroon" title="c10">p2'</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E6:40_1_1_3"/><i><font color="Green" title="E28">A10</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p1'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p2'</font> &amp; <font color="Maroon" title="c9">p1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c10">p2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c9">p1'</font>,<font color="Maroon" title="c10">p2'</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E4:40_1_1"><i><font color="Green" title="E24">A4</font></i></a></i>;<br/><a NAME="E7:40_1_1_3"/>
( <font color="Maroon" title="c7">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p1'</font> &amp; <font color="Maroon" title="c8">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p2'</font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E6:40_1_1_3"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/><b>hence </b><a NAME="E8:40_1_1_3"/>
<font color="Maroon" title="c6">v</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="c5">f</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:40_1_1_3"><i><font color="Green" title="E26">A8</font></i></a>, <a class="txt" href="groeb_2.html#E4:40_1_1_3"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="groeb_2.html#E6:40_1_1_3"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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>
<a NAME="E7:40_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="groeb_2.html#K4" title="GROEB_2:func.4">S-Poly</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E5:40_1_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E8:40_1_1"/>
 <a href="groeb_2.html#K4" title="GROEB_2:func.4">S-Poly</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="groeb_2.html#E4:40_1_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</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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