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<span class="kw">defpred </span><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> ] <span class="kw">means </span> ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b1">p</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> );<br/>
<span class="kw">set </span><font color="Maroon" title="c5">A</font> =  <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">B</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">R</font> being   <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <span class="p1">(<span class="default"><a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:76_1_1"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="relset_1.html#S1" title="RELSET_1:sch.1">RELSET_1:sch 1</a>();<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c7">R</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font> iff <font color="Olive" title="b1">p</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">q</font> be   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font> iff <font color="Maroon" title="c8">p</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> )</span><br/><div><span class="lab"><font color="Green" title="E59">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:76_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A3</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">pp</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E3:76_1_1_1_1"/>
<font color="Maroon" title="c10">pp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font>
 <span class="kw">and </span><br/><a NAME="E4:76_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c10">pp</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D7" target="_self" title="POLYRED:def.7">Def7</a></span>;<br/><a NAME="E5:76_1_1_1_1"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> st <font color="Maroon" title="c8">p</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c10">pp</font>,<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:76_1_1_1_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="polyred.html#D6" target="_self" title="POLYRED:def.6">Def6</a></span>;<br/><a NAME="E6:76_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D5" target="_self" title="POLYRED:def.5">Def5</a></span>;<br/><a NAME="E7:76_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A5</font></span>: 
not <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E8:76_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A6</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><a NAME="E9:76_1_1_1_1"/>
(  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> &amp; <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><a NAME="E10:76_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:76_1_1_1_1"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E11:76_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76_1_1"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E1:76_1_1_1_1"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E8:76_1_1_1_1"><span class="lab"><font color="Green" title="E63">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:76_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">p</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span><br/><a NAME="E2:76_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76_1_1"><span class="lab"><font color="Green" title="E58">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:76_1_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:76_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font> iff <font color="Maroon" title="c8">p</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76_1_1_1"><span class="lab"><font color="Green" title="E59">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:76_1_1"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font> iff <font color="Olive" title="b1">p</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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