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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="pboole.html#M1">ManySortedSet</a> of  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a>  = <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">P</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:34"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E3:34"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> holds <br/><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:34_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">y'</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM5">Function</a> of  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">R</font> ;<br/><a NAME="E6:34_1_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">y'</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a>, <a class="ref" href="polynom1.html#T55">POLYNOM1:55</a></i>;<br/><a NAME="E7:34_1_1"/><b>then </b>
<font color="Maroon">y'</font> is  <a href="polynom1.html#V2">finite-Support</a> <a href="polynom1.html#NM3">Series</a> of 0,<font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D10">POLYNOM1:def 10</a></i>;<br/><a NAME="E8:34_1_1"/><b>then </b>
<font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> 0,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>hence </b><a NAME="E9:34_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> 0,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:34_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> 0,<font color="Maroon">R</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM5">Function</a> of  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E4:34_1"/><i><font color="Green">E47</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:34_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T14">FUNCT_1:14</a>, <a class="ref" href="polynom1.html#T55">POLYNOM1:55</a></i>;<br/><a NAME="E6:34_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T15">FUNCOP_1:15</a></i>;<br/><a NAME="E7:34_1"/>
<font color="Maroon">y'</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">y'</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/><a NAME="E9:34_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E10:34_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:34"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> 0,<font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">R</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> 0,<font color="Maroon">R</font></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>reconsider </b><font color="Maroon">Px</font> = <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>, <font color="Maroon">Py</font> = <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>, <font color="Maroon">Pxy</font> = <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> as   <a href="polynom1.html#NM4">Polynomial</a> of 0,<font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be   <a href="polynom1.html#NM1">bag</a> of 0;<br/><a NAME="E1:34_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><i><font color="Green">E53</font></i>: <font color="Maroon">Pxy</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><i><font color="Green">E54</font></i>: <font color="Maroon">Px</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><font color="Maroon">Py</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E5:34_2_1"/>
<font color="Maroon">Pxy</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Px</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">Py</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E3:34_2"/><b>then </b>
<font color="Maroon">Pxy</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">Px</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">Py</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D21">POLYNOM1:def 21</a></i>;<br/><b>hence </b><a NAME="E4:34_2"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:34"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">P</font> is <a href="grcat_1.html#V1">additive</a>
 
<b>by </b><i><a class="ref" href="grcat_1.html#D13">GRCAT_1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>reconsider </b><font color="Maroon">Px</font> = <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>, <font color="Maroon">Py</font> = <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>, <font color="Maroon">Pxy</font> = <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> as   <a href="polynom1.html#NM4">Polynomial</a> of 0,<font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of 0;<br/><b>reconsider </b><font color="Maroon">s</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> ;<br/><b>take </b><font color="Maroon">s</font> = <font color="Maroon">s</font>;<br/><a NAME="E2:34_3_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>thus </b><font color="Maroon">Pxy</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
;<br/><a NAME="E4:34_3_1"/><i><font color="Green">E59</font></i>: 
 <a href="polynom1.html#K21">decomp</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:34_3_1"/><i><font color="Green">E75</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E6:34_3_1"/><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><b>thus </b><a NAME="E7:34_3_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:34_3_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E2:34_3_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:34_3_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>set </b><font color="Maroon">b1</font> =  <a href="xboole_0.html#K1">{}</a> ;<br/><b>set </b><font color="Maroon">b2</font> =  <a href="xboole_0.html#K1">{}</a> ;<br/><a NAME="E4:34_3_1_2"/><i><font color="Green">E121</font></i>: 
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="polynom1.html#T55">POLYNOM1:55</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">b1</font> =  <a href="xboole_0.html#K1">{}</a> , <font color="Maroon">b2</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="polynom1.html#NM1">bag</a> of 0 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>take </b><font color="Maroon">b1</font> = <font color="Maroon">b1</font>;<br/><b>take </b><font color="Maroon">b2</font> = <font color="Maroon">b2</font>;<br/><a NAME="E6:34_3_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/><br/><i><font color="Green">E138</font></i>: <font color="Maroon">Px</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><i><font color="Green">E151</font></i>: <font color="Maroon">Py</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b2</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>thus </b><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> = 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Px</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">Py</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i>, <a class="txt" href="hilbasis.html#E2:34"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:34_3_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> holds <br/> ex <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="polynom1.html#NM1">bag</a> of 0 st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b1</font>,<font color="Olive">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Px</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">Py</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b2</font></span>)</span> )
 ;<br/></div><b>end;</b></div><a NAME="E3:34_3"/><b>then </b>
<font color="Maroon">Pxy</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">Px</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">Py</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D26">POLYNOM1:def 26</a></i>;<br/><b>hence </b><a NAME="E4:34_3"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:34"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">P</font> is <a href="group_6.html#V1">multiplicative</a>
 
<b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>
<a NAME="E11:34"/>
 <a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="borsuk_1.html#K1">--&gt;</a> 0
 
<b>by </b><i><a class="ref" href="polynom1.html#D15">POLYNOM1:def 15</a></i>;<br/>
<a NAME="E12:34"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T16">FUNCOP_1:16</a></i>;<br/>
<a NAME="E13:34"/><b>then </b><i><font color="Green">E153</font></i>: 
 <a href="polynom1.html#K16">EmptyBag</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<a NAME="E14:34"/><i><font color="Green">E154</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:34"/><b>then </b><i><font color="Green">E155</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E16:34"/><i><font color="Green">E156</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> 0,<font color="Maroon">R</font></span>)</span> = 
 <a href="polynom1.html#K27">1_</a> 0,<font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="polynom1.html#T90">POLYNOM1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> 0,<font color="Maroon">R</font></span>)</span> <a href="polynom1.html#K1">+*</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom1.html#D25">POLYNOM1:def 25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="polynom1.html#K1">+*</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T55">POLYNOM1:55</a>, <a class="ref" href="polynom1.html#D24">POLYNOM1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_7.html#D3">FUNCT_7:def 3</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T20">FUNCT_4:20</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom1.html#T55">POLYNOM1:55</a></i>
;<br/>
<a NAME="E18:34"/><b>then </b>
<font color="Maroon">P</font> is <a href="endalg.html#V1">unity-preserving</a>
 
<b>by </b><i><a class="ref" href="endalg.html#D8">ENDALG:def 8</a></i>;<br/>
<a NAME="E19:34"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">P</font> is <a href="quofield.html#V1">RingHomomorphism</a>
 
<b>by </b><i>, , <a class="ref" href="quofield.html#D21">QUOFIELD:def 21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:34_5"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> )
 ;<br/><b>assume </b><a NAME="E2:34_5"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/><a NAME="E3:34_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:34_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x2</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:34_5"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T7">FUNCT_4:7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:34"/><b>then </b>
<font color="Maroon">P</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E22:34"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">P</font> is <a href="quofield.html#V3">RingMonomorphism</a>
 
<b>by </b><i>, <a class="ref" href="quofield.html#D23">QUOFIELD:def 23</a></i>;<br/>
<a NAME="E23:34"/>
<font color="Maroon">P</font> is <a href="quofield.html#V2">RingEpimorphism</a>
 
<b>by </b><i>, , <a class="ref" href="quofield.html#D22">QUOFIELD:def 22</a></i>;<br/>
<b>hence </b><a NAME="E24:34"/>
<font color="Maroon">P</font> is <a href="quofield.html#V4">RingIsomorphism</a>
 <b>by </b><i>, <a class="ref" href="quofield.html#D24">QUOFIELD:def 24</a></i>;<br/>


</div>
