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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">well-unital</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  = <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c2">P</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:34"/><i><font color="Green" title="E31">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 <b>and </b><br/><a NAME="E3:34"/><i><font color="Green" title="E32">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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> holds <br/><font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">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" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:34_1_1"/>
<font color="Maroon" title="c3">y</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="c2">P</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:34_1_1"/><i><font color="Green" title="E33">A3</font></i>: 
( <font color="Maroon" title="c4">x</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 <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E2:34"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="hilbasis.html#E3:34_1_1"><i><font color="Green" title="E33">A3</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">y'</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">R</font> ;<br/><a NAME="E6:34_1_1"/>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c6">y'</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="polynom1.html#T55" title="POLYNOM1:th.55">POLYNOM1:55</a></i>;<br/><a NAME="E7:34_1_1"/><b>then </b>
<font color="Maroon" title="c6">y'</font> is  <a href="polynom1.html#V2" title="POLYNOM1:attr.2">finite-Support</a> <a href="polynom1.html#NM3" title="POLYNOM1:NM.3">Series</a> of 0,<font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></i>;<br/><a NAME="E8:34_1_1"/><b>then </b>
<font color="Maroon" title="c6">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#K30" title="POLYNOM1:func.30">Polynom-Ring</a> 0,<font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>hence </b><a NAME="E9:34_1_1"/>
<font color="Maroon" title="c3">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#K30" title="POLYNOM1:func.30">Polynom-Ring</a> 0,<font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:34"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="hilbasis.html#E3:34_1_1"><i><font color="Green" title="E33">A3</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:34_1"/>
<font color="Maroon" title="c3">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#K30" title="POLYNOM1:func.30">Polynom-Ring</a> 0,<font color="Maroon" title="c1">R</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c4">y'</font> = <font color="Maroon" title="c3">y</font> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E4:34_1"/><i><font color="Green" title="E33">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:34_1"/><b>then </b><i><font color="Green" title="E34">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T14" title="FUNCT_1:th.14">FUNCT_1:14</a>, <a class="ref" href="polynom1.html#T55" title="POLYNOM1:th.55">POLYNOM1:55</a></i>;<br/><a NAME="E6:34_1"/><b>then </b><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c4">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:34_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="funcop_1.html#T15" title="FUNCOP_1:th.15">FUNCOP_1:15</a></i>;<br/><a NAME="E7:34_1"/>
<font color="Maroon" title="c4">y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">y'</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E5:34_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/><a NAME="E9:34_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:34"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="hilbasis.html#E6:34_1"><i><font color="Green" title="E35">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E10:34_1"/>
<font color="Maroon" title="c3">y</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="c2">P</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E2:34"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:34"/><b>then </b><i><font color="Green" title="E33">A7</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K30" title="POLYNOM1:func.30">Polynom-Ring</a> 0,<font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">P</font> = <font color="Maroon" title="c2">P</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">R</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> 0,<font color="Maroon" title="c1">R</font></span>)</span> <b>by </b><i><a class="txt" href="hilbasis.html#E2:34"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">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" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/><b>reconsider </b><font color="Maroon" title="c6">Px</font> = <font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">Py</font> = <font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c8">Pxy</font> = <font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">y</font></span>)</span> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of 0,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">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" title="c9">z</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of 0;<br/><a NAME="E1:34_2_1"/><i><font color="Green" title="E34">A8</font></i>: 
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><i><font color="Green" title="E35">A9</font></i>: <font color="Maroon" title="c8">Pxy</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">z</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:34"><i><font color="Green" title="E32">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">y</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E1:34_2_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><i><font color="Green" title="E36">A10</font></i>: <font color="Maroon" title="c6">Px</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">z</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:34"><i><font color="Green" title="E32">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E1:34_2_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><font color="Maroon" title="c7">Py</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">z</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:34"><i><font color="Green" title="E32">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">y</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E1:34_2_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E5:34_2_1"/>
<font color="Maroon" title="c8">Pxy</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Px</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">z</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Py</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E2:34_2_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="hilbasis.html#E3:34_2_1"><i><font color="Green" title="E36">A10</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:34_2"/><b>then </b>
<font color="Maroon" title="c8">Pxy</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">Px</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c7">Py</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>;<br/><b>hence </b><a NAME="E4:34_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:34"/><b>then </b><i><font color="Green" title="E34">A11</font></i>: 
<font color="Maroon" title="c3">P</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>
 
<b>by </b><i><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">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" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/><b>reconsider </b><font color="Maroon" title="c6">Px</font> = <font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">Py</font> = <font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c8">Pxy</font> = <font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">y</font></span>)</span> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of 0,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">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" title="c9">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of 0;<br/><b>reconsider </b><font color="Maroon" title="c10">s</font> = <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> ;<br/><b>take </b><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c10">s</font>;<br/><a NAME="E2:34_3_1"/><i><font color="Green" title="E35">A12</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>thus </b><font color="Maroon" title="c8">Pxy</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">b</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">b</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:34"><i><font color="Green" title="E32">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">y</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E2:34_3_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c11">s</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61" title="RLVECT_1:th.61">RLVECT_1:61</a></i>
;<br/><a NAME="E4:34_3_1"/><i><font color="Green" title="E36">A13</font></i>: 
 <a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="hilbasis.html#T2" target="_self" title="HILBASIS:th.2">Th2</a></i>;<br/><a NAME="E5:34_3_1"/><i><font color="Green" title="E37">A14</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/><a NAME="E6:34_3_1"/><i><font color="Green" title="E38">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:34_3_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/><b>thus </b><a NAME="E7:34_3_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:34_3_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="hilbasis.html#E5:34_3_1"><i><font color="Green" title="E37">A14</font></i></a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">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" title="c12">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:34_3_1_2"/><i><font color="Green" title="E39">A16</font></i>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">s</font>
 ;<br/><a NAME="E2:34_3_1_2"/><b>then </b><i><font color="Green" title="E40">A17</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="hilbasis.html#E5:34_3_1"><i><font color="Green" title="E37">A14</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:34_3_1_2"/><b>then </b><i><font color="Green" title="E41">A18</font></i>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>set </b><font color="Maroon" title="c13">b1</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><b>set </b><font color="Maroon" title="c14">b2</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><a NAME="E4:34_3_1_2"/><i><font color="Green" title="E42">A19</font></i>: 
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="polynom1.html#T55" title="POLYNOM1:th.55">POLYNOM1:55</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">b1</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <font color="Maroon" title="c16">b2</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of 0 <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>take </b><font color="Maroon" title="c17">b1</font> = <font color="Maroon" title="c15">b1</font>;<br/><b>take </b><font color="Maroon" title="c18">b2</font> = <font color="Maroon" title="c16">b2</font>;<br/><a NAME="E6:34_3_1_2"/>
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E6:34_3_1"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="hilbasis.html#E2:34_3_1_2"><i><font color="Green" title="E40">A17</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">k</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="hilbasis.html#E3:34_3_1_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">b1</font>,<font color="Maroon" title="c18">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<b>by </b><i><a class="txt" href="hilbasis.html#E4:34_3_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/><br/><i><font color="Green" title="E43">A20</font></i>: <font color="Maroon" title="c6">Px</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c17">b1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">b1</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:34"><i><font color="Green" title="E32">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E4:34_3_1_2"><i><font color="Green" title="E42">A19</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><i><font color="Green" title="E44">A21</font></i>: <font color="Maroon" title="c7">Py</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c18">b2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b2</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:34"><i><font color="Green" title="E32">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">y</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E4:34_3_1_2"><i><font color="Green" title="E42">A19</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>thus </b><font color="Maroon" title="c11">s</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">k</font> = 
<font color="Maroon" title="c11">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="hilbasis.html#E1:34_3_1_2"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="hilbasis.html#E3:34_3_1_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">Px</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c17">b1</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Py</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c18">b2</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E8:34_3_1_2"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="hilbasis.html#E9:34_3_1_2"><i><font color="Green" title="E44">A21</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:34_3_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">s</font> holds <br/> ex <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of 0 st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c11">s</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Px</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Py</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b3">b2</font></span>)</span> )
 ;<br/></div><b>end;</b></div><a NAME="E3:34_3"/><b>then </b>
<font color="Maroon" title="c8">Pxy</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">Px</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c7">Py</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D26" title="POLYNOM1:def.26">POLYNOM1:def 26</a></i>;<br/><b>hence </b><a NAME="E4:34_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:34"/><b>then </b><i><font color="Green" title="E35">A22</font></i>: 
<font color="Maroon" title="c3">P</font> is <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>
 
<b>by </b><i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>;<br/>
<a NAME="E11:34"/><i><font color="Green" title="E36">A23</font></i>: 
 <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> 0
 
<b>by </b><i><a class="ref" href="polynom1.html#D15" title="POLYNOM1:def.15">POLYNOM1:def 15</a></i>;<br/>
<a NAME="E12:34"/><i><font color="Green" title="E37">A24</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:34"/><b>then </b><i><font color="Green" title="E38">A25</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E14:34"/><i><font color="Green" title="E39">A26</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> 0,<font color="Maroon" title="c1">R</font></span>)</span> = 
 <a href="polynom1.html#K27" title="POLYNOM1:func.27">1_</a> 0,<font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="ref" href="polynom1.html#T90" title="POLYNOM1:th.90">POLYNOM1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> 0,<font color="Maroon" title="c1">R</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E11:34"><i><font color="Green" title="E36">A23</font></i></a>, <a class="ref" href="polynom1.html#D25" title="POLYNOM1:def.25">POLYNOM1:def 25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T55" title="POLYNOM1:th.55">POLYNOM1:55</a>, <a class="ref" href="polynom1.html#D24" title="POLYNOM1:def.24">POLYNOM1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E13:34"><i><font color="Green" title="E38">A25</font></i></a>, <a class="ref" href="funct_7.html#D3" title="FUNCT_7:def.3">FUNCT_7:def 3</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E12:34"><i><font color="Green" title="E37">A24</font></i></a>, <a class="txt" href="hilbasis.html#E14:34"><i><font color="Green" title="E39">A26</font></i></a>, <a class="ref" href="funct_4.html#T20" title="FUNCT_4:th.20">FUNCT_4:20</a></i>
<br/>.= 
<font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:34"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="polynom1.html#T55" title="POLYNOM1:th.55">POLYNOM1:55</a></i>
;<br/>
<a NAME="E16:34"/><b>then </b>
<font color="Maroon" title="c3">P</font> is <a href="group_1.html#V7" title="GROUP_1:attr.7">unity-preserving</a>
 
<b>by </b><i><a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a></i>;<br/>
<a NAME="E17:34"/><b>then </b><i><font color="Green" title="E40">A27</font></i>: 
<font color="Maroon" title="c3">P</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E8:34"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="hilbasis.html#E10:34"><i><font color="Green" title="E35">A22</font></i></a>, <a class="ref" href="quofield.html#D21" title="QUOFIELD:def.21">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" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:34_5"/><i><font color="Green" title="E41">A28</font></i>: 
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">P</font> )
 ;<br/><b>assume </b><a NAME="E2:34_5"/>
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font>
 ;<br/><a NAME="E3:34_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:34"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="hilbasis.html#E1:34_5"><i><font color="Green" title="E41">A28</font></i></a></i>;<br/><a NAME="E4:34_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c5">x2</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:34"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="hilbasis.html#E1:34_5"><i><font color="Green" title="E41">A28</font></i></a></i>;<br/><b>hence </b><a NAME="E5:34_5"/>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T7" title="FUNCT_4:th.7">FUNCT_4:7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:34"/><b>then </b>
<font color="Maroon" title="c3">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E20:34"/><b>then </b><i><font color="Green" title="E41">A29</font></i>: 
<font color="Maroon" title="c3">P</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E17:34"><i><font color="Green" title="E40">A27</font></i></a>, <a class="ref" href="quofield.html#D23" title="QUOFIELD:def.23">QUOFIELD:def 23</a></i>;<br/>
<a NAME="E21:34"/>
<font color="Maroon" title="c3">P</font> is <a href="quofield.html#V2" title="QUOFIELD:attr.2">RingEpimorphism</a>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E5:34"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="hilbasis.html#E17:34"><i><font color="Green" title="E40">A27</font></i></a>, <a class="ref" href="quofield.html#D22" title="QUOFIELD:def.22">QUOFIELD:def 22</a></i>;<br/>
<b>hence </b><a NAME="E22:34"/>
<font color="Maroon" title="c3">P</font> is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a>
 <b>by </b><i><a class="txt" href="hilbasis.html#E20:34"><i><font color="Green" title="E41">A29</font></i></a>, <a class="ref" href="quofield.html#D24" title="QUOFIELD:def.24">QUOFIELD:def 24</a></i>;<br/>


</div>
