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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</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#V2" title="GROUP_1:attr.2">unital</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a> non <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">PN1R</font> =  <a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c4">CPRPNR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16" title="POLYNOM3:func.16">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">CPN1R</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#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">P</font> =  <a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font>;<br/>
<a NAME="E1:40"/><i><font color="Green" title="E36">A1</font></i>: 
 <a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a>
 
<b>by </b><i><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="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1"/>
<font color="Maroon" title="c7">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#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">p'</font> = <font color="Maroon" title="c7">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span> ;<br/><a NAME="E3:40_1"/><i><font color="Green" title="E37">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <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="polynom3.html#K16" title="POLYNOM3:func.16">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</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="E4:40_1"/>
<span class="p1">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K7" title="HILBASIS:func.7">mpu</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p'</font>
 <b>by </b><i><a class="ref" href="hilbasis.html#T30" target="_self" title="HILBASIS:th.30">Th30</a></i>;<br/><b>hence </b><a NAME="E5:40_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:40_1"><i><font color="Green" title="E37">A2</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="E3:40"/><b>then </b>
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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:40"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E5:40"/><b>then </b><i><font color="Green" title="E37">A3</font></i>: 
 <a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> is <a href="quofield.html#V2" title="QUOFIELD:attr.2">RingEpimorphism</a>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E1:40"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="quofield.html#D22" title="QUOFIELD:def.22">QUOFIELD:def 22</a></i>;<br/>
<a NAME="E6:40"/>
 <a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E1:40"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="quofield.html#D23" title="QUOFIELD:def.23">QUOFIELD:def 23</a></i>;<br/>
<a NAME="E7:40"/><b>then </b>
 <a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E5:40"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="quofield.html#D24" title="QUOFIELD:def.24">QUOFIELD:def 24</a></i>;<br/>
<b>hence </b><a NAME="E8:40"/>
 ex <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16" title="POLYNOM3:func.16">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span> st <font color="Olive" title="b1">P</font> is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a>
 ;<br/>


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