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<div class="add">

<b>let </b><font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">P</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font>;<br/>



<b>assume </b><a NAME="E1:31"/>
<font color="Maroon" title="c3">P</font> is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a>
 ;<br/>

<a NAME="E2:31"/><b>then </b>
( <font color="Maroon" title="c3">P</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a> &amp; <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="ref" href="quofield.html#D24" title="QUOFIELD:def.24">QUOFIELD:def 24</a></i>;<br/>
<a NAME="E3:31"/><b>then </b><i><font color="Green" title="E28">A1</font></i>: 
( <font color="Maroon" title="c3">P</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a> &amp; <font color="Maroon" title="c3">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">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="c2">S</font> )
 
<b>by </b><i><a class="ref" href="quofield.html#D22" title="QUOFIELD:def.22">QUOFIELD:def 22</a>, <a class="ref" href="quofield.html#D23" title="QUOFIELD:def.23">QUOFIELD:def 23</a></i>;<br/>
<a NAME="E4:31"/><b>then </b><i><font color="Green" title="E29">A2</font></i>: 
( <font color="Maroon" title="c3">P</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> &amp; <font color="Maroon" title="c3">P</font> is <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a> &amp; <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="quofield.html#D21" title="QUOFIELD:def.21">QUOFIELD:def 21</a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green" title="E30">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E3:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E6:31"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </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="c2">S</font>;<br/>

<b>consider </b><font color="Maroon" title="c6">x'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:31_1_1"/><i><font color="Green" title="E32">A5</font></i>: 
( <font color="Maroon" title="c6">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">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x'</font> = <font color="Maroon" title="c6">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#E2:31_1_1"><i><font color="Green" title="E32">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">y'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:31_1_1"/><i><font color="Green" title="E33">A6</font></i>: 
( <font color="Maroon" title="c8">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 <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">y'</font> = <font color="Maroon" title="c8">y'</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#E5:31_1_1"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/>
<i><font color="Green" title="E34">A7</font></i>: <font color="Maroon" title="c7">x'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_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="c7">x'</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="hilbasis.html#E2:31_1_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>
;<br/>
<i><font color="Green" title="E35">A8</font></i>: <font color="Maroon" title="c9">y'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_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="c9">y'</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="hilbasis.html#E5:31_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>
;<br/>
<i><font color="Green" title="E36">A9</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2: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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x'</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c9">y'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E4:31"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="hilbasis.html#E2:31_1_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="hilbasis.html#E5:31_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x'</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c9">y'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="hilbasis.html#E7:31_1_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="hilbasis.html#E8:31_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></i>
;<br/>
<i><font color="Green" title="E37">A10</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2: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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">y'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E4:31"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="hilbasis.html#E2:31_1_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="hilbasis.html#E5:31_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">y'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="hilbasis.html#E7:31_1_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="hilbasis.html#E8:31_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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="c2">S</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2: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> <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#E4:31"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_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> <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#E3:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></i>
;<br/>
<b>hence </b><a NAME="E12:31_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <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="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E9:31_1_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="txt" href="hilbasis.html#E10:31_1_1"><i><font color="Green" title="E37">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:31_1"/><b>then </b>
( <font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> &amp; <font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a> &amp; <font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="group_1.html#V7" title="GROUP_1:attr.7">unity-preserving</a> )
 
<b>by </b><i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a>, <a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a>, <a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E3:31_1"/>
<font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  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><b>end;</b></div>
<a NAME="E7:31"/>
<font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E3:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="hilbasis.html#E5:31"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="tops_2.html#T63" title="TOPS_2:th.63">TOPS_2:63</a></i>;<br/>
<a NAME="E8:31"/><b>then </b><i><font color="Green" title="E32">A11</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E6:31"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="quofield.html#D23" title="QUOFIELD:def.23">QUOFIELD:def 23</a></i>;<br/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:31"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="hilbasis.html#E5:31"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="tops_2.html#T62" title="TOPS_2:th.62">TOPS_2:62</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E10:31"/><b>then </b>
<font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="quofield.html#V2" title="QUOFIELD:attr.2">RingEpimorphism</a>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E6:31"><i><font color="Green" title="E31">A4</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="E11:31"/>
<font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a>
 <b>by </b><i><a class="txt" href="hilbasis.html#E8:31"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="quofield.html#D24" title="QUOFIELD:def.24">QUOFIELD:def 24</a></i>;<br/>


</div>
