<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> , <font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>given </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">R</font>,<font color="Maroon">S</font><b> such that </b><a NAME="E1:94_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">f</font> is <a href="quofield.html#V4" target="_self">RingIsomorphism</a>
 ;<br/>

<a NAME="E2:94_1_1"/>
( <font color="Maroon">f</font> is <a href="quofield.html#V3" target="_self">RingMonomorphism</a> &amp; <font color="Maroon">f</font> is <a href="quofield.html#V2" target="_self">RingEpimorphism</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:94_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
( <font color="Maroon">f</font> is <a href="quofield.html#V1" target="_self">RingHomomorphism</a> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:94_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">f</font> is <a href="grcat_1.html#V1">additive</a> &amp; <font color="Maroon">f</font> is <a href="group_6.html#V1">multiplicative</a> &amp; <font color="Maroon">f</font> is <a href="endalg.html#V1">unity-preserving</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:94_1_1"/><i><font color="Green">E25</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> ;<br/>
<a NAME="E6:94_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="quofield.html#V1" target="_self">RingHomomorphism</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:94_1_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1_1_1"><b>proof </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">S</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>

<b>consider </b><font color="Maroon">x'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:94_1_1_1_1"/><i><font color="Green">E39</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">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T17">FUNCT_2:17</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><a class="ref" href="quofield.html#D11" target="_self">Def11</a></i>;<br/>
<b>consider </b><font color="Maroon">y'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:94_1_1_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">y'</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">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T17">FUNCT_2:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="quofield.html#T13" target="_self">Th13</a></i>;<br/>
<i><font color="Green">E41</font></i>: <font color="Maroon">x'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T32">FUNCT_2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="quofield.html#D11" target="_self">Def11</a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
;<br/>
<i><font color="Green">E53</font></i>: <font color="Maroon">y'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T32">FUNCT_2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="quofield.html#T13" target="_self">Th13</a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
;<br/>
<i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quofield.html#D8" target="_self">Def8</a>, <a class="ref" href="quofield.html#D11" target="_self">Def11</a>, <a class="ref" href="quofield.html#T13" target="_self">Th13</a>, <a class="ref" href="grcat_1.html#D13">GRCAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y'</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="quofield.html#T14" target="_self">Th14</a>, <a class="ref" href="quofield.html#T15" target="_self">Th15</a>, <a class="ref" href="funct_2.html#T32">FUNCT_2:32</a></i>
;<br/>
<i><font color="Green">E55</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <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> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</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>
<b>by </b><i><a class="ref" href="quofield.html#D8" target="_self">Def8</a>, <a class="ref" href="quofield.html#D11" target="_self">Def11</a>, <a class="ref" href="quofield.html#T13" target="_self">Th13</a>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</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>
<b>by </b><i>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="quofield.html#T14" target="_self">Th14</a>, <a class="ref" href="quofield.html#T15" target="_self">Th15</a>, <a class="ref" href="funct_2.html#T32">FUNCT_2:32</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">S</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</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="quofield.html#D8" target="_self">Def8</a>, <a class="ref" href="endalg.html#D8">ENDALG:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</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="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T32">FUNCT_2:32</a></i>
;<br/>
<b>hence </b><a NAME="E12:94_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="quofield.html#T16" target="_self">Th16</a>, <a class="ref" href="quofield.html#T17" target="_self">Th17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:94_1_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="grcat_1.html#V1">additive</a> &amp; <font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="group_6.html#V1">multiplicative</a> &amp; <font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="endalg.html#V1">unity-preserving</a> )
 
<b>by </b><i><a class="ref" href="endalg.html#D8">ENDALG:def 8</a>, <a class="ref" href="group_6.html#D7">GROUP_6:def 7</a>, <a class="ref" href="grcat_1.html#D13">GRCAT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E3:94_1_1_1"/>
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="quofield.html#V1" target="_self">RingHomomorphism</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:94_1_1"/>
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="quofield.html#D9" target="_self">Def9</a>, <a class="ref" href="tops_2.html#T63">TOPS_2:63</a></i>;<br/>
<a NAME="E8:94_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="quofield.html#V3" target="_self">RingMonomorphism</a>
 
<b>by </b><i><a class="ref" href="quofield.html#D10" target="_self">Def10</a>, </i>;<br/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">R</font>
<b>by </b><i>, <a class="ref" href="quofield.html#D9" target="_self">Def9</a>, <a class="ref" href="tops_2.html#T62">TOPS_2:62</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E10:94_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="quofield.html#V2" target="_self">RingEpimorphism</a>
 
<b>by </b><i><a class="ref" href="quofield.html#D10" target="_self">Def10</a>, </i>;<br/>
<a NAME="E11:94_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="quofield.html#V4" target="_self">RingIsomorphism</a>
 
<b>by </b><i><a class="ref" href="quofield.html#D11" target="_self">Def11</a>, </i>;<br/>
<b>hence </b><a NAME="E12:94_1_1"/>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">R</font> st <font color="Olive">f</font> is <a href="quofield.html#V4" target="_self">RingIsomorphism</a>
 ;<br/>


</div>
