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

<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">I</font> be   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font>;<br/>



<b>set </b><font color="Maroon" title="c3">E</font> =  <a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font>;<br/>
<a NAME="E1:34"/><i><font color="Green" title="E22">A1</font></i>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<span class="p1">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ring_1.html#D6" target="_self" title="RING_1:def.6">Def6</a></i>;<br/>
<b>thus </b><a NAME="E2:34"/>
( <font color="Maroon" title="c2">I</font> is <a href="ring_1.html#V1" title="RING_1:attr.1">quasi-prime</a> implies <font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font> is <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">domRing-like</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c2">I</font> is <a href="ring_1.html#V1" title="RING_1:attr.1">quasi-prime</a>
 ;<br/>

<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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a><br/>

<b>assume </b><a NAME="E2:34_1"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E4:34_1"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="ref" href="ring_1.html#T11" target="_self" title="RING_1:th.11">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E6:34_1"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="ref" href="ring_1.html#T11" target="_self" title="RING_1:th.11">Th11</a></i>;<br/>
<a NAME="E7:34_1"/><i><font color="Green" title="E27">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>;<br/>
<a NAME="E8:34_1"/><i><font color="Green" title="E28">A7</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c7">b</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>;<br/>
<a NAME="E9:34_1"/>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ring_1.html#E4:34_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="ring_1.html#E6:34_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="ring_1.html#T14" target="_self" title="RING_1:th.14">Th14</a></i>;<br/>
<a NAME="E10:34_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:34"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="ring_1.html#E2:34_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="ring_1.html#T6" target="_self" title="RING_1:th.6">Th6</a></i>;<br/>
<a NAME="E11:34_1"/><b>then </b>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:34_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="ring_1.html#E7:34_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="ring_1.html#D1" target="_self" title="RING_1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E12:34_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span> or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span> )
 <b>by </b><i><a class="txt" href="ring_1.html#E1:34"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="ring_1.html#E4:34_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="ring_1.html#E6:34_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="ring_1.html#E8:34_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="ring_1.html#T6" target="_self" title="RING_1:th.6">Th6</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:34"/><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font> is <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">domRing-like</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ring_1.html#D1" target="_self" title="RING_1:def.1">RING_1:def 1</a><br/>

<b>reconsider </b><font color="Maroon" title="c6">x</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c7">y</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<font color="Maroon" title="c5">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span> <b>by </b><i><a class="ref" href="ring_1.html#T12" target="_self" title="RING_1:th.12">Th12</a></i>;<br/>
<b>assume </b><a NAME="E5:34"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ;<br/>

<a NAME="E6:34"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>;<br/>
<a NAME="E7:34"/><b>then </b><i><font color="Green" title="E25">A10</font></i>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<span class="p1">(<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>
 
<b>by </b><i><a class="txt" href="ring_1.html#E5:34"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="ring_1.html#T6" target="_self" title="RING_1:th.6">Th6</a></i>;<br/>
<a NAME="E8:34"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y</font>
 
<b>by </b><i><a class="ref" href="ring_1.html#T14" target="_self" title="RING_1:th.14">Th14</a></i>;<br/>
<a NAME="E9:34"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span> or <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:34"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="ring_1.html#E3:34"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="ring_1.html#E7:34"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></i>;<br/>
<a NAME="E10:34"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c5">b</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )
 
<b>by </b><i><a class="txt" href="ring_1.html#E1:34"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="ring_1.html#T6" target="_self" title="RING_1:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E11:34"/>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>;<br/>


</div>
