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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ideal_1.html#V1">add-closed</a> <a href="ideal_1.html#V3">right-ideal</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>





<b>assume </b><a NAME="E1:92"/>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="polyred.html#R11" target="_self">are_congruent_mod</a> <font color="Maroon">I</font> &amp; <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="polyred.html#R11" target="_self">are_congruent_mod</a> <font color="Maroon">I</font> )
 ;<br/>

<a NAME="E2:92"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">c</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:92"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="ideal_1.html#D3">IDEAL_1:def 3</a></i>;<br/>
<a NAME="E4:92"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c</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">c</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#D1">IDEAL_1:def 1</a></i>;<br/>
<i><font color="Green">E24</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D12">VECTSP_1:def 12</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D12">VECTSP_1:def 12</a></i>
;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c</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">c</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T41">VECTSP_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="algstr_1.html#D5">ALGSTR_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T41">VECTSP_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
;<br/>
<b>hence </b><a NAME="E8:92"/>
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font>,<font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d</font> <a href="polyred.html#R11" target="_self">are_congruent_mod</a> <font color="Maroon">I</font>
 <b>by </b><i>, </i>;<br/>


</div>
