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<span class="kw">let </span><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="zmodul01.html#NM1" title="ZMODUL01:NM.1">Z_Module</a><br/>  for <font color="Olive" title="b2">W</font> being   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b3">x</font> being   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font>)</span>)</span>  st <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Olive" title="b1">V</font> holds <br/><font color="Maroon" title="c2">i</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="int_3.html#K5" title="INT_3:func.5">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b3">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">V</font> be   <a href="zmodul01.html#NM1" title="ZMODUL01:NM.1">Z_Module</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W</font> being   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c3">V</font><br/>  for <font color="Olive" title="b2">x</font> being   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">V</font>,<font color="Olive" title="b1">W</font>)</span>)</span>  st <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c3">V</font> holds <br/><font color="Maroon" title="c2">i</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="int_3.html#K5" title="INT_3:func.5">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b2">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">W</font> be   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Maroon" title="c3">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>)</span>)</span>  st <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c3">V</font> holds <br/><font color="Maroon" title="c2">i</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="int_3.html#K5" title="INT_3:func.5">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">x</font> be   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c3">V</font> implies <font color="Maroon" title="c2">i</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="int_3.html#K5" title="INT_3:func.5">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">x</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:8"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:8"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c3">V</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">i</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="int_3.html#K5" title="INT_3:func.5">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">x</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a><span class="kw"> such that </span><br/><a NAME="E4:8"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> <a href="int_3.html#K4" title="INT_3:func.4">div</a> <font color="Maroon" title="c1">p</font>
 ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c2">i</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="int_3.html#K5" title="INT_3:func.5">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E4:8"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="int_1.html#T59" onmouseover="rs('int_1/T59')" onmouseout="rh()">INT_1:59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="int_3.html#K5" title="INT_3:func.5">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D15" onmouseover="rs('vectsp_1/D15')" onmouseout="rh()">VECTSP_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="int_3.html#K5" title="INT_3:func.5">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" onmouseover="rs('vectsp_1/D16')" onmouseout="rh()">VECTSP_1:def 16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="int_3.html#K5" title="INT_3:func.5">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="zmodul02.html#T1" target="_self" onmouseover="rs('zmodul02/T1')" onmouseout="rh()">Th1</a>, <a class="ref" href="zmodul01.html#T1" onmouseover="rs('zmodul01/T1')" onmouseout="rh()">ZMODUL01:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="int_3.html#K5" title="INT_3:func.5">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" onmouseover="rs('rlvect_1/T4')" onmouseout="rh()">RLVECT_1:4</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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