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

<span class="kw">set </span><font color="Maroon" title="c3">M</font> =  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c4">F</font> =  <a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c1">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">AD</font> =  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c2">V</font></span>)</span>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">ML</font> =  <a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</a> (<font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c2">V</font></span>)</span>);<br/>
<span class="kw">thus </span><a NAME="E1:14_1_1"/>
 <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</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"> (  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c1">p</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D14" onmouseover="rs('vectsp_1/D14')" onmouseout="rh()">VECTSP_1:def 14</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">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c9">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font></span>)</span></span><br/>



<span class="kw">consider </span><font color="Maroon" title="c10">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E2:14_1_1_1"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#T13" onmouseover="rs('ec_pf_1/T13')" onmouseout="rh()">EC_PF_1:13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">i</font> = <font color="Maroon" title="c10">i</font> as   <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">by</span> <span class="lab"><a class="txt" href="zmodul02.html#E13"><span class="lab"><font color="Green" title="E3">Lem1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:14_1_1_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#T13" onmouseover="rs('ec_pf_1/T13')" onmouseout="rh()">EC_PF_1:13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c12">j</font> as   <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">by</span> <span class="lab"><a class="txt" href="zmodul02.html#E13"><span class="lab"><font color="Green" title="E3">Lem1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">v1</font> = <font color="Maroon" title="c9">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c2">V</font></span>)</span>)</span>)</span> ;<br/>
<span class="lab"><font color="Green" title="E7">A3</font></span>: <font color="Maroon" title="c7">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="int_3.html#K5" title="INT_3:func.5">mod</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E5:14_1_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="ec_pf_1.html#T15" onmouseover="rs('ec_pf_1/T15')" onmouseout="rh()">EC_PF_1:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</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="int_3.html#K5" title="INT_3:func.5">mod</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" onmouseover="rs('nat_d/T66')" onmouseout="rh()">NAT_D:66</a></span>
;<br/>
<a NAME="E9:14_1_1_1"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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="c14">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#D11" target="_self" onmouseover="rs('zmodul02/D11')" onmouseout="rh()">Def11</a>, <a class="txt" href="#E2:14_1_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>;<br/>
<a NAME="E10:14_1_1_1"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</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="c14">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#D11" target="_self" onmouseover="rs('zmodul02/D11')" onmouseout="rh()">Def11</a>, <a class="txt" href="#E5:14_1_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">j</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="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="c14">v1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</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="c14">v1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T2" target="_self" onmouseover="rs('zmodul02/T2')" onmouseout="rh()">Th2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c14">v1</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="c13">j</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="c14">v1</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="kw">hence </span><a NAME="E12:14_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:14_1_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E9:14_1_1_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E10:14_1_1_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="zmodul02.html#D11" target="_self" onmouseover="rs('zmodul02/D11')" onmouseout="rh()">Def11</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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:14_1_1"/>
 <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</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"> (  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c1">p</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D13" onmouseover="rs('vectsp_1/D13')" onmouseout="rh()">VECTSP_1:def 13</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">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> holds  <font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c9">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</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="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c8">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">w</font></span>)</span></span><br/>



<span class="kw">consider </span><font color="Maroon" title="c10">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E2:14_1_1_2"/><span class="lab"><font color="Green" title="E5">A6</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#T13" onmouseover="rs('ec_pf_1/T13')" onmouseout="rh()">EC_PF_1:13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">i</font> = <font color="Maroon" title="c10">i</font> as   <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">by</span> <span class="lab"><a class="txt" href="zmodul02.html#E13"><span class="lab"><font color="Green" title="E3">Lem1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">v1</font> = <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c13">w1</font> = <font color="Maroon" title="c9">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c2">V</font></span>)</span>)</span>)</span> ;<br/>
<a NAME="E5:14_1_1_2"/><span class="lab"><font color="Green" title="E6">A7</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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="c13">w1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#D11" target="_self" onmouseover="rs('zmodul02/D11')" onmouseout="rh()">Def11</a>, <a class="txt" href="#E2:14_1_1_2"><span class="lab"><font color="Green" title="E5">A6</font></span></a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">w</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">w1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#D11" target="_self" onmouseover="rs('zmodul02/D11')" onmouseout="rh()">Def11</a>, <a class="txt" href="#E2:14_1_1_2"><span class="lab"><font color="Green" title="E5">A6</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c12">v1</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="c11">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="c13">w1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D14" onmouseover="rs('vectsp_1/D14')" onmouseout="rh()">VECTSP_1:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c8">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_1_2"><span class="lab"><font color="Green" title="E5">A6</font></span></a>, <a class="txt" href="#E5:14_1_1_2"><span class="lab"><font color="Green" title="E6">A7</font></span></a>, <a class="ref" href="zmodul02.html#D11" target="_self" onmouseover="rs('zmodul02/D11')" onmouseout="rh()">Def11</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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:14_1_1"/>
 <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</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"> (  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c1">p</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D15" onmouseover="rs('vectsp_1/D15')" onmouseout="rh()">VECTSP_1:def 15</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">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c9">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font></span>)</span></span><br/>



<span class="kw">consider </span><font color="Maroon" title="c10">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E2:14_1_1_3"/><span class="lab"><font color="Green" title="E5">A8</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#T13" onmouseover="rs('ec_pf_1/T13')" onmouseout="rh()">EC_PF_1:13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">i</font> = <font color="Maroon" title="c10">i</font> as   <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">by</span> <span class="lab"><a class="txt" href="zmodul02.html#E13"><span class="lab"><font color="Green" title="E3">Lem1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:14_1_1_3"/><span class="lab"><font color="Green" title="E6">A9</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#T13" onmouseover="rs('ec_pf_1/T13')" onmouseout="rh()">EC_PF_1:13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c12">j</font> as   <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">by</span> <span class="lab"><a class="txt" href="zmodul02.html#E13"><span class="lab"><font color="Green" title="E3">Lem1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">v1</font> = <font color="Maroon" title="c9">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c2">V</font></span>)</span>)</span>)</span> ;<br/>
<a NAME="E8:14_1_1_3"/><span class="lab"><font color="Green" title="E7">A10</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="int_3.html#K5" title="INT_3:func.5">mod</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_1_3"><span class="lab"><font color="Green" title="E5">A8</font></span></a>, <a class="txt" href="#E5:14_1_1_3"><span class="lab"><font color="Green" title="E6">A9</font></span></a>, <a class="ref" href="ec_pf_1.html#T18" onmouseover="rs('ec_pf_1/T18')" onmouseout="rh()">EC_PF_1:18</a></span>;<br/>
<a NAME="E9:14_1_1_3"/><span class="lab"><font color="Green" title="E8">A11</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</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="c14">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#D11" target="_self" onmouseover="rs('zmodul02/D11')" onmouseout="rh()">Def11</a>, <a class="txt" href="#E5:14_1_1_3"><span class="lab"><font color="Green" title="E6">A9</font></span></a></span>;<br/>
<a NAME="E10:14_1_1_3"/><span class="lab"><font color="Green" title="E9">A12</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">j</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="c14">v1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_1_3"><span class="lab"><font color="Green" title="E5">A8</font></span></a>, <a class="txt" href="#E9:14_1_1_3"><span class="lab"><font color="Green" title="E8">A11</font></span></a>, <a class="ref" href="zmodul02.html#D11" target="_self" onmouseover="rs('zmodul02/D11')" onmouseout="rh()">Def11</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c13">j</font></span>)</span> <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="c14">v1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c13">j</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c14">v1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T2" target="_self" onmouseover="rs('zmodul02/T2')" onmouseout="rh()">Th2</a></span>
<br/>.= 
<font color="Maroon" title="c11">i</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c14">v1</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/>.= 
<font color="Maroon" title="c11">i</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">j</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="c14">v1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T2" target="_self" onmouseover="rs('zmodul02/T2')" onmouseout="rh()">Th2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">j</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="c14">v1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T2" target="_self" onmouseover="rs('zmodul02/T2')" onmouseout="rh()">Th2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:14_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:14_1_1_3"><span class="lab"><font color="Green" title="E7">A10</font></span></a>, <a class="txt" href="#E10:14_1_1_3"><span class="lab"><font color="Green" title="E9">A12</font></span></a>, <a class="ref" href="zmodul02.html#D11" target="_self" onmouseover="rs('zmodul02/D11')" onmouseout="rh()">Def11</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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:14_1_1"/>
 <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</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"> (  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D16" onmouseover="rs('vectsp_1/D16')" onmouseout="rh()">VECTSP_1:def 16</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"> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</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="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c7">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c2">V</font></span>)</span>)</span>)</span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:14_1_1_4"/><span class="lab"><font color="Green" title="E5">A13</font></span>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ec_pf_1.html#T13" onmouseover="rs('ec_pf_1/T13')" onmouseout="rh()">EC_PF_1:13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">i</font> = <font color="Maroon" title="c9">i</font> as   <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">by</span> <span class="lab"><a class="txt" href="zmodul02.html#E13"><span class="lab"><font color="Green" title="E3">Lem1</font></span></a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="ec_pf_1.html#NK2" title="EC_PF_1:NK.2">GF</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="c7">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">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="c8">v1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#D11" target="_self" onmouseover="rs('zmodul02/D11')" onmouseout="rh()">Def11</a>, <a class="txt" href="#E3:14_1_1_4"><span class="lab"><font color="Green" title="E5">A13</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c8">v1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_1_1_4"><span class="lab"><font color="Green" title="E5">A13</font></span></a>, <a class="ref" href="ec_pf_1.html#T12" onmouseover="rs('ec_pf_1/T12')" onmouseout="rh()">EC_PF_1:12</a></span>
<br/>.= 
<font color="Maroon" title="c7">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"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:14_1_1"/>
 <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</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"> (  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">u</font>, <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c9">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">w</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">u1</font> = <font color="Maroon" title="c7">u</font>, <font color="Maroon" title="c11">v1</font> = <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c12">w1</font> = <font color="Maroon" title="c9">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c2">V</font></span>)</span>)</span>)</span> ;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c7">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">w1</font>

<br/>.= 
<font color="Maroon" title="c10">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">w1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c7">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">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"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E6:14_1_1"/>
 <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</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"> (  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</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"> <font color="Maroon" title="c7">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <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="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">u1</font> = <font color="Maroon" title="c7">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c2">V</font></span>)</span>)</span>)</span> ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c7">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <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="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>)</span>)</span></span>)</span> = 
<font color="Maroon" title="c8">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <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="c2">V</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c2">V</font></span>)</span>)</span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c7">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E7:14_1_1"/>
 <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</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">  <a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D16" onmouseover="rs('algstr_0/D16')" onmouseout="rh()">ALGSTR_0:def 16</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"> <font color="Maroon" title="c7">v</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c7">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c2">V</font></span>)</span>)</span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">w</font> =  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">v1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">w</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D11" onmouseover="rs('algstr_0/D11')" onmouseout="rh()">ALGSTR_0:def 11</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"> <font color="Maroon" title="c7">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">w</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"><a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>)</span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c7">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">w</font> = 
<font color="Maroon" title="c8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">v1</font></span>)</span>

<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c2">V</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T5" onmouseover="rs('rlvect_1/T5')" onmouseout="rh()">RLVECT_1:5</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</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"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c8">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D2" onmouseover="rs('rlvect_1/D2')" onmouseout="rh()">RLVECT_1:def 2</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"> <font color="Maroon" title="c7">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">v1</font> = <font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c10">w1</font> = <font color="Maroon" title="c8">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c2">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c2">V</font></span>)</span>)</span>)</span> ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c7">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">w</font> = 
<font color="Maroon" title="c9">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">w1</font>

<br/>.= 
<font color="Maroon" title="c10">w1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">v1</font>

<br/>.= 
<font color="Maroon" title="c8">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">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"> verum</span><br/>


</div>
