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

<span class="kw">set </span><font color="Maroon" title="c3">S</font> =  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font>;<br/>
<div><span class="lab"><font color="Green" title="E18">A1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font></span>)</span><br/>  for <font color="Olive" title="b3">K</font>, <font color="Olive" title="b4">L</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">L</font> holds <br/><font color="Olive" title="b1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Olive" title="b4">L</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">u</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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">  for <font color="Olive" title="b1">K</font>, <font color="Olive" title="b2">L</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font>  st <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">L</font> holds <br/><font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">K</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Olive" title="b2">L</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">K</font>, <font color="Maroon" title="c7">L</font> be    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">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="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">K</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L</font> implies <font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">K</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c7">L</font> )</span><br/><a NAME="E1:37_1_1_1"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
(  <a href="zmodul02.html#K12" title="ZMODUL02:func.12">@</a> <span class="p1">(<span class="default"><a href="zmodul02.html#K13" title="ZMODUL02:func.13">@</a> <font color="Maroon" title="c6">K</font></span>)</span> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c6">K</font> &amp;  <a href="zmodul02.html#K12" title="ZMODUL02:func.12">@</a> <span class="p1">(<span class="default"><a href="zmodul02.html#K13" title="ZMODUL02:func.13">@</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c7">L</font> )
 ;<br/><span class="kw">assume </span><a NAME="E2:37_1_1_1"/>
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">K</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L</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="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">K</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c7">L</font></span><br/><span class="kw">hence </span><a NAME="E3:37_1_1_1"/>
<font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">K</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c7">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="zmodul02.html#D32" target="_self" onmouseover="rs('zmodul02/D32')" onmouseout="rh()">Def32</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="E2:37_1_1"/>
 <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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="c4">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">u</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">K</font> = <font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c7">L</font> = <font color="Maroon" title="c5">v</font> as    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#D29" target="_self" onmouseover="rs('zmodul02/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c4">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">v</font> = 
<font color="Maroon" title="c6">K</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c7">L</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">L</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c6">K</font>

<br/>.= 
<font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></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="E3:37_1_1"/>
 <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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="c4">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">L</font> = <font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c8">K</font> = <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c9">M</font> = <font color="Maroon" title="c6">w</font> as    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#D29" target="_self" onmouseover="rs('zmodul02/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<a NAME="E2:37_1_1_3"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">K</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c9">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>;<br/>
<a NAME="E3:37_1_1_3"/>
<font color="Maroon" title="c4">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c8">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c8">K</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c9">M</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">L</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">K</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c9">M</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T28" target="_self" onmouseover="rs('zmodul02/T28')" onmouseout="rh()">Th28</a></span>
<br/>.= 
<font color="Maroon" title="c4">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:37_1_1_3"><span class="lab"><font color="Green" title="E19">A3</font></span></a></span>
;<br/> <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="E4:37_1_1"/>
 <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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="c4">v</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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">K</font> = <font color="Maroon" title="c4">v</font> as    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#D29" target="_self" onmouseover="rs('zmodul02/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c4">v</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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c5">K</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K2" title="VECTSP_6:func.2">ZeroLC</a> <font color="Maroon" title="c2">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">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="E5:37_1_1"/>
 <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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="c4">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="c5">K</font> = <font color="Maroon" title="c4">v</font> as    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#D29" target="_self" onmouseover="rs('zmodul02/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<a NAME="E2:37_1_1_5"/>
 <a href="vectsp_6.html#K7" title="VECTSP_6:func.7">-</a> <font color="Maroon" title="c5">K</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#D29" target="_self" onmouseover="rs('zmodul02/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<a NAME="E3:37_1_1_5"/><span class="kw">then </span>
 <a href="vectsp_6.html#K7" title="VECTSP_6:func.7">-</a> <font color="Maroon" title="c5">K</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font>
 
;<br/>
<a NAME="E4:37_1_1_5"/><span class="kw">then </span>
 <a href="vectsp_6.html#K7" title="VECTSP_6:func.7">-</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> (<span class="p1">(<span class="default"><a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_6.html#K7" title="VECTSP_6:func.7">-</a> <font color="Maroon" title="c5">K</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D1" onmouseover="rs('rlvect_2/D1')" onmouseout="rh()">RLVECT_2:def 1</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> (<span class="p2">(<span class="default"><a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="vectsp_6.html#K7" title="VECTSP_6:func.7">-</a> <font color="Maroon" title="c5">K</font></span>)</span>)</span>)</span> = 
<font color="Maroon" title="c5">K</font> <a href="vectsp_6.html#K8" title="VECTSP_6:func.8">-</a> <font color="Maroon" title="c5">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T42" target="_self" onmouseover="rs('zmodul02/T42')" onmouseout="rh()">Th42</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:37_1_1_5"/>
 ex <font color="Olive" title="b1">w</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font></span>)</span> st <font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font></span>)</span>
 ; <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"> verum</span><br/>


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

<div><span class="lab"><font color="Green" title="E19">A4</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1_6"><span class="kw">now </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">  for <font color="Olive" title="b1">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font></span>)</span><br/>  for <font color="Olive" title="b2">L</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">L</font> holds <br/><font color="Olive" title="b3">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Olive" title="b2">L</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">v</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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">  for <font color="Olive" title="b1">L</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> holds <br/><font color="Olive" title="b2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Olive" title="b1">L</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">L</font> be    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">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">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> holds <br/><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">L</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</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="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> implies <font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">L</font> )</span><br/><a NAME="E1:37_1_1_6"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
 <a href="zmodul02.html#K12" title="ZMODUL02:func.12">@</a> <span class="p1">(<span class="default"><a href="zmodul02.html#K13" title="ZMODUL02:func.13">@</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c5">L</font>
 ;<br/><span class="kw">assume </span><a NAME="E2:37_1_1_6"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</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="c6">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">L</font></span><br/><span class="kw">hence </span><a NAME="E3:37_1_1_6"/>
<font color="Maroon" title="c6">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1_6"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="zmodul02.html#D33" target="_self" onmouseover="rs('zmodul02/D33')" onmouseout="rh()">Def33</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="E7:37_1_1"/>
 <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font></span>)</span> holds  <font color="Maroon" title="c4">a</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="c4">a</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="c4">a</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="c5">v</font>, <font color="Maroon" title="c6">w</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">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="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">w</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c7">K</font> = <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c8">M</font> = <font color="Maroon" title="c6">w</font> as    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#D29" target="_self" onmouseover="rs('zmodul02/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<a NAME="E2:37_1_1_7"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
( <font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c7">K</font> &amp; <font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c8">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>;<br/>
<a NAME="E3:37_1_1_7"/>
<font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">K</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c8">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span> = 
<font color="Maroon" title="c4">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">K</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c8">M</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c7">K</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c8">M</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T33" target="_self" onmouseover="rs('zmodul02/T33')" onmouseout="rh()">Th33</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">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="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:37_1_1_7"><span class="lab"><font color="Green" title="E20">A6</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:37_1_1_7"/>
<font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c5">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="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">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="E8:37_1_1"/>
 <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp;  <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">b</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="c4">a</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="c5">b</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="c6">v</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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="c4">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">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="c5">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">v</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c7">K</font> = <font color="Maroon" title="c6">v</font> as    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#D29" target="_self" onmouseover="rs('zmodul02/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<a NAME="E2:37_1_1_8"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
( <font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c7">K</font> &amp; <font color="Maroon" title="c5">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c7">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c7">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c7">K</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c7">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T32" target="_self" onmouseover="rs('zmodul02/T32')" onmouseout="rh()">Th32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">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="c5">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:37_1_1_8"><span class="lab"><font color="Green" title="E20">A7</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:37_1_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">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="c5">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">v</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="E9:37_1_1"/>
 <a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <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#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="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="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</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="c6">v</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">v</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c7">K</font> = <font color="Maroon" title="c6">v</font> as    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#D29" target="_self" onmouseover="rs('zmodul02/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<a NAME="E2:37_1_1_9"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c7">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c7">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c7">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#T34" target="_self" onmouseover="rs('zmodul02/T34')" onmouseout="rh()">Th34</a></span>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E2:37_1_1_9"><span class="lab"><font color="Green" title="E20">A8</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:37_1_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">v</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="c4">v</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K16" title="ZMODUL02:func.16">LC_Z_Module</a> <font color="Maroon" title="c2">V</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> <font color="Maroon" title="c1">R</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">K</font> = <font color="Maroon" title="c4">v</font> as    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul02.html#D29" target="_self" onmouseover="rs('zmodul02/D29')" onmouseout="rh()">Def29</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> <font color="Maroon" title="c1">R</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">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>
