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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">L1</font>, <font color="Olive" title="b3">L2</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Olive" title="b1">V</font> holds  <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="Olive" title="b1">V</font></span>)</span>,<font color="Olive" title="b2">L1</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="Olive" title="b1">V</font></span>)</span>,<font color="Olive" title="b3">L2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">L1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Olive" title="b3">L2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</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">  for <font color="Olive" title="b1">L1</font>, <font color="Olive" title="b2">L2</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font> holds  <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>,<font color="Olive" title="b1">L1</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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>,<font color="Olive" title="b2">L2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Olive" title="b2">L2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">L1</font>, <font color="Maroon" title="c4">L2</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"> <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>,<font color="Maroon" title="c3">L1</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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>,<font color="Maroon" title="c4">L2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">L1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c4">L2</font></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">v2</font> =  <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>,<font color="Maroon" title="c4">L2</font>);<br/>
<a NAME="E1:38"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
( <font color="Maroon" title="c3">L1</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a>  <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="c3">L1</font></span>)</span> &amp; <font color="Maroon" title="c4">L2</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a>  <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="c4">L2</font></span>)</span> )
 
;<br/>
<a NAME="E2:38"/>
<font color="Maroon" title="c4">L2</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="E3:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c4">L2</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="E4:38"/>
<font color="Maroon" title="c3">L1</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="E5:38"/><span class="kw">then </span>
<font color="Maroon" title="c3">L1</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/>
<span class="kw">hence </span><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>,<font color="Maroon" title="c3">L1</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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>,<font color="Maroon" title="c4">L2</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="zmodul02.html#K14" title="ZMODUL02:func.14">LCAdd</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">L1</font>,<span class="p2">(<span class="default"><a href="rlvect_2.html#K1" title="RLVECT_2:func.1">vector</a> (<span class="p3">(<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>,<font color="Maroon" title="c4">L2</font>)</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></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="p1">(<span class="default"><a href="zmodul02.html#K14" title="ZMODUL02:func.14">LCAdd</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="zmodul02.html#K13" title="ZMODUL02:func.13">@</a> <font color="Maroon" title="c3">L1</font></span>)</span>,<span class="p1">(<span class="default"><a href="zmodul02.html#K13" title="ZMODUL02:func.13">@</a> <font color="Maroon" title="c4">L2</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="rlvect_2.html#D1" onmouseover="rs('rlvect_2/D1')" onmouseout="rh()">RLVECT_2:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c3">L1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c4">L2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="zmodul02.html#D32" target="_self" onmouseover="rs('zmodul02/D32')" onmouseout="rh()">Def32</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/>


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