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<span class="kw">consider </span><font color="Maroon" title="c6">a1</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E2:49_1_1"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a1</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lmod_7.html#T15" target="_self" onmouseover="rs('lmod_7/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E20">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_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">a</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</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="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> )</span><br/><span class="kw">assume </span><a NAME="E1:49_1_1_1"/>
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</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"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font></span><br/><a NAME="E2:49_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">a1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_1_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="lmod_7.html#T11" target="_self" onmouseover="rs('lmod_7/T11')" onmouseout="rh()">Th11</a></span>;<br/><a NAME="E3:49_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T21" onmouseover="rs('vectsp_4/T21')" onmouseout="rh()">VECTSP_4:21</a></span>;<br/><a NAME="E4:49_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T23" onmouseover="rs('vectsp_1/T23')" onmouseout="rh()">VECTSP_1:23</a></span>;<br/><span class="kw">hence </span><a NAME="E5:49_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lmod_7.html#T11" target="_self" onmouseover="rs('lmod_7/T11')" onmouseout="rh()">Th11</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">take </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</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="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font></span><br/>

<span class="kw">thus </span><a NAME="E4:49_1_1"/>
 for <font color="Olive" title="b1">a</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1_1"><span class="lab"><font color="Green" title="E20">A2</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/>


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