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

<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="E2:41_2_1"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<font color="Maroon" title="c4">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a1</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lmod_7.html#T12" target="_self" onmouseover="rs('lmod_7/T12')" onmouseout="rh()">Th12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">a2</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="E4:41_2_1"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<font color="Maroon" title="c5">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a2</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lmod_7.html#T12" target="_self" onmouseover="rs('lmod_7/T12')" onmouseout="rh()">Th12</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E17">A8</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_2_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">b1</font>, <font color="Olive" title="b2">b2</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="c4">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c5">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b2</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</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="Olive" title="b1">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">b2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">b1</font>, <font color="Maroon" title="c9">b2</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="c4">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c5">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b2</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</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="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">b2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:41_2_1_1"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
<font color="Maroon" title="c4">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">and </span><br/><a NAME="E2:41_2_1_1"/><span class="lab"><font color="Green" title="E19">A10</font></span>: 
<font color="Maroon" title="c5">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b2</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</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="c6">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</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="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">b2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font></span><br/><a NAME="E3:41_2_1_1"/><span class="lab"><font color="Green" title="E20">A11</font></span>: 
<font color="Maroon" title="c6">a1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">b1</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="#E2:41_2_1"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E1:41_2_1_1"><span class="lab"><font color="Green" title="E18">A9</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="E4:41_2_1_1"/>
<font color="Maroon" title="c7">a2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c9">b2</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="#E4:41_2_1"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E2:41_2_1_1"><span class="lab"><font color="Green" title="E19">A10</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="E5:41_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c9">b2</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="txt" href="#E3:41_2_1_1"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="ref" href="vectsp_4.html#T20" onmouseover="rs('vectsp_4/T20')" onmouseout="rh()">VECTSP_4:20</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c9">b2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c9">b2</font>
<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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c9">b2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="lmod_7.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">a2</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="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">b2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T17" onmouseover="rs('vectsp_1/T17')" onmouseout="rh()">VECTSP_1:17</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:41_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</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="c8">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">b2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2_1_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <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="c6">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</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">a1</font>, <font color="Olive" title="b2">a2</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="c4">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c5">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a2</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</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="Olive" title="b1">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">a2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font></span><br/>

<span class="kw">thus </span><a NAME="E6:41_2_1"/>
 for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</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="c4">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c5">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a2</font> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">a2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</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="Olive" title="b1">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">a2</font></span>)</span> <a href="lmod_7.html#K8" title="LMOD_7:func.8">..</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2_1"><span class="lab"><font color="Green" title="E17">A8</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>
