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

<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">W</font> being    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b3">Ws</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b4">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">Ws</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b2">W</font> holds <br/>( <font color="Olive" title="b4">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Olive" title="b2">W</font> iff <font color="Olive" title="b4">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Olive" title="b3">Ws</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">W</font> being    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b2">Ws</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b3">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">Ws</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b1">W</font> holds <br/>( <font color="Olive" title="b3">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Olive" title="b1">W</font> iff <font color="Olive" title="b3">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Olive" title="b2">Ws</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">W</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</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">Ws</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b2">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">Ws</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W</font> holds <br/>( <font color="Olive" title="b2">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W</font> iff <font color="Olive" title="b2">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Olive" title="b1">Ws</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">Ws</font> be   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</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="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Maroon" title="c4">Ws</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W</font> holds <br/>( <font color="Olive" title="b1">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W</font> iff <font color="Olive" title="b1">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c4">Ws</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">Ws</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W</font> implies ( <font color="Maroon" title="c5">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W</font> iff <font color="Maroon" title="c5">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c4">Ws</font> ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:58"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c4">Ws</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</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"> ( <font color="Maroon" title="c5">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W</font> iff <font color="Maroon" title="c5">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c4">Ws</font> )</span><br/>

<div><span class="kw">hereby </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="c5">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c4">Ws</font> implies <font color="Maroon" title="c5">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:58_1"/>
<font color="Maroon" title="c5">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <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"> <font color="Maroon" title="c5">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c4">Ws</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">v</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="E3:58_1"/><span class="lab"><font color="Green" title="E33">B1</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D6" onmouseover="rs('vectsp_4/D6')" onmouseout="rh()">VECTSP_4:def 6</a></span>;<br/><a NAME="E4:58_1"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c4">Ws</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E3:58_1"><span class="lab"><font color="Green" title="E33">B1</font></span></a>, <a class="ref" href="zmodul07.html#T32" target="_self" onmouseover="rs('zmodul07/T32')" onmouseout="rh()">LmStrict11a</a></span>;<br/><span class="kw">hence </span><a NAME="E5:58_1"/>
<font color="Maroon" title="c5">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c4">Ws</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D6" onmouseover="rs('vectsp_4/D6')" onmouseout="rh()">VECTSP_4:def 6</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">assume </span><a NAME="E3:58"/>
<font color="Maroon" title="c5">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c4">Ws</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">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">v</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="E5:58"/><span class="lab"><font color="Green" title="E33">B1</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c4">Ws</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D6" onmouseover="rs('vectsp_4/D6')" onmouseout="rh()">VECTSP_4:def 6</a></span>;<br/>
<a NAME="E6:58"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E5:58"><span class="lab"><font color="Green" title="E33">B1</font></span></a>, <a class="ref" href="zmodul07.html#T32" target="_self" onmouseover="rs('zmodul07/T32')" onmouseout="rh()">LmStrict11a</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:58"/>
<font color="Maroon" title="c5">A</font> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D6" onmouseover="rs('vectsp_4/D6')" onmouseout="rh()">VECTSP_4:def 6</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>
