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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">W1</font>, <font color="Olive" title="b3">W2</font>, <font color="Olive" title="b4">W3</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b2">W1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Olive" title="b3">W2</font> holds <br/><font color="Olive" title="b3">W2</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Olive" title="b4">W3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Olive" title="b3">W2</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">W2</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Olive" title="b4">W3</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font>, <font color="Olive" title="b3">W3</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">W1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Olive" title="b2">W2</font> holds <br/><font color="Olive" title="b2">W2</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Olive" title="b3">W3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Olive" title="b2">W2</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">W2</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Olive" title="b3">W3</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font>, <font color="Maroon" title="c5">W3</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">W1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">W2</font> implies <font color="Maroon" title="c4">W2</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c5">W3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c5">W3</font></span>)</span> )</span><br/>









<a NAME="E1:38"/>
( <font color="Maroon" title="c3">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">W2</font> implies <font color="Maroon" title="c4">W2</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c5">W3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c5">W3</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#T33" title="VECTSP_5:th.33">VECTSP_5:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:38"/>
( <font color="Maroon" title="c3">W1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">W2</font> implies <font color="Maroon" title="c4">W2</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c5">W3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W2</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c5">W3</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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