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<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a><br/> for <font color="Olive" title="b2">M</font>, <font color="Olive" title="b3">N</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Olive" title="b1">K</font>  st <font color="Olive" title="b2">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Olive" title="b3">N</font> holds <br/>( ( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">M</font> implies <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b3">N</font> ) &amp; ( <font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b2">M</font> implies <font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b3">N</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">N</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b1">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Olive" title="b2">N</font> holds <br/>( ( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">M</font> implies <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">N</font> ) &amp; ( <font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b1">M</font> implies <font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b2">N</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c4">N</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c2">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">N</font> implies ( ( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> ) &amp; ( <font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">N</font> ) ) )</span><br/>







<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> ) &amp; ( <font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">N</font> ) )</span><br/>

<b>thus </b><a NAME="E2:19"/>
( <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">N</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E3:19_1"/>
<font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">M'</font>
 
<b>by </b><i><a class="txt" href="#E1:19_1"><i><font color="Green" title="E11">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:19_1"/>
<font color="Maroon" title="c1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T17" title="VECTSP_4:th.17">VECTSP_4:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:19"/>
( <font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">N</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_2"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">N</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E3:19_2"/>
<font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c5">M'</font>
 
<b>by </b><i><a class="txt" href="#E1:19_2"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:19_2"/>
<font color="Maroon" title="c1">x</font> is   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">N</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T18" title="VECTSP_4:th.18">VECTSP_4:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


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