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<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c3">N</font>, <font color="Maroon" title="c4">M2</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font>;<br/>







<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E11">A1</font></i>: 
( <font color="Maroon" title="c2">M1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">M2</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c3">N</font> )
 ;<br/>

<b>thus </b><a NAME="E2:22"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">M2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c5">M1'</font> = <font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c6">M2'</font> = <font color="Maroon" title="c4">M2</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:22"><i><font color="Green" title="E11">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="E2:22_1"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c5">M1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c6">M2'</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T20" title="VECTSP_4:th.20">VECTSP_4:20</a></i>;<br/>
<b>hence </b><a NAME="E3:22_1"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">M2</font>
 ;<br/>


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

<b>thus </b><a NAME="E3:22"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">M1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">M2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c5">M1'</font> = <font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c6">M2'</font> = <font color="Maroon" title="c4">M2</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:22"><i><font color="Green" title="E11">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="E2:22_2"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c5">M1'</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c6">M2'</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T26" title="VECTSP_4:th.26">VECTSP_4:26</a></i>;<br/>
<b>hence </b><a NAME="E3:22_2"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">M1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">M2</font>
 ;<br/>


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

<b>thus </b><a NAME="E4:22"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">M2</font> implies <font color="Maroon" title="c2">M1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">M2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_3"/><i><font color="Green" title="E12">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">M2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">M1'</font> = <font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c6">M2'</font> = <font color="Maroon" title="c4">M2</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:22"><i><font color="Green" title="E11">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="E3:22_3"/>
<font color="Maroon" title="c5">M1'</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">M2'</font>
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:22_3"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="vectsp_4.html#T35" title="VECTSP_4:th.35">VECTSP_4:35</a></i>;<br/>
<b>hence </b><a NAME="E4:22_3"/>
<font color="Maroon" title="c2">M1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">M2</font>
 <b>by </b><i><a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>


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

<b>thus </b><a NAME="E5:22"/>
( ( for <font color="Olive" title="b1">n</font> being  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c3">N</font>  st <font color="Olive" title="b1">n</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">M1</font> holds <br/><font color="Olive" title="b1">n</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">M2</font> ) implies <font color="Maroon" title="c2">M1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">M2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_4"/><i><font color="Green" title="E12">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c3">N</font>  st <font color="Olive" title="b1">n</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">M1</font> holds <br/><font color="Olive" title="b1">n</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">M2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">M1'</font> = <font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c6">M2'</font> = <font color="Maroon" title="c4">M2</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:22"><i><font color="Green" title="E11">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="E3:22_4"/>
<font color="Maroon" title="c5">M1'</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">M2'</font>
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:22_4"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="vectsp_4.html#T36" title="VECTSP_4:th.36">VECTSP_4:36</a></i>;<br/>
<b>hence </b><a NAME="E4:22_4"/>
<font color="Maroon" title="c2">M1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">M2</font>
 <b>by </b><i><a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>


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

<b>thus </b><a NAME="E6:22"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">M2</font> &amp; <font color="Maroon" title="c2">M1</font> is <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a> &amp; <font color="Maroon" title="c4">M2</font> is <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a> implies <font color="Maroon" title="c2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_5"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:22_5"/><i><font color="Green" title="E12">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">M2</font>
 <b>and </b><br/><a NAME="E2:22_5"/><i><font color="Green" title="E13">A5</font></i>: 
( <font color="Maroon" title="c2">M1</font> is <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a> &amp; <font color="Maroon" title="c4">M2</font> is <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">M1'</font> = <font color="Maroon" title="c2">M1</font> as  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:22"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="lmod_6.html#E2:22_5"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">M2'</font> = <font color="Maroon" title="c4">M2</font> as  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:22"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="lmod_6.html#E2:22_5"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E5:22_5"/>
<font color="Maroon" title="c5">M1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M2'</font>
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:22_5"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="vectsp_4.html#T37" title="VECTSP_4:th.37">VECTSP_4:37</a></i>;<br/>
<b>hence </b><a NAME="E6:22_5"/>
<font color="Maroon" title="c2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M2</font>
 ;<br/>


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

<b>thus </b><a NAME="E7:22"/>
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">M1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">M2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_6"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c5">M1'</font> = <font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c6">M2'</font> = <font color="Maroon" title="c4">M2</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:22"><i><font color="Green" title="E11">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="E2:22_6"/>
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c5">M1'</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">M2'</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T51" title="VECTSP_4:th.51">VECTSP_4:51</a></i>;<br/>
<b>hence </b><a NAME="E3:22_6"/>
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">M1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">M2</font>
 <b>by </b><i><a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>


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


</div>
