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<div class="add">

<b>let </b><font color="Maroon">K</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">M1</font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">M2</font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">K</font>;<br/>







<b>assume </b><a NAME="E1:22"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">M1</font> <a href="lmod_6.html#R1" target="_self">c=</a> <font color="Maroon">N</font> &amp; <font color="Maroon">M2</font> <a href="lmod_6.html#R1" target="_self">c=</a> <font color="Maroon">N</font> )
 ;<br/>

<b>thus </b><a NAME="E2:22"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">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">M1'</font> = <font color="Maroon">M1</font>, <font color="Maroon">M2'</font> = <font color="Maroon">M2</font> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">N</font> <b>by </b><i>, </i>;<br/>
<a NAME="E2:22_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">M1'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">M2'</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T20">VECTSP_4:20</a></i>;<br/>
<b>hence </b><a NAME="E3:22_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">M2</font>
 ;<br/>


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

<b>thus </b><a NAME="E3:22"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">M1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">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">M1'</font> = <font color="Maroon">M1</font>, <font color="Maroon">M2'</font> = <font color="Maroon">M2</font> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">N</font> <b>by </b><i>, </i>;<br/>
<a NAME="E2:22_2"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">M1'</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">M2'</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T26">VECTSP_4:26</a></i>;<br/>
<b>hence </b><a NAME="E3:22_2"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">M1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">M2</font>
 ;<br/>


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

<b>thus </b><a NAME="E4:22"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M2</font> implies <font color="Maroon">M1</font> <a href="lmod_6.html#R1" target="_self">c=</a> <font color="Maroon">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">E47</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">M1'</font> = <font color="Maroon">M1</font>, <font color="Maroon">M2'</font> = <font color="Maroon">M2</font> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">N</font> <b>by </b><i>, </i>;<br/>
<a NAME="E3:22_3"/>
<font color="Maroon">M1'</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">M2'</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#T35">VECTSP_4:35</a></i>;<br/>
<b>hence </b><a NAME="E4:22_3"/>
<font color="Maroon">M1</font> <a href="lmod_6.html#R1" target="_self">c=</a> <font color="Maroon">M2</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E5:22"/>
( ( for <font color="Olive">n</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">N</font>  st <font color="Olive">n</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">M1</font> holds <br/><font color="Olive">n</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">M2</font> ) implies <font color="Maroon">M1</font> <a href="lmod_6.html#R1" target="_self">c=</a> <font color="Maroon">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">E52</font></i>: 
for <font color="Olive">n</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">N</font>  st <font color="Olive">n</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">M1</font> holds <br/><font color="Olive">n</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">M2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">M1'</font> = <font color="Maroon">M1</font>, <font color="Maroon">M2'</font> = <font color="Maroon">M2</font> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">N</font> <b>by </b><i>, </i>;<br/>
<a NAME="E3:22_4"/>
<font color="Maroon">M1'</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">M2'</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#T36">VECTSP_4:36</a></i>;<br/>
<b>hence </b><a NAME="E4:22_4"/>
<font color="Maroon">M1</font> <a href="lmod_6.html#R1" target="_self">c=</a> <font color="Maroon">M2</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E6:22"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M2</font> &amp; <font color="Maroon">M1</font> is <a href="vectsp_1.html#V11">strict</a> &amp; <font color="Maroon">M2</font> is <a href="vectsp_1.html#V11">strict</a> implies <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">E63</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M2</font>
 <b>and </b><br/><a NAME="E2:22_5"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">M1</font> is <a href="vectsp_1.html#V11">strict</a> &amp; <font color="Maroon">M2</font> is <a href="vectsp_1.html#V11">strict</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">M1'</font> = <font color="Maroon">M1</font> as  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">N</font> <b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">M2'</font> = <font color="Maroon">M2</font> as  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">N</font> <b>by </b><i>, , </i>;<br/>
<a NAME="E5:22_5"/>
<font color="Maroon">M1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2'</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#T37">VECTSP_4:37</a></i>;<br/>
<b>hence </b><a NAME="E6:22_5"/>
<font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font>
 ;<br/>


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

<b>thus </b><a NAME="E7:22"/>
 <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">M1</font> <a href="lmod_6.html#R1" target="_self">c=</a> <font color="Maroon">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">M1'</font> = <font color="Maroon">M1</font>, <font color="Maroon">M2'</font> = <font color="Maroon">M2</font> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">N</font> <b>by </b><i>, </i>;<br/>
<a NAME="E2:22_6"/>
 <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">M1'</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">M2'</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T51">VECTSP_4:51</a></i>;<br/>
<b>hence </b><a NAME="E3:22_6"/>
 <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">M1</font> <a href="lmod_6.html#R1" target="_self">c=</a> <font color="Maroon">M2</font>
 <b>by </b><i/>;<br/>


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


</div>
