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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">M1</font>, <font color="Olive" title="b2">N</font>, <font color="Olive" title="b3">M2</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">M1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b3">M2</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Olive" title="b2">N</font> holds <br/>(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b3">M2</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">M1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b3">M2</font> &amp; ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">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="Olive" title="b3">M2</font> implies <font color="Olive" title="b1">M1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Olive" title="b3">M2</font> ) &amp; ( ( for <font color="Olive" title="b4">n</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b2">N</font>  st <font color="Olive" title="b4">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">M1</font> holds <br/><font color="Olive" title="b4">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b3">M2</font> ) implies <font color="Olive" title="b1">M1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Olive" title="b3">M2</font> ) &amp; ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">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="Olive" title="b3">M2</font> &amp; <font color="Olive" title="b1">M1</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  &amp; <font color="Olive" title="b3">M2</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  implies <font color="Olive" title="b1">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">M2</font> ) &amp;  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Olive" title="b1">M1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Olive" title="b3">M2</font> )</span><br/><span class="kw">let </span><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>; <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="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> implies (  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">M2</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">M2</font> &amp; ( 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> ) &amp; ( ( for <font color="Olive" title="b1">n</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">N</font>  st <font color="Olive" title="b1">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">M1</font> holds <br/><font color="Olive" title="b1">n</font> <a href="struct_0.html#R1" title="STRUCT_0: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> ) &amp; ( 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#V7" title="VECTSP_1:attr.7">strict</a>  &amp; <font color="Maroon" title="c4">M2</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">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> ) &amp;  <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> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:23"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">M2</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">M2</font> &amp; ( 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> ) &amp; ( ( for <font color="Olive" title="b1">n</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">N</font>  st <font color="Olive" title="b1">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">M1</font> holds <br/><font color="Olive" title="b1">n</font> <a href="struct_0.html#R1" title="STRUCT_0: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> ) &amp; ( 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#V7" title="VECTSP_1:attr.7">strict</a>  &amp; <font color="Maroon" title="c4">M2</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">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> ) &amp;  <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> )</span><br/>

<span class="kw">thus </span><a NAME="E3:23"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">M2</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">M2</font> &amp; ( 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> ) &amp; ( ( for <font color="Olive" title="b1">n</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">N</font>  st <font color="Olive" title="b1">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">M1</font> holds <br/><font color="Olive" title="b1">n</font> <a href="struct_0.html#R1" title="STRUCT_0: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> ) &amp; ( 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#V7" title="VECTSP_1:attr.7">strict</a>  &amp; <font color="Maroon" title="c4">M2</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">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> ) &amp;  <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c5">M19</font> = <font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c6">M29</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></span>;<br/>
<a NAME="E2:23_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">M19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c6">M29</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T20" title="VECTSP_4:th.20">VECTSP_4:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:23_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">M2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:23"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">M2</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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> ) &amp; ( ( for <font color="Olive" title="b1">n</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">N</font>  st <font color="Olive" title="b1">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">M1</font> holds <br/><font color="Olive" title="b1">n</font> <a href="struct_0.html#R1" title="STRUCT_0: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> ) &amp; ( 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#V7" title="VECTSP_1:attr.7">strict</a>  &amp; <font color="Maroon" title="c4">M2</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">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> ) &amp;  <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c5">M19</font> = <font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c6">M29</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></span>;<br/>
<a NAME="E2:23_2"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">M19</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">M29</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T26" title="VECTSP_4:th.26">VECTSP_4:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:23_2"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">M1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">M2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:23"/>
( 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> )
  <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">n</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">N</font>  st <font color="Olive" title="b1">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">M1</font> holds <br/><font color="Olive" title="b1">n</font> <a href="struct_0.html#R1" title="STRUCT_0: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> ) &amp; ( 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#V7" title="VECTSP_1:attr.7">strict</a>  &amp; <font color="Maroon" title="c4">M2</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">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> ) &amp;  <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c5">M19</font> = <font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c6">M29</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:23_3"/>
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>
 ; <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="c2">M1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">M2</font></span><br/>

<a NAME="E3:23_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">M19</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">M29</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T35" title="VECTSP_4:th.35">VECTSP_4:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:23_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>
 <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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E6:23"/>
( ( for <font color="Olive" title="b1">n</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">N</font>  st <font color="Olive" title="b1">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">M1</font> holds <br/><font color="Olive" title="b1">n</font> <a href="struct_0.html#R1" title="STRUCT_0: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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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#V7" title="VECTSP_1:attr.7">strict</a>  &amp; <font color="Maroon" title="c4">M2</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">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> ) &amp;  <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c5">M19</font> = <font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c6">M29</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:23_4"/>
for <font color="Olive" title="b1">n</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">N</font>  st <font color="Olive" title="b1">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">M1</font> holds <br/><font color="Olive" title="b1">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">M2</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="c2">M1</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">M2</font></span><br/>

<a NAME="E3:23_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">M19</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">M29</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T36" title="VECTSP_4:th.36">VECTSP_4:36</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:23_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>
 <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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E7:23"/>
( 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#V7" title="VECTSP_1:attr.7">strict</a>  &amp; <font color="Maroon" title="c4">M2</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23_5"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
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>
 <span class="kw">and </span><br/><a NAME="E2:23_5"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c2">M1</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a> 
 <span class="kw">and </span><br/><a NAME="E3:23_5"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c4">M2</font> is  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a> 
 ; <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="c2">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">M29</font> = <font color="Maroon" title="c4">M2</font> as   <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="c3">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E3:23_5"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">M19</font> = <font color="Maroon" title="c2">M1</font> as   <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="c3">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E2:23_5"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></span>;<br/>
<a NAME="E6:23_5"/>
<font color="Maroon" title="c6">M19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_5"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="vectsp_4.html#T37" title="VECTSP_4:th.37">VECTSP_4:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:23_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E8:23"/>
 <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>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c5">M19</font> = <font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c6">M29</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></span>;<br/>
<a NAME="E2:23_6"/>
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c5">M19</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">M29</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T51" title="VECTSP_4:th.51">VECTSP_4:51</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:23_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>
 <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/>


</div><span class="kw">end;</span></div>


</div>
