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<b>let </b><font color="Maroon">K</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">V1</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">V2</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">W</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W1</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W</font>, <font color="Maroon">W2</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W</font>;<br/>







<b>assume </b><a NAME="E1:24"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V1</font> &amp; <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">W3</font> = <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T34">VECTSP_4:34</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:24_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W3</font>
 ;<br/><b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">v</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">w1</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font>, <font color="Maroon">w2</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E4:24_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">w1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">w2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> )
 <b>and </b><br/><a NAME="E5:24_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w2</font>
 <b>by </b><i>, <a class="ref" href="vectsp_5.html#T5">VECTSP_5:5</a></i>;<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">w1</font>, <font color="Maroon">v2</font> = <font color="Maroon">w2</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T18">VECTSP_4:18</a></i>;<br/><a NAME="E7:24_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font>
 <b>by </b><i>, <a class="ref" href="vectsp_4.html#T21">VECTSP_4:21</a></i>;<br/><b>hence </b><a NAME="E8:24_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">V1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">V2</font>
 <b>by </b><i>, , <a class="ref" href="vectsp_5.html#T5">VECTSP_5:5</a></i>;<br/></div>
<b>end;</b></div><a NAME="E2:24_1"/><i><font color="Green">E47</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> )
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/><b>assume </b><a NAME="E3:24_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">V1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">V2</font>
 ;<br/><b>then consider </b><font color="Maroon">v1</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">v2</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:24_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">V1</font> &amp; <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">V2</font> )
 <b>and </b><br/><a NAME="E6:24_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T5">VECTSP_5:5</a></i>;<br/><a NAME="E7:24_1"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> )
 <b>by </b><i>, , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">v1</font>, <font color="Maroon">w2</font> = <font color="Maroon">v2</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font> <b>by </b><i/>;<br/><a NAME="E9:24_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w2</font>
 <b>by </b><i>, <a class="ref" href="vectsp_4.html#T21">VECTSP_4:21</a></i>;<br/><b>hence </b><a NAME="E10:24_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W3</font>
 <b>by </b><i>, , <a class="ref" href="vectsp_5.html#T5">VECTSP_5:5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:24"/>
<font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">V2</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T38">VECTSP_4:38</a></i>;<br/>


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