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<b>let </b><font color="Maroon" title="c1">K</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c3">V1</font>, <font color="Maroon" title="c4">V2</font>, <font color="Maroon" title="c5">W</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>;<br/><b>let </b><font color="Maroon" title="c6">W1</font>, <font color="Maroon" title="c7">W2</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c5">W</font>;<br/>







<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c6">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">V1</font> &amp; <font color="Maroon" title="c7">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">V2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">W3</font> = <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T34" title="VECTSP_4:th.34">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" title="c9">v</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:24_1_1"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c9">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">W3</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">w</font> = <font color="Maroon" title="c9">v</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c5">W</font> <b>by </b><i><a class="ref" href="vectsp10.html#T12" target="_self" title="VECTSP10:th.12">Th12</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">w1</font>, <font color="Maroon" title="c12">w2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c5">W</font><b> such that </b><br/><a NAME="E4:24_1_1"/><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c11">w1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> &amp; <font color="Maroon" title="c12">w2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">W2</font> )
 <b>and </b><br/><a NAME="E5:24_1_1"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c10">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">w1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">w2</font>
 <b>by </b><i><a class="txt" href="vectsp10.html#E1:24_1_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="vectsp_5.html#T5" title="VECTSP_5:th.5">VECTSP_5:5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">v1</font> = <font color="Maroon" title="c11">w1</font>, <font color="Maroon" title="c14">v2</font> = <font color="Maroon" title="c12">w2</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T18" title="VECTSP_4:th.18">VECTSP_4:18</a></i>;<br/><a NAME="E7:24_1_1"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">v2</font>
 <b>by </b><i><a class="txt" href="vectsp10.html#E5:24_1_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="vectsp_4.html#T21" title="VECTSP_4:th.21">VECTSP_4:21</a></i>;<br/><b>hence </b><a NAME="E8:24_1_1"/>
<font color="Maroon" title="c9">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">V1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">V2</font>
 <b>by </b><i><a class="txt" href="vectsp10.html#E1:24"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="vectsp10.html#E4:24_1_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="vectsp_5.html#T5" title="VECTSP_5:th.5">VECTSP_5:5</a></i>;<br/></div>
<b>end;</b></div><a NAME="E2:24_1"/><i><font color="Green" title="E13">A5</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">W1</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="c5">W</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">W2</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="c5">W</font> )
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/><b>assume </b><a NAME="E3:24_1"/>
<font color="Maroon" title="c9">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">V1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">V2</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">v1</font>, <font color="Maroon" title="c11">v2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E5:24_1"/><i><font color="Green" title="E14">A6</font></i>: 
( <font color="Maroon" title="c10">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">V1</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">V2</font> )
 <b>and </b><br/><a NAME="E6:24_1"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">v2</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T5" title="VECTSP_5:th.5">VECTSP_5:5</a></i>;<br/><a NAME="E7:24_1"/>
( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">W1</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">W2</font> )
 <b>by </b><i><a class="txt" href="vectsp10.html#E1:24"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="vectsp10.html#E5:24_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">w1</font> = <font color="Maroon" title="c10">v1</font>, <font color="Maroon" title="c13">w2</font> = <font color="Maroon" title="c11">v2</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c5">W</font> <b>by </b><i><a class="txt" href="vectsp10.html#E2:24_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/><a NAME="E9:24_1"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">w2</font>
 <b>by </b><i><a class="txt" href="vectsp10.html#E6:24_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="vectsp_4.html#T21" title="VECTSP_4:th.21">VECTSP_4:21</a></i>;<br/><b>hence </b><a NAME="E10:24_1"/>
<font color="Maroon" title="c9">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">W3</font>
 <b>by </b><i><a class="txt" href="vectsp10.html#E1:24"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="vectsp10.html#E5:24_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="vectsp_5.html#T5" title="VECTSP_5:th.5">VECTSP_5:5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:24"/>
<font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">V1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">V2</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T38" title="VECTSP_4:th.38">VECTSP_4:38</a></i>;<br/>


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