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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">V1</font>, <font color="Maroon" title="c3">V2</font>, <font color="Maroon" title="c4">W</font> be    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c5">W1</font>, <font color="Maroon" title="c6">W2</font> be    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c4">W</font>;<br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c5">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V1</font> &amp; <font color="Maroon" title="c6">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">V2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">W3</font> = <font color="Maroon" title="c5">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c6">W2</font> as    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlsub_1.html#T35" title="RLSUB_1:th.35">RLSUB_1:35</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:19_1_1"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c8">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">W3</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c9">w</font> = <font color="Maroon" title="c8">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">W</font> <b>by </b><i><a class="ref" href="hahnban.html#T21" target="_self" title="HAHNBAN:th.21">Th21</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">w1</font>, <font color="Maroon" title="c11">w2</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">W</font><b> such that </b><br/><a NAME="E4:19_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
( <font color="Maroon" title="c10">w1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c5">W1</font> &amp; <font color="Maroon" title="c11">w2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c6">W2</font> )
 <b>and </b><br/><a NAME="E5:19_1_1"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c9">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">w1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">w2</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E1:19_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="rlsub_2.html#T5" title="RLSUB_2:th.5">RLSUB_2:5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">v1</font> = <font color="Maroon" title="c10">w1</font>, <font color="Maroon" title="c13">v2</font> = <font color="Maroon" title="c11">w2</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlsub_1.html#T18" title="RLSUB_1:th.18">RLSUB_1:18</a></i>;<br/><a NAME="E7:19_1_1"/>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c13">v2</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E5:19_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="rlsub_1.html#T21" title="RLSUB_1:th.21">RLSUB_1:21</a></i>;<br/><b>hence </b><a NAME="E8:19_1_1"/>
<font color="Maroon" title="c8">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">V1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c3">V2</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="hahnban.html#E4:19_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="rlsub_2.html#T5" title="RLSUB_2:th.5">RLSUB_2:5</a></i>;<br/></div>
<b>end;</b></div><a NAME="E2:19_1"/><i><font color="Green" title="E18">A5</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">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="c4">W</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">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="c4">W</font> )
 <b>by </b><i><a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></i>;<br/><b>assume </b><a NAME="E3:19_1"/>
<font color="Maroon" title="c8">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">V1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c3">V2</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c10">v2</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E5:19_1"/><i><font color="Green" title="E19">A6</font></i>: 
( <font color="Maroon" title="c9">v1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">V1</font> &amp; <font color="Maroon" title="c10">v2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">V2</font> )
 <b>and </b><br/><a NAME="E6:19_1"/><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">v2</font>
 <b>by </b><i><a class="ref" href="rlsub_2.html#T5" title="RLSUB_2:th.5">RLSUB_2:5</a></i>;<br/><a NAME="E7:19_1"/>
( <font color="Maroon" title="c9">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="c5">W1</font> &amp; <font color="Maroon" title="c10">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="c6">W2</font> )
 <b>by </b><i><a class="txt" href="hahnban.html#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="hahnban.html#E5:19_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">w1</font> = <font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c12">w2</font> = <font color="Maroon" title="c10">v2</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">W</font> <b>by </b><i><a class="txt" href="hahnban.html#E2:19_1"><i><font color="Green" title="E18">A5</font></i></a></i>;<br/><a NAME="E9:19_1"/>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">w1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c12">w2</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E6:19_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="rlsub_1.html#T21" title="RLSUB_1:th.21">RLSUB_1:21</a></i>;<br/><b>hence </b><a NAME="E10:19_1"/>
<font color="Maroon" title="c8">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c7">W3</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="hahnban.html#E5:19_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="rlsub_2.html#T5" title="RLSUB_2:th.5">RLSUB_2:5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:19"/>
<font color="Maroon" title="c5">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c6">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c3">V2</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T39" title="RLSUB_1:th.39">RLSUB_1:39</a></i>;<br/>


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