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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</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">V1</font>, <font color="Olive" title="b2">V2</font>, <font color="Olive" title="b3">W</font> being   <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b4">W1</font>, <font color="Olive" title="b5">W2</font> being   <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Olive" title="b3">W</font>  st <font color="Olive" title="b4">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">V1</font> &amp; <font color="Olive" title="b5">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">V2</font> holds <br/><font color="Olive" title="b4">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Olive" title="b5">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">V1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Olive" title="b2">V2</font></span><br/><span class="kw">let </span><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>; <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">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c4">W</font>  st <font color="Olive" title="b1">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V1</font> &amp; <font color="Olive" title="b2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">V2</font> holds <br/><font color="Olive" title="b1">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Olive" title="b2">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></span><br/><span class="kw">let </span><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>; <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="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> implies <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> )</span><br/>





<span class="kw">assume </span><a NAME="E1:22"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
( <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> )
 ; <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="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></span><br/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#T35" title="RLSUB_1:th.35">RLSUB_1:35</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <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="c8">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">W3</font> implies <font color="Maroon" title="c8">v</font> <a href="struct_0.html#R1" title="STRUCT_0: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> ) &amp; ( <font color="Maroon" title="c8">v</font> <a href="struct_0.html#R1" title="STRUCT_0: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> implies <font color="Maroon" title="c8">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">W3</font> ) )</span><br/><a NAME="E1:22_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
( 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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#D2" title="RLSUB_1:def.2">RLSUB_1:def 2</a></span>;<br/><div><span class="kw">hereby </span> <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="c8">v</font> <a href="struct_0.html#R1" title="STRUCT_0: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> implies <font color="Maroon" title="c8">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">W3</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:22_1_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c8">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">W3</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="c8">v</font> <a href="struct_0.html#R1" title="STRUCT_0: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></span><br/><span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban.html#T21" target="_self" title="HAHNBAN:th.21">Th21</a></span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:22_1_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
( <font color="Maroon" title="c10">w1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">W1</font> &amp; <font color="Maroon" title="c11">w2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W2</font> )
 <span class="kw">and </span><br/><a NAME="E5:22_1_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<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#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="rlsub_2.html#T5" title="RLSUB_2:th.5">RLSUB_2:5</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#T18" title="RLSUB_1:th.18">RLSUB_1:18</a></span>;<br/><a NAME="E7:22_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#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c13">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="rlsub_1.html#T21" title="RLSUB_1:th.21">RLSUB_1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E8:22_1_1"/>
<font color="Maroon" title="c8">v</font> <a href="struct_0.html#R1" title="STRUCT_0: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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="rlsub_2.html#T5" title="RLSUB_2:th.5">RLSUB_2:5</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">assume </span><a NAME="E3:22_1"/>
<font color="Maroon" title="c8">v</font> <a href="struct_0.html#R1" title="STRUCT_0: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>
 ; <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="c8">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">W3</font></span><br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:22_1"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
( <font color="Maroon" title="c9">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">V1</font> &amp; <font color="Maroon" title="c10">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">V2</font> )
 <span class="kw">and </span><br/><a NAME="E6:22_1"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<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#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#T5" title="RLSUB_2:th.5">RLSUB_2:5</a></span>;<br/><a NAME="E7:22_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E5:22_1"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/><a NAME="E9:22_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#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="rlsub_1.html#T21" title="RLSUB_1:th.21">RLSUB_1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E10:22_1"/>
<font color="Maroon" title="c8">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">W3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E5:22_1"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="rlsub_2.html#T5" title="RLSUB_2:th.5">RLSUB_2:5</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">hence </span><a NAME="E4:22"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#T39" title="RLSUB_1:th.39">RLSUB_1:39</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/>


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