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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">y</font>, <font color="Olive" title="b2">u</font>, <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">w</font> holds <br/><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b2">u</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</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="c2">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">w</font> implies <font color="Maroon" title="c2">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">u</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:10"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c2">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">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="c2">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">u</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c2">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c3">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#T78" title="RLSUB_2:th.78">RLSUB_2:78</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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