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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">OAS</font> be   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</a>;<br/>



<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c2">OAS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">OAS</font><b> such that </b><br/><a NAME="E3:7"/><i><font color="Green" title="E6">A2</font></i>: 
( not <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font> &amp; not <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">u</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c8">v</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c9">w</font> = <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c6">d</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="txt" href="papdesaf.html#E1:7"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="papdesaf.html#T4" target="_self" title="PAPDESAF:th.4">Th4</a></i>;<br/>
<b>take </b><font color="Maroon" title="c11">z1</font> = <font color="Maroon" title="c8">v</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c7">u</font>;<br/>

<b>take </b><font color="Maroon" title="c12">z2</font> = <font color="Maroon" title="c10">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c9">w</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">r1</font>, <font color="Maroon" title="c14">r2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:7_1"/><i><font color="Green" title="E7">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c11">z1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r2</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 ;<br/><b>assume </b><a NAME="E2:7_1"/>
( <font color="Maroon" title="c13">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or <font color="Maroon" title="c14">r2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 ;<br/><a NAME="E3:7_1"/><b>then </b><i><font color="Green" title="E8">A4</font></i>: 
( <font color="Maroon" title="c13">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c14">r2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 ;<br/><font color="Maroon" title="c13">r1</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c11">z1</font> = 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r2</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">z2</font></span>)</span>
<b>by </b><i><a class="txt" href="papdesaf.html#E1:7_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="rlvect_1.html#T19" title="RLVECT_1:th.19">RLVECT_1:19</a></i>
<br/>.= 
<font color="Maroon" title="c14">r2</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c12">z2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T39" title="RLVECT_1:th.39">RLVECT_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c14">r2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">z2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T38" title="RLVECT_1:th.38">RLVECT_1:38</a></i>
;<br/><a NAME="E5:7_1"/><b>then </b>
( <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c10">y</font> or <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c9">w</font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E3:7_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="analmetr.html#T18" title="ANALMETR:th.18">ANALMETR:18</a></i>;<br/><b>hence </b><a NAME="E6:7_1"/>
( <font color="Maroon" title="c13">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c14">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:7"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="papdesaf.html#E3:7"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="geomtrap.html#T2" title="GEOMTRAP:th.2">GEOMTRAP:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:7"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c11">z1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c12">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/>


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