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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">w</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">u</font>, <font color="Olive" title="b4">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st  <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font> holds <br/>( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K2" title="GEOMTRAP:func.2">pr1</a> <font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Olive" title="b4">v</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K2" title="GEOMTRAP:func.2">pr1</a> <font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">u</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K2" title="GEOMTRAP:func.2">pr1</a> <font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b4">v</font></span>)</span> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K3" title="GEOMTRAP:func.3">pr2</a> <font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Olive" title="b4">v</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K3" title="GEOMTRAP:func.3">pr2</a> <font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">u</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K3" title="GEOMTRAP:func.3">pr2</a> <font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b4">v</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">w</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font> implies ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K2" title="GEOMTRAP:func.2">pr1</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K2" title="GEOMTRAP:func.2">pr1</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">u</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K2" title="GEOMTRAP:func.2">pr1</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">v</font></span>)</span> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K3" title="GEOMTRAP:func.3">pr2</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K3" title="GEOMTRAP:func.3">pr2</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">u</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K3" title="GEOMTRAP:func.3">pr2</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">v</font></span>)</span> ) )</span><br/>









<b>assume </b><a NAME="E1:51"/><i><font color="Green" title="E49">A1</font></i>: 
 <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K2" title="GEOMTRAP:func.2">pr1</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K2" title="GEOMTRAP:func.2">pr1</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">u</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K2" title="GEOMTRAP:func.2">pr1</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">v</font></span>)</span> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K3" title="GEOMTRAP:func.3">pr2</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K3" title="GEOMTRAP:func.3">pr2</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">u</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K3" title="GEOMTRAP:func.3">pr2</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">v</font></span>)</span> )</span><br/>

<b>set </b><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font>;<br/>
2 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</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="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c4">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">v</font>
<b>by </b><i><a class="ref" href="geomtrap.html#D2" target="_self" title="GEOMTRAP:def.2">Def2</a></i>
;<br/>
<a NAME="E3:51"/><b>then </b>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K2" title="GEOMTRAP:func.2">pr1</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="geomtrap.html#K2" title="GEOMTRAP:func.2">pr1</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">v</font></span>)</span> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K3" title="GEOMTRAP:func.3">pr2</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="geomtrap.html#K3" title="GEOMTRAP:func.3">pr2</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">v</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:51"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="geomtrap.html#E52"><i><font color="Green" title="E47">Lm16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:51"/>
( 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K2" title="GEOMTRAP:func.2">pr1</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K2" title="GEOMTRAP:func.2">pr1</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">u</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K2" title="GEOMTRAP:func.2">pr1</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">v</font></span>)</span> &amp; 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K3" title="GEOMTRAP:func.3">pr2</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K3" title="GEOMTRAP:func.3">pr2</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">u</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="geomtrap.html#K3" title="GEOMTRAP:func.3">pr2</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">v</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:51"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="geomtrap.html#E52"><i><font color="Green" title="E47">Lm16</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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