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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="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>









<span class="kw">assume </span><a NAME="E1:51"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
 <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">set </span><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>
<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/>.= 
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#D2" target="_self" title="GEOMTRAP:def.2">Def2</a></span>
;<br/>
<a NAME="E3:51"/><span class="kw">then </span>
( 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>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="geomtrap.html#E50"><span class="lab"><font color="Green" title="E47">Lm17</font></span></a></span>;<br/>
<span class="kw">hence </span><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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="geomtrap.html#E50"><span class="lab"><font color="Green" title="E47">Lm17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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