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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> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="geomtrap.html#K7" title="GEOMTRAP:func.7">DTrSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font></span>)</span>  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/><span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b4">b</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b6">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b5">c</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b6">d</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">w</font>, <font color="Maroon" title="c3">y</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"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="geomtrap.html#K7" title="GEOMTRAP:func.7">DTrSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font></span>)</span>  st  <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> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b4">d</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="geomtrap.html#K7" title="GEOMTRAP:func.7">DTrSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font></span>)</span>; <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 <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">d</font></span>)</span> )</span><br/>













<b>assume </b><a NAME="E1:86"/>
 <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"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">d</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c8">u</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c9">u1</font> = <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c10">v</font> = <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c11">v1</font> = <font color="Maroon" title="c7">d</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<b>set </b><font color="Maroon" title="c12">ab</font> = <font color="Maroon" title="c4">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c5">b</font>;<br/>
<b>set </b><font color="Maroon" title="c13">cd</font> = <font color="Maroon" title="c6">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">d</font>;<br/>
<b>set </b><font color="Maroon" title="c14">ac</font> = <font color="Maroon" title="c4">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c6">c</font>;<br/>
<b>set </b><font color="Maroon" title="c15">bd</font> = <font color="Maroon" title="c5">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">d</font>;<br/>
<b>set </b><font color="Maroon" title="c16">uu1</font> = <font color="Maroon" title="c8">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c9">u1</font>;<br/>
<b>set </b><font color="Maroon" title="c17">vv1</font> = <font color="Maroon" title="c10">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c11">v1</font>;<br/>
<b>set </b><font color="Maroon" title="c18">uv</font> = <font color="Maroon" title="c8">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c10">v</font>;<br/>
<b>set </b><font color="Maroon" title="c19">u1v1</font> = <font color="Maroon" title="c9">u1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c11">v1</font>;<br/>
<a NAME="E3:86"/><i><font color="Green" title="E75">A1</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c9">u1</font> &amp; <font color="Maroon" title="c6">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c11">v1</font> &amp; <font color="Maroon" title="c4">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c10">v</font> &amp; <font color="Maroon" title="c5">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">u1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c11">v1</font> )
 
<b>by </b><i><a class="ref" href="geomtrap.html#D8" target="_self" title="GEOMTRAP:def.8">Def8</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">d</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c9">u1</font></span>)</span> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c11">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="geomtrap.html#D8" target="_self" title="GEOMTRAP:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">u</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c11">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="geomtrap.html#T8" target="_self" title="GEOMTRAP:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">d</font></span>)</span>
<b>by </b><i><a class="txt" href="geomtrap.html#E3:86"><i><font color="Green" title="E75">A1</font></i></a>, <a class="ref" href="geomtrap.html#D8" target="_self" title="GEOMTRAP:def.8">Def8</a></i>
;<br/> <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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