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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">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>, <font color="Olive" title="b7">a1</font>, <font color="Olive" title="b8">p</font>, <font color="Olive" title="b9">b1</font>, <font color="Olive" title="b10">c1</font>, <font color="Olive" title="b11">d1</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 <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">d</font> &amp; <font color="Olive" title="b3">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b7">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">p</font> &amp; <font color="Olive" title="b4">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b9">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">p</font> &amp; <font color="Olive" title="b5">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b10">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">p</font> &amp; <font color="Olive" title="b6">d</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b11">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">p</font> holds <br/><font color="Olive" title="b7">a1</font>,<font color="Olive" title="b9">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b10">c1</font>,<font color="Olive" title="b11">d1</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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>, <font color="Olive" title="b5">a1</font>, <font color="Olive" title="b6">p</font>, <font color="Olive" title="b7">b1</font>, <font color="Olive" title="b8">c1</font>, <font color="Olive" title="b9">d1</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 <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> &amp; <font color="Olive" title="b1">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">p</font> &amp; <font color="Olive" title="b2">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b7">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">p</font> &amp; <font color="Olive" title="b3">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b8">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">p</font> &amp; <font color="Olive" title="b4">d</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b9">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">p</font> holds <br/><font color="Olive" title="b5">a1</font>,<font color="Olive" title="b7">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">c1</font>,<font color="Olive" title="b9">d1</font></span><br/><span class="kw">let </span><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>, <font color="Maroon" title="c8">a1</font>, <font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">b1</font>, <font color="Maroon" title="c11">c1</font>, <font color="Maroon" title="c12">d1</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> &amp; <font color="Maroon" title="c4">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c5">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c6">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c7">d</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> implies <font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> )</span><br/>























<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:94"/><span class="lab"><font color="Green" title="E80">A1</font></span>: 
( <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> &amp; <font color="Maroon" title="c4">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )
 <span class="kw">and </span><br/><a NAME="E2:94"/><span class="lab"><font color="Green" title="E81">A2</font></span>: 
( <font color="Maroon" title="c5">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c6">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )
 <span class="kw">and </span><br/><a NAME="E3:94"/><span class="lab"><font color="Green" title="E82">A3</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</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="c8">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">u1</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c14">u2</font> = <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c15">v1</font> = <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c16">v2</font> = <font color="Maroon" title="c7">d</font>, <font color="Maroon" title="c17">u</font> = <font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c18">t1</font> = <font color="Maroon" title="c8">a1</font>, <font color="Maroon" title="c19">t2</font> = <font color="Maroon" title="c10">b1</font>, <font color="Maroon" title="c20">w1</font> = <font color="Maroon" title="c11">c1</font>, <font color="Maroon" title="c21">w2</font> = <font color="Maroon" title="c12">d1</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E5:94"/><span class="lab"><font color="Green" title="E83">A4</font></span>: 
( <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u2</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c19">t2</font> &amp; <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c20">w1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:94"><span class="lab"><font color="Green" title="E81">A2</font></span></a>, <a class="ref" href="geomtrap.html#D8" target="_self" title="GEOMTRAP:def.8">Def8</a></span>;<br/>
<a NAME="E6:94"/><span class="lab"><font color="Green" title="E84">A5</font></span>: 
<font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">v2</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c21">w2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:94"><span class="lab"><font color="Green" title="E82">A3</font></span></a>, <a class="ref" href="geomtrap.html#D8" target="_self" title="GEOMTRAP:def.8">Def8</a></span>;<br/>
<a NAME="E7:94"/>
( <font color="Maroon" title="c13">u1</font>,<font color="Maroon" title="c14">u2</font>,<font color="Maroon" title="c15">v1</font>,<font color="Maroon" title="c16">v2</font> <a href="geomtrap.html#R2" title="GEOMTRAP:pred.2">are_DTr_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">u1</font> <a href="geomtrap.html#K1" title="GEOMTRAP:func.1">#</a> <font color="Maroon" title="c18">t1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94"><span class="lab"><font color="Green" title="E80">A1</font></span></a>, <a class="ref" href="geomtrap.html#D8" target="_self" title="GEOMTRAP:def.8">Def8</a>, <a class="ref" href="geomtrap.html#T47" target="_self" title="GEOMTRAP:th.47">Th47</a></span>;<br/>
<a NAME="E8:94"/><span class="kw">then </span>
<font color="Maroon" title="c18">t1</font>,<font color="Maroon" title="c19">t2</font>,<font color="Maroon" title="c20">w1</font>,<font color="Maroon" title="c21">w2</font> <a href="geomtrap.html#R2" title="GEOMTRAP:pred.2">are_DTr_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:94"><span class="lab"><font color="Green" title="E83">A4</font></span></a>, <a class="txt" href="#E6:94"><span class="lab"><font color="Green" title="E84">A5</font></span></a>, <a class="ref" href="geomtrap.html#T30" target="_self" title="GEOMTRAP:th.30">Th30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:94"/>
<font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="geomtrap.html#T47" target="_self" title="GEOMTRAP:th.47">Th47</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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