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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>, <font color="Olive" title="b5">u1</font>, <font color="Olive" title="b6">v1</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/>for <font color="Olive" title="b7">p</font>, <font color="Olive" title="b8">q</font>, <font color="Olive" title="b9">p1</font>, <font color="Olive" title="b10">q1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <span class="p2">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span></span>)</span>  st <font color="Olive" title="b7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">u</font> &amp; <font color="Olive" title="b8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v</font> &amp; <font color="Olive" title="b9">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">u1</font> &amp; <font color="Olive" title="b10">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">v1</font> holds <br/>( <font color="Olive" title="b7">p</font>,<font color="Olive" title="b8">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b9">p1</font>,<font color="Olive" title="b10">q1</font> iff <font color="Olive" title="b3">u</font>,<font color="Olive" title="b4">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Olive" title="b5">u1</font>,<font color="Olive" title="b6">v1</font> )</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>, <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c7">v1</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 for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">q1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <span class="p2">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> &amp; <font color="Olive" title="b3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u1</font> &amp; <font color="Olive" title="b4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v1</font> holds <br/>( <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">q1</font> iff <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c7">v1</font> ) )</span><br/>













<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E5">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"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">q1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <span class="p2">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> &amp; <font color="Olive" title="b3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u1</font> &amp; <font color="Olive" title="b4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v1</font> holds <br/>( <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">q1</font> iff <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c7">v1</font> )</span><br/>

<b>let </b><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">q</font>, <font color="Maroon" title="c10">p1</font>, <font color="Maroon" title="c11">q1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <span class="p2">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> &amp; <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> &amp; <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u1</font> &amp; <font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v1</font> implies ( <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">q1</font> iff <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c7">v1</font> ) )</span><br/>

<b>assume </b><a NAME="E2:4"/><i><font color="Green" title="E6">A2</font></i>: 
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> &amp; <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> &amp; <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u1</font> &amp; <font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">q1</font> iff <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c7">v1</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c12">S</font> =  <a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font> as   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</a> <b>by </b><i><a class="txt" href="geomtrap.html#E1:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="geomtrap.html#T1" target="_self" title="GEOMTRAP:th.1">Th1</a></i>;<br/>
<a NAME="E4:4"/>
 <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="diraf.html#K1" title="DIRAF:func.1">lambda</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <span class="p2">(<span class="default"><a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="diraf.html#D2" title="DIRAF:def.2">DIRAF:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">p'</font> = <font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c14">q'</font> = <font color="Maroon" title="c9">q</font>, <font color="Maroon" title="c15">p1'</font> = <font color="Maroon" title="c10">p1</font>, <font color="Maroon" title="c16">q1'</font> = <font color="Maroon" title="c11">q1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">S</font> ;<br/>
<div><i><font color="Green" title="E7">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c7">v1</font></span><br/><a NAME="E2:4_1"/><b>then </b>
<font color="Maroon" title="c13">p'</font>,<font color="Maroon" title="c14">q'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c15">p1'</font>,<font color="Maroon" title="c16">q1'</font>
 <b>by </b><i><a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></i>;<br/><a NAME="E3:4_1"/><b>then </b>
( <font color="Maroon" title="c13">p'</font>,<font color="Maroon" title="c14">q'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">p1'</font>,<font color="Maroon" title="c16">q1'</font> or <font color="Maroon" title="c13">p'</font>,<font color="Maroon" title="c14">q'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">q1'</font>,<font color="Maroon" title="c15">p1'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/><a NAME="E4:4_1"/><b>then </b>
( <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c7">v1</font> or <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c6">u1</font> )
 <b>by </b><i><a class="txt" href="geomtrap.html#E2:4"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="geomtrap.html#T2" target="_self" title="GEOMTRAP:th.2">Th2</a></i>;<br/><b>hence </b><a NAME="E5:4_1"/>
<font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c7">v1</font>
 <b>by </b><i><a class="ref" href="geomtrap.html#D1" target="_self" title="GEOMTRAP:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_2"/>
<font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c7">v1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">q1</font></span><br/><a NAME="E2:4_2"/><b>then </b>
( <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c7">v1</font> or <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c6">u1</font> )
 <b>by </b><i><a class="ref" href="geomtrap.html#D1" target="_self" title="GEOMTRAP:def.1">Def1</a></i>;<br/><a NAME="E3:4_2"/><b>then </b>
( <font color="Maroon" title="c13">p'</font>,<font color="Maroon" title="c14">q'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">p1'</font>,<font color="Maroon" title="c16">q1'</font> or <font color="Maroon" title="c13">p'</font>,<font color="Maroon" title="c14">q'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">q1'</font>,<font color="Maroon" title="c15">p1'</font> )
 <b>by </b><i><a class="txt" href="geomtrap.html#E2:4"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="geomtrap.html#T2" target="_self" title="GEOMTRAP:th.2">Th2</a></i>;<br/><a NAME="E4:4_2"/><b>then </b>
<font color="Maroon" title="c13">p'</font>,<font color="Maroon" title="c14">q'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c15">p1'</font>,<font color="Maroon" title="c16">q1'</font>
 <b>by </b><i><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/><b>hence </b><a NAME="E5:4_2"/>
<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">q1</font>
 <b>by </b><i><a class="ref" href="diraf.html#T45" title="DIRAF:th.45">DIRAF:45</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:4"/>
( <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">p1</font>,<font color="Maroon" title="c11">q1</font> iff <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="geomtrap.html#R1" title="GEOMTRAP:pred.1">'||'</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c7">v1</font> )
 <b>by </b><i><a class="txt" href="geomtrap.html#E6:4"><i><font color="Green" title="E7">A3</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/>


</div>
