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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">u1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>













<b>assume </b><a NAME="E1:4"/><i><font color="Green">E45</font></i>: 
 <a href="analmetr.html#R1">Gen</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <span class="p2">(<span class="default"><a href="analoaf.html#K2">OASpace</a> <font color="Maroon">V</font></span>)</span></span>)</span>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <span class="p2">(<span class="default"><a href="analoaf.html#K2">OASpace</a> <font color="Maroon">V</font></span>)</span></span>)</span>, <font color="Maroon">p1</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <span class="p2">(<span class="default"><a href="analoaf.html#K2">OASpace</a> <font color="Maroon">V</font></span>)</span></span>)</span>, <font color="Maroon">q1</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="diraf.html#K2">Lambda</a> <span class="p2">(<span class="default"><a href="analoaf.html#K2">OASpace</a> <font color="Maroon">V</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E2:4"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">S</font> =  <a href="analoaf.html#K2">OASpace</a> <font color="Maroon">V</font> as   <a href="analoaf.html#NM1">OAffinSpace</a> <b>by </b><i>, </i>;<br/>
<a NAME="E4:4"/>
 <a href="diraf.html#K2">Lambda</a> <span class="p1">(<span class="default"><a href="analoaf.html#K2">OASpace</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="analoaf.html#G1">AffinStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2">OASpace</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="diraf.html#K1">lambda</a> the <a href="analoaf.html#U1">CONGR</a> of <span class="p2">(<span class="default"><a href="analoaf.html#K2">OASpace</a> <font color="Maroon">V</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="diraf.html#D2">DIRAF:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font>, <font color="Maroon">p1'</font> = <font color="Maroon">p1</font>, <font color="Maroon">q1'</font> = <font color="Maroon">q1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<div><i><font color="Green">E47</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">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font>
 ;<br/><a NAME="E2:4_1"/><b>then </b>
<font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p1'</font>,<font color="Maroon">q1'</font>
 <b>by </b><i><a class="ref" href="diraf.html#T45">DIRAF:45</a></i>;<br/><a NAME="E3:4_1"/><b>then </b>
( <font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p1'</font>,<font color="Maroon">q1'</font> or <font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q1'</font>,<font color="Maroon">p1'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/><a NAME="E4:4_1"/><b>then </b>
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> or <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v1</font>,<font color="Maroon">u1</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:4_1"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font>
 <b>by </b><i/>;<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">u</font>,<font color="Maroon">v</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font>
 ;<br/><a NAME="E2:4_2"/><b>then </b>
( <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> or <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v1</font>,<font color="Maroon">u1</font> )
 <b>by </b><i/>;<br/><a NAME="E3:4_2"/><b>then </b>
( <font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p1'</font>,<font color="Maroon">q1'</font> or <font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q1'</font>,<font color="Maroon">p1'</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:4_2"/><b>then </b>
<font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="diraf.html#R2">'||'</a> <font color="Maroon">p1'</font>,<font color="Maroon">q1'</font>
 <b>by </b><i><a class="ref" href="diraf.html#D4">DIRAF:def 4</a></i>;<br/><b>hence </b><a NAME="E5:4_2"/>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font>
 <b>by </b><i><a class="ref" href="diraf.html#T45">DIRAF:45</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:4"/>
( <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font> iff <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="geomtrap.html#R1" target="_self">'||'</a> <font color="Maroon">u1</font>,<font color="Maroon">v1</font> )
 <b>by </b><i/>;<br/>


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