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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">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">u1</font>, <font color="Olive" title="b4">v1</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="b5">p</font>, <font color="Olive" title="b6">q</font>, <font color="Olive" title="b7">p1</font>, <font color="Olive" title="b8">q1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>  st <font color="Olive" title="b5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> &amp; <font color="Olive" title="b7">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">u1</font> &amp; <font color="Olive" title="b8">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v1</font> holds <br/>( <font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">p1</font>,<font color="Olive" title="b8">q1</font> iff <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Olive" title="b3">u1</font>,<font color="Olive" title="b4">v1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">u1</font>, <font color="Maroon" title="c5">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Olive" title="b3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u1</font> &amp; <font color="Olive" title="b4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">u1</font>,<font color="Maroon" title="c5">v1</font> )</span><br/>









<a NAME="E1:3"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
 <a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font> <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 <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="analoaf.html#K1" title="ANALOAF:func.1">DirPar</a> <font color="Maroon" title="c1">V</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#D4" title="ANALOAF:def.4">ANALOAF:def 4</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">q</font>, <font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c9">q1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>; <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="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> &amp; <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u1</font> &amp; <font color="Maroon" title="c9">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> implies ( <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">q1</font> iff <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">u1</font>,<font color="Maroon" title="c5">v1</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:3"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> &amp; <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u1</font> &amp; <font color="Maroon" title="c9">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</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="c6">p</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">q1</font> iff <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">u1</font>,<font color="Maroon" title="c5">v1</font> )</span><br/>

<div><span class="lab"><font color="Green" title="E6">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:3_1"/>
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">u1</font>,<font color="Maroon" title="c5">v1</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="c6">p</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">q1</font></span><br/><a NAME="E2:3_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">q1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</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="kw">by</span> <span class="lab"><a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="analoaf.html#T33" title="ANALOAF:th.33">ANALOAF:33</a></span>;<br/><span class="kw">hence </span><a NAME="E3:3_1"/>
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#D2" title="ANALOAF:def.2">ANALOAF:def 2</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/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:3_2"/>
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">q1</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="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">u1</font>,<font color="Maroon" title="c5">v1</font></span><br/><a NAME="E2:3_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">u1</font>,<font color="Maroon" title="c5">v1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="analoaf.html#K1" title="ANALOAF:func.1">DirPar</a> <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="analoaf.html#D2" title="ANALOAF:def.2">ANALOAF:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:3_2"/>
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">u1</font>,<font color="Maroon" title="c5">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#T33" title="ANALOAF:th.33">ANALOAF:33</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:3"/>
( <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">q1</font> iff <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">u1</font>,<font color="Maroon" title="c5">v1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E6">A3</font></span></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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