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<b>consider </b><font color="Maroon" title="c1">V</font> being   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a><b> such that </b><br/><a NAME="E2:34_1_1"/><i><font color="Green" title="E27">A1</font></i>: 
 ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> st <br/>( ( for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> holds <br/>( <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) &amp; ( for <font color="Olive" title="b3">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  ex <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Olive" title="b3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="funcsdom.html#T37" title="FUNCSDOM:th.37">FUNCSDOM:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">OAS</font> =  <a href="analoaf.html#K2" title="ANALOAF:func.2">OASpace</a> <font color="Maroon" title="c1">V</font> as   <a href="analoaf.html#NM2" title="ANALOAF:NM.2">OAffinPlane</a> <b>by </b><i><a class="txt" href="papdesaf.html#E2:34_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="analoaf.html#T51" title="ANALOAF:th.51">ANALOAF:51</a></i>;<br/>
<b>take </b><font color="Maroon" title="c3">X</font> =  <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">OAS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">X</font> is <a href="analoaf.html#V1" title="ANALOAF:attr.1">strict</a> &amp; <font color="Maroon" title="c3">X</font> <a href="papdesaf.html#NR5" title="PAPDESAF:NR.5">satisfies_Fano</a><a href="papdesaf.html#NV1" title="PAPDESAF:NV.1">satisfies_Fano</a>  &amp; <font color="Maroon" title="c3">X</font> <a href="papdesaf.html#NR1" title="PAPDESAF:NR.1">satisfies_PAP'</a>  &amp; <font color="Maroon" title="c3">X</font> <a href="papdesaf.html#NR2" title="PAPDESAF:NR.2">satisfies_DES'</a>  &amp; <font color="Maroon" title="c3">X</font> <a href="papdesaf.html#NR3" title="PAPDESAF:NR.3">satisfies_TDES'</a>  &amp; <font color="Maroon" title="c3">X</font> <a href="papdesaf.html#NR4" title="PAPDESAF:NR.4">satisfies_des'</a>  )</span><br/>

<a NAME="E4:34_1_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="papdesaf.html#NR1" title="PAPDESAF:NR.1">satisfies_PAP'</a> 
 
<b>by </b><i><a class="ref" href="papdesaf.html#T19" target="_self" title="PAPDESAF:th.19">Th19</a></i>;<br/>
<a NAME="E5:34_1_1"/><b>then </b>
<font color="Maroon" title="c3">X</font> <a href="papdesaf.html#NR3" title="PAPDESAF:NR.3">satisfies_TDES'</a> 
 
<b>by </b><i><a class="ref" href="aff_2.html#T25" title="AFF_2:th.25">AFF_2:25</a>, <a class="ref" href="aff_2.html#T26" title="AFF_2:th.26">AFF_2:26</a></i>;<br/>
<b>hence </b><a NAME="E6:34_1_1"/>
( <font color="Maroon" title="c3">X</font> is <a href="analoaf.html#V1" title="ANALOAF:attr.1">strict</a> &amp; <font color="Maroon" title="c3">X</font> <a href="papdesaf.html#NR5" title="PAPDESAF:NR.5">satisfies_Fano</a><a href="papdesaf.html#NV1" title="PAPDESAF:NV.1">satisfies_Fano</a>  &amp; <font color="Maroon" title="c3">X</font> <a href="papdesaf.html#NR1" title="PAPDESAF:NR.1">satisfies_PAP'</a>  &amp; <font color="Maroon" title="c3">X</font> <a href="papdesaf.html#NR2" title="PAPDESAF:NR.2">satisfies_DES'</a>  &amp; <font color="Maroon" title="c3">X</font> <a href="papdesaf.html#NR3" title="PAPDESAF:NR.3">satisfies_TDES'</a>  &amp; <font color="Maroon" title="c3">X</font> <a href="papdesaf.html#NR4" title="PAPDESAF:NR.4">satisfies_des'</a>  )
 <b>by </b><i><a class="txt" href="papdesaf.html#E4:34_1_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="papdesaf.html#T24" target="_self" title="PAPDESAF:th.24">Th24</a>, <a class="ref" href="aff_2.html#T25" title="AFF_2:th.25">AFF_2:25</a>, <a class="ref" href="aff_2.html#T28" title="AFF_2:th.28">AFF_2:28</a></i>; <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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