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<span class="kw">let </span><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</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">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><br/> for <font color="Olive" title="b2">X</font>, <font color="Olive" title="b3">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b2">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Olive" title="b3">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Olive" title="b2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b3">Y</font> holds <br/><font color="Olive" title="b1">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Olive" title="b3">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</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">X</font>, <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Olive" title="b2">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Olive" title="b2">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</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="c3">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Maroon" title="c4">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Maroon" title="c3">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">Y</font> implies <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c4">Y</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:86"/><span class="lab"><font color="Green" title="E68">A1</font></span>: 
<font color="Maroon" title="c3">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> 
 <span class="kw">and </span><br/><a NAME="E2:86"/><span class="lab"><font color="Green" title="E69">A2</font></span>: 
<font color="Maroon" title="c4">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> 
 <span class="kw">and </span><br/><a NAME="E3:86"/><span class="lab"><font color="Green" title="E70">A3</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">Y</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">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c4">Y</font></span><br/>

<a NAME="E4:86"/><span class="lab"><font color="Green" title="E71">A4</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c3">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="aff_4.html#D6" target="_self" title="AFF_4:def.6">Def6</a></span>;<br/>
<a NAME="E5:86"/><span class="lab"><font color="Green" title="E72">A5</font></span>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c4">Y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="aff_4.html#D6" target="_self" title="AFF_4:def.6">Def6</a></span>;<br/>
<a NAME="E6:86"/><span class="lab"><font color="Green" title="E73">A6</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c4">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="aff_4.html#D6" target="_self" title="AFF_4:def.6">Def6</a></span>;<br/>
<a NAME="E7:86"/>
<font color="Maroon" title="c3">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="aff_4.html#D6" target="_self" title="AFF_4:def.6">Def6</a></span>;<br/>
<a NAME="E8:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A7</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c3">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="txt" href="#E3:86"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="txt" href="#E4:86"><span class="lab"><font color="Green" title="E71">A4</font></span></a>, <a class="ref" href="aff_4.html#T61" target="_self" title="AFF_4:th.61">Th61</a></span>;<br/>
<a NAME="E9:86"/>
<font color="Maroon" title="c4">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c4">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="aff_4.html#D6" target="_self" title="AFF_4:def.6">Def6</a></span>;<br/>
<a NAME="E10:86"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c3">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c4">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="txt" href="#E4:86"><span class="lab"><font color="Green" title="E71">A4</font></span></a>, <a class="txt" href="#E6:86"><span class="lab"><font color="Green" title="E73">A6</font></span></a>, <a class="txt" href="#E8:86"><span class="lab"><font color="Green" title="E74">A7</font></span></a>, <a class="ref" href="aff_4.html#T61" target="_self" title="AFF_4:th.61">Th61</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:86"/>
<font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:86"><span class="lab"><font color="Green" title="E72">A5</font></span></a>, <a class="txt" href="#E4:86"><span class="lab"><font color="Green" title="E71">A4</font></span></a>, <a class="txt" href="#E6:86"><span class="lab"><font color="Green" title="E73">A6</font></span></a>, <a class="txt" href="aff_4.html#E79"><span class="lab"><font color="Green" title="E64">Lm13</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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