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<span class="kw">let </span><font color="Maroon" title="c1">SAS</font> be   <a href="semi_af1.html#NM1" title="SEMI_AF1:NM.1">Semi_Affine_Space</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">r</font>, <font color="Olive" title="b4">s</font>, <font color="Olive" title="b5">o</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>  st <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">r</font>,<font color="Olive" title="b4">s</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>  <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">r</font>,<font color="Olive" title="b5">o</font>), <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Olive" title="b2">q</font>,<font color="Olive" title="b4">s</font>,<font color="Olive" title="b5">o</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</font>, <font color="Maroon" title="c5">s</font>, <font color="Maroon" title="c6">o</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font> implies <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a>  <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c6">o</font>), <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">o</font>) )</span><br/>











<span class="kw">assume </span><a NAME="E1:90"/><span class="lab"><font color="Green" title="E72">A1</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a>  <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c6">o</font>), <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">o</font>)</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c7">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font><span class="kw"> such that </span><br/><a NAME="E2:90_1"/><span class="lab"><font color="Green" title="E73">A2</font></span>: 
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c7">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T63" target="_self" title="SEMI_AF1:th.63">Th82</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:90_1"/><span class="lab"><font color="Green" title="E74">A3</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">and </span><br/><a NAME="E4:90_1"/><span class="lab"><font color="Green" title="E75">A4</font></span>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a>  <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c6">o</font>), <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">o</font>)</span><br/><a NAME="E5:90_1"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">o</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#D5" target="_self" title="SEMI_AF1:def.5">Def5</a></span>;<br/><a NAME="E6:90_1"/><span class="kw">then </span>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c5">s</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">o</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:90_1"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="ref" href="semi_af1.html#T65" target="_self" title="SEMI_AF1:th.65">Th85</a></span>;<br/><a NAME="E7:90_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A5</font></span>: 
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c7">t</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">o</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T69" target="_self" title="SEMI_AF1:th.69">Th89</a></span>;<br/><a NAME="E8:90_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A6</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">o</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_1"><span class="lab"><font color="Green" title="E75">A4</font></span></a>, <a class="ref" href="semi_af1.html#T55" target="_self" title="SEMI_AF1:th.55">Th74</a></span>;<br/><a NAME="E9:90_1"/>
<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">t</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">o</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90_1"><span class="lab"><font color="Green" title="E76">A5</font></span></a>, <a class="ref" href="semi_af1.html#T57" target="_self" title="SEMI_AF1:th.57">Th76</a></span>;<br/><a NAME="E10:90_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A7</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">t</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">o</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="txt" href="#E4:90_1"><span class="lab"><font color="Green" title="E75">A4</font></span></a>, <a class="ref" href="semi_af1.html#T8" target="_self" title="SEMI_AF1:th.8">Th20</a></span>;<br/><a NAME="E11:90_1"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c6">o</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#D5" target="_self" title="SEMI_AF1:def.5">Def5</a></span>;<br/><a NAME="E12:90_1"/><span class="kw">then </span>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">o</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c6">o</font>),<font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T69" target="_self" title="SEMI_AF1:th.69">Th89</a></span>;<br/><a NAME="E13:90_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a>  <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c6">o</font>),<font color="Maroon" title="c7">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:90_1"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="ref" href="semi_af1.html#T57" target="_self" title="SEMI_AF1:th.57">Th76</a>, <a class="ref" href="semi_af1.html#T70" target="_self" title="SEMI_AF1:th.70">Th90</a></span>;<br/><a NAME="E14:90_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">t</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c6">o</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T6" target="_self" title="SEMI_AF1:th.6">Th17</a></span>;<br/><a NAME="E15:90_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">o</font>) <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">t</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c6">o</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1"><span class="lab"><font color="Green" title="E74">A3</font></span></a>, <a class="txt" href="#E10:90_1"><span class="lab"><font color="Green" title="E78">A7</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></span>;<br/><a NAME="E16:90_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">o</font>) <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a>  <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c6">o</font>), <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">o</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T7" target="_self" title="SEMI_AF1:th.7">Th18</a></span>;<br/><span class="kw">hence </span><a NAME="E17:90_1"/>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a>  <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c6">o</font>), <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">o</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:90_1"><span class="lab"><font color="Green" title="E77">A6</font></span></a>, <a class="txt" href="#E10:90_1"><span class="lab"><font color="Green" title="E78">A7</font></span></a>, <a class="ref" href="semi_af1.html#T8" target="_self" title="SEMI_AF1:th.8">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:90"/>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a>  <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c6">o</font>), <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">o</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T3" target="_self" title="SEMI_AF1:th.3">Th14</a>, <a class="ref" href="semi_af1.html#T85" target="_self" title="SEMI_AF1:th.85">Th110</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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