<?xml version="1.0"?>
<div class="add">

<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">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">o</font>, <font color="Olive" title="b4">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font> holds   <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<span class="p1">(<span class="default"><a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">o</font>)</span>)</span>,<font color="Olive" title="b4">c</font>,<font color="Olive" title="b3">o</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">c</font>,<font color="Olive" title="b3">o</font>)</span>)</span>,<font color="Olive" title="b3">o</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">o</font>, <font color="Maroon" title="c5">c</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">  <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<span class="p1">(<span class="default"><a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">o</font>)</span>)</span>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">o</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">o</font>)</span>)</span>,<font color="Maroon" title="c4">o</font>)</span><br/>









<a NAME="E1:81"/>
(  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">o</font>) &amp;  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">a</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">o</font>), <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">o</font>)</span>)</span>,<font color="Maroon" title="c4">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="E2:81"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A1</font></span>: 
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c3">b</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">o</font>), <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">o</font>), <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">o</font>)</span>)</span>,<font color="Maroon" title="c4">o</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T65" target="_self" title="SEMI_AF1:th.65">Th85</a></span>;<br/>
<a NAME="E3:81"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">c</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">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="E4:81"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A2</font></span>: 
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">o</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">o</font>),<font color="Maroon" title="c5">c</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="E5:81"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">o</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">o</font>),<font color="Maroon" title="c5">c</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<span class="p1">(<span class="default"><a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">o</font>)</span>)</span>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">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="E6:81"/><span class="kw">then </span>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c3">b</font>, <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">o</font>), <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">o</font>), <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<span class="p1">(<span class="default"><a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">o</font>)</span>)</span>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">o</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="semi_af1.html#T70" target="_self" title="SEMI_AF1:th.70">Th90</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:81"/>
 <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<span class="p1">(<span class="default"><a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">o</font>)</span>)</span>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">o</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="semi_af1.html#K1" title="SEMI_AF1:func.1">sum</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">o</font>)</span>)</span>,<font color="Maroon" title="c4">o</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="semi_af1.html#T62" target="_self" title="SEMI_AF1:th.62">Th81</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>
