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<b>let </b><font color="Maroon" title="c1">SAS</font> be   <a href="semi_af1.html#NM1" title="SEMI_AF1:NM.1">Semi_Affine_Space</a>;<br/><b>let </b><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>;<br/>











<b>assume </b><a NAME="E1:90"/><i><font color="Green" title="E72">A1</font></i>: 
<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>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:90_1"/><i><font color="Green" title="E73">A2</font></i>: 
( <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> &amp; <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> )
 ;<br/><b>consider </b><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><b> such that </b><br/><a NAME="E3:90_1"/><i><font color="Green" title="E74">A3</font></i>: 
 <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>
 <b>by </b><i><a class="ref" href="semi_af1.html#T82" target="_self" title="SEMI_AF1:th.82">Th82</a></i>;<br/><a NAME="E4:90_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="c4">r</font>,<font color="Maroon" title="c7">t</font> &amp;  <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> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E3:90_1"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="semi_af1.html#D5" target="_self" title="SEMI_AF1:def.5">Def5</a></i>;<br/><a NAME="E5:90_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="ref" href="semi_af1.html#T85" target="_self" title="SEMI_AF1:th.85">Th85</a></i>;<br/><a NAME="E6:90_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="ref" href="semi_af1.html#T89" target="_self" title="SEMI_AF1:th.89">Th89</a></i>;<br/><a NAME="E7:90_1"/><b>then </b>
( <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:90_1"><i><font color="Green" title="E73">A2</font></i></a>, <a class="ref" href="semi_af1.html#T74" target="_self" title="SEMI_AF1:th.74">Th74</a>, <a class="ref" href="semi_af1.html#T76" target="_self" title="SEMI_AF1:th.76">Th76</a></i>;<br/><a NAME="E8:90_1"/><b>then </b><i><font color="Green" title="E75">A4</font></i>: 
( <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:90"><i><font color="Green" title="E72">A1</font></i></a>, <a class="txt" href="semi_af1.html#E1:90_1"><i><font color="Green" title="E73">A2</font></i></a>, <a class="ref" href="semi_af1.html#T20" target="_self" title="SEMI_AF1:th.20">Th20</a></i>;<br/><a NAME="E9:90_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>
 <b>by </b><i><a class="ref" href="semi_af1.html#D5" target="_self" title="SEMI_AF1:def.5">Def5</a></i>;<br/><a NAME="E10:90_1"/><b>then </b>
(  <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> &amp;  <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> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E3:90_1"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="semi_af1.html#T89" target="_self" title="SEMI_AF1:th.89">Th89</a></i>;<br/><a NAME="E11:90_1"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="semi_af1.html#T76" target="_self" title="SEMI_AF1:th.76">Th76</a>, <a class="ref" href="semi_af1.html#T90" target="_self" title="SEMI_AF1:th.90">Th90</a></i>;<br/><a NAME="E12:90_1"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="semi_af1.html#T17" target="_self" title="SEMI_AF1:th.17">Th17</a></i>;<br/><a NAME="E13:90_1"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:90_1"><i><font color="Green" title="E73">A2</font></i></a>, <a class="txt" href="semi_af1.html#E8:90_1"><i><font color="Green" title="E75">A4</font></i></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></i>;<br/><a NAME="E14:90_1"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="semi_af1.html#T18" target="_self" title="SEMI_AF1:th.18">Th18</a></i>;<br/><b>hence </b><a NAME="E15:90_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>
 <b>by </b><i><a class="txt" href="semi_af1.html#E8:90_1"><i><font color="Green" title="E75">A4</font></i></a>, <a class="ref" href="semi_af1.html#T20" target="_self" title="SEMI_AF1:th.20">Th20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><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>
 <b>by </b><i><a class="ref" href="semi_af1.html#T14" target="_self" title="SEMI_AF1:th.14">Th14</a>, <a class="ref" href="semi_af1.html#T110" target="_self" title="SEMI_AF1:th.110">Th110</a></i>;<br/>


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