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<b>let </b><font color="Maroon">SAS</font> be   <a href="semi_af1.html#NM1">Semi_Affine_Space</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SAS</font>;<br/>







<a NAME="E1:91"/><i><font color="Green">E29</font></i>: 
(  <a href="semi_af1.html#K3" target="_self">diff</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> implies <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_1"/>
 <a href="semi_af1.html#K3" target="_self">diff</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 ;<br/>

<a NAME="E2:91_1"/><b>then </b>
(  <a href="semi_af1.html#K1" target="_self">sum</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="semi_af1.html#K2" target="_self">opposite</a> <font color="Maroon">b</font>,<font color="Maroon">o</font></span>)</span>,<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp;  <a href="semi_af1.html#K1" target="_self">sum</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="semi_af1.html#K2" target="_self">opposite</a> <font color="Maroon">a</font>,<font color="Maroon">o</font></span>)</span>,<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> )
 
<b>by </b><i><a class="ref" href="semi_af1.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E3:91_1"/><b>then </b>
 <a href="semi_af1.html#K2" target="_self">opposite</a> <font color="Maroon">a</font>,<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="semi_af1.html#K2" target="_self">opposite</a> <font color="Maroon">b</font>,<font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="semi_af1.html#T44" target="_self">Th44</a></i>;<br/>
<b>hence </b><a NAME="E4:91_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T50" target="_self">Th50</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:91"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> implies  <a href="semi_af1.html#K3" target="_self">diff</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> )
 
<b>by </b><i><a class="ref" href="semi_af1.html#T41" target="_self">Th41</a></i>;<br/>
<b>hence </b><a NAME="E3:91"/>
(  <a href="semi_af1.html#K3" target="_self">diff</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> iff <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T54" target="_self">Th54</a></i>;<br/>


</div>
