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

<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">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">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/>







<a NAME="E1:91"/><i><font color="Green" title="E74">A1</font></i>: 
(  <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</font> implies <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</font>
 ;<br/>

<a NAME="E2:91_1"/><b>then </b>
(  <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#K2" title="SEMI_AF1:func.2">opposite</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">o</font></span>)</span>,<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</font> &amp;  <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#K2" title="SEMI_AF1:func.2">opposite</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">o</font></span>)</span>,<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</font> )
 
<b>by </b><i><a class="ref" href="semi_af1.html#D6" target="_self" title="SEMI_AF1:def.6">Def6</a></i>;<br/>
<a NAME="E3:91_1"/><b>then </b>
 <a href="semi_af1.html#K2" title="SEMI_AF1:func.2">opposite</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="semi_af1.html#K2" title="SEMI_AF1:func.2">opposite</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">o</font>
 
<b>by </b><i><a class="ref" href="semi_af1.html#T104" target="_self" title="SEMI_AF1:th.104">Th104</a></i>;<br/>
<b>hence </b><a NAME="E4:91_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T107" target="_self" title="SEMI_AF1:th.107">Th107</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:91"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> implies  <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</font> )
 
<b>by </b><i><a class="ref" href="semi_af1.html#D6" target="_self" title="SEMI_AF1:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E3:91"/>
(  <a href="semi_af1.html#K3" title="SEMI_AF1:func.3">diff</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</font> iff <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:91"><i><font color="Green" title="E74">A1</font></i></a></i>;<br/>


</div>
