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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>  st  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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> )</span><br/>





<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E49">A1</font></i>: 
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:59_1"/><b>then </b>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font> st <br/>(  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="#E1:59"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E3:59_1"/>
contradiction
 <b>by </b><i><a class="ref" href="semi_af1.html#T65" target="_self" title="SEMI_AF1:th.65">Th65</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:59"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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