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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">r</font>, <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">d</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">r</font>,<font color="Olive" title="b2">s</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> &amp;  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font>,<font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">d</font> holds <br/> <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">d</font></span><br/><b>let </b><font color="Maroon" title="c2">r</font>, <font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">d</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">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp;  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> implies  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> )</span><br/>













<b>assume </b><a NAME="E1:68"/><i><font color="Green" title="E58">A1</font></i>: 
(  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp;  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</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="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font></span><br/>

<div><i><font color="Green" title="E59">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_1"/>
<font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</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="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font></span><br/><a NAME="E2:68_1"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">d</font> )
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="semi_af1.html#T73" target="_self" title="SEMI_AF1:th.73">Th73</a></i>;<br/><b>hence </b><a NAME="E3:68_1"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</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>
<div><i><font color="Green" title="E60">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_2"/><i><font color="Green" title="E61">A4</font></i>: 
( <font color="Maroon" title="c2">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">a</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  )
 ; <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="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font></span><br/><b>then consider </b><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">q</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="E3:68_2"/><i><font color="Green" title="E62">A5</font></i>: 
(  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> )
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</a></i>;<br/><a NAME="E4:68_2"/>
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="#E1:68_2"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="#E3:68_2"><i><font color="Green" title="E62">A5</font></i></a>, <a class="ref" href="semi_af1.html#T80" target="_self" title="SEMI_AF1:th.80">Th80</a></i>;<br/><b>hence </b><a NAME="E5:68_2"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <b>by </b><i><a class="txt" href="#E3:68_2"><i><font color="Green" title="E62">A5</font></i></a>, <a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</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>
<div><i><font color="Green" title="E61">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_3"/><i><font color="Green" title="E62">A7</font></i>: 
( <font color="Maroon" title="c2">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c6">c</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  )
 ; <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="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font></span><br/><b>then consider </b><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">q</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="E3:68_3"/><i><font color="Green" title="E63">A8</font></i>: 
(  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</a></i>;<br/><a NAME="E4:68_3"/>
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="#E1:68_3"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="#E3:68_3"><i><font color="Green" title="E63">A8</font></i></a>, <a class="ref" href="semi_af1.html#T80" target="_self" title="SEMI_AF1:th.80">Th80</a></i>;<br/><b>hence </b><a NAME="E5:68_3"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <b>by </b><i><a class="txt" href="#E3:68_3"><i><font color="Green" title="E63">A8</font></i></a>, <a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_4"/>
( <font color="Maroon" title="c2">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">s</font> &amp; not <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">a</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; not <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c6">c</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  )
 ; <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="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font></span><br/><a NAME="E2:68_4"/><b>then </b>
(  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp;  <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> )
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="semi_af1.html#T78" target="_self" title="SEMI_AF1:th.78">Th78</a></i>;<br/><b>hence </b><a NAME="E3:68_4"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</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="E6:68"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 <b>by </b><i><a class="txt" href="#E2:68"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="#E3:68"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="#E4:68"><i><font color="Green" title="E61">A6</font></i></a></i>; <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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