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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">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">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> holds <br/> <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">d</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c4">c</font>,<font color="Maroon" title="c5">d</font> implies  <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:91"/><span class="lab"><font color="Green" title="E61">A1</font></span>: 
 <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="c4">c</font>,<font color="Maroon" title="c5">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font></span><br/>

<div><span class="lab"><font color="Green" title="E62">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:91_1"/><span class="lab"><font color="Green" title="E63">A3</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font></span><br/><a NAME="E2:91_1"/>
 <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="c2">a</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T84" target="_self" title="SEMI_AF1:th.84">Th84</a></span>;<br/><a NAME="E3:91_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E1:91_1"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="ref" href="semi_af1.html#T81" target="_self" title="SEMI_AF1:th.81">Th81</a></span>;<br/><span class="kw">hence </span><a NAME="E4:91_1"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E63">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:91_2"/><span class="lab"><font color="Green" title="E64">A5</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:91_2"/><span class="lab"><font color="Green" title="E65">A6</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">and </span><br/><a NAME="E3:91_2"/><span class="lab"><font color="Green" title="E66">A7</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font></span><br/><a NAME="E4:91_2"/><span class="lab"><font color="Green" title="E67">A8</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_2"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font><span class="kw"> such that </span><br/><a NAME="E6:91_2"/><span class="lab"><font color="Green" title="E68">A9</font></span>: 
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">and </span><br/><a NAME="E7:91_2"/><span class="lab"><font color="Green" title="E69">A10</font></span>: 
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E1:91_2"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</a></span>;<br/><a NAME="E8:91_2"/><span class="lab"><font color="Green" title="E70">A11</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T12" target="_self" title="SEMI_AF1:th.12">Th12</a></span>;<br/><a NAME="E9:91_2"/>
( not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">p</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:91_2"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="ref" href="semi_af1.html#T54" target="_self" title="SEMI_AF1:th.54">Th54</a>, <a class="ref" href="semi_af1.html#T56" target="_self" title="SEMI_AF1:th.56">Th56</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font><span class="kw"> such that </span><br/><a NAME="E11:91_2"/><span class="lab"><font color="Green" title="E71">A12</font></span>: 
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c8">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_2"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E4:91_2"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E8:91_2"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="ref" href="semi_af1.html#T39" target="_self" title="SEMI_AF1:th.39">Th39</a>, <a class="ref" href="semi_af1.html#T62" target="_self" title="SEMI_AF1:th.62">Th62</a></span>;<br/><a NAME="E12:91_2"/><span class="lab"><font color="Green" title="E72">A13</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c8">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:91_2"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="semi_af1.html#D3" target="_self" title="SEMI_AF1:def.3">Def3</a></span>;<br/><a NAME="E13:91_2"/><span class="lab"><font color="Green" title="E73">A14</font></span>: 
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:91_2"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="ref" href="semi_af1.html#D3" target="_self" title="SEMI_AF1:def.3">Def3</a></span>;<br/><a NAME="E14:91_2"/>
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:91_2"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="ref" href="semi_af1.html#D3" target="_self" title="SEMI_AF1:def.3">Def3</a>, <a class="ref" href="semi_af1.html#T54" target="_self" title="SEMI_AF1:th.54">Th54</a></span>;<br/><a NAME="E15:91_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A15</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:91_2"><span class="lab"><font color="Green" title="E73">A14</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></span>;<br/><a NAME="E16:91_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_2"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E3:91_2"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="semi_af1.html#T49" target="_self" title="SEMI_AF1:th.49">Th49</a></span>;<br/><a NAME="E17:91_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A16</font></span>: 
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c8">r</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_2"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E12:91_2"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></span>;<br/><a NAME="E18:91_2"/>
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:91_2"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="semi_af1.html#T61" target="_self" title="SEMI_AF1:th.61">Th61</a></span>;<br/><a NAME="E19:91_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A17</font></span>: 
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#D3" target="_self" title="SEMI_AF1:def.3">Def3</a></span>;<br/><a NAME="E20:91_2"/>
( <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:91_2"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="ref" href="semi_af1.html#D3" target="_self" title="SEMI_AF1:def.3">Def3</a>, <a class="ref" href="semi_af1.html#T54" target="_self" title="SEMI_AF1:th.54">Th54</a></span>;<br/><a NAME="E21:91_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A18</font></span>: 
<font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:91_2"><span class="lab"><font color="Green" title="E76">A17</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></span>;<br/><a NAME="E22:91_2"/>
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:91_2"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="ref" href="semi_af1.html#D3" target="_self" title="SEMI_AF1:def.3">Def3</a></span>;<br/><a NAME="E23:91_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A19</font></span>: 
<font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c5">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T17" target="_self" title="SEMI_AF1:th.17">Th17</a></span>;<br/><a NAME="E24:91_2"/>
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:91_2"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="ref" href="semi_af1.html#D3" target="_self" title="SEMI_AF1:def.3">Def3</a></span>;<br/><a NAME="E25:91_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A20</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T17" target="_self" title="SEMI_AF1:th.17">Th17</a></span>;<br/><a NAME="E26:91_2"/><span class="lab"><font color="Green" title="E80">A21</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c8">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:91_2"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="semi_af1.html#D3" target="_self" title="SEMI_AF1:def.3">Def3</a></span>;<br/><a NAME="E27:91_2"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_2"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a></span>;<br/><a NAME="E28:91_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_2"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E25:91_2"><span class="lab"><font color="Green" title="E79">A20</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></span>;<br/><a NAME="E29:91_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c8">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_2"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E26:91_2"><span class="lab"><font color="Green" title="E80">A21</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></span>;<br/><a NAME="E30:91_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A22</font></span>: 
<font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T18" target="_self" title="SEMI_AF1:th.18">Th18</a></span>;<br/><a NAME="E31:91_2"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_2"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="semi_af1.html#T37" target="_self" title="SEMI_AF1:th.37">Th37</a></span>;<br/><a NAME="E32:91_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A23</font></span>: 
not <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c3">b</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:91_2"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="semi_af1.html#T57" target="_self" title="SEMI_AF1:th.57">Th57</a></span>;<br/><a NAME="E33:91_2"/><span class="lab"><font color="Green" title="E83">A24</font></span>: 
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:91_2"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="semi_af1.html#T61" target="_self" title="SEMI_AF1:th.61">Th61</a></span>;<br/><a NAME="E34:91_2"/>
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_2"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E3:91_2"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="txt" href="#E15:91_2"><span class="lab"><font color="Green" title="E74">A15</font></span></a>, <a class="ref" href="semi_af1.html#T50" target="_self" title="SEMI_AF1:th.50">Th50</a></span>;<br/><a NAME="E35:91_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T18" target="_self" title="SEMI_AF1:th.18">Th18</a></span>;<br/><a NAME="E36:91_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:91_2"><span class="lab"><font color="Green" title="E81">A22</font></span></a>, <a class="txt" href="#E21:91_2"><span class="lab"><font color="Green" title="E77">A18</font></span></a>, <a class="txt" href="#E23:91_2"><span class="lab"><font color="Green" title="E78">A19</font></span></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a></span>;<br/><a NAME="E37:91_2"/><span class="kw">then </span>
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:91_2"><span class="lab"><font color="Green" title="E82">A23</font></span></a>, <a class="txt" href="#E17:91_2"><span class="lab"><font color="Green" title="E75">A16</font></span></a>, <a class="ref" href="semi_af1.html#D3" target="_self" title="SEMI_AF1:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E38:91_2"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:91_2"><span class="lab"><font color="Green" title="E83">A24</font></span></a>, <a class="ref" href="semi_af1.html#D4" target="_self" title="SEMI_AF1:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E64">A25</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:91_3"/>
<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>
 <span class="kw">and </span><br/><a NAME="E2:91_3"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">and </span><br/><a NAME="E3:91_3"/><span class="lab"><font color="Green" title="E65">A26</font></span>: 
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font></span><br/><a NAME="E4:91_3"/>
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E3:91_3"><span class="lab"><font color="Green" title="E65">A26</font></span></a>, <a class="ref" href="semi_af1.html#T78" target="_self" title="SEMI_AF1:th.78">Th78</a></span>;<br/><a NAME="E5:91_3"/><span class="kw">then </span>
 <a href="semi_af1.html#R2" title="SEMI_AF1:pred.2">parallelogram</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T61" target="_self" title="SEMI_AF1:th.61">Th61</a></span>;<br/><span class="kw">hence </span><a NAME="E6:91_3"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="semi_af1.html#T79" target="_self" title="SEMI_AF1:th.79">Th79</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:91_4"/><span class="lab"><font color="Green" title="E65">A27</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font></span><br/><a NAME="E2:91_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="semi_af1.html#T73" target="_self" title="SEMI_AF1:th.73">Th73</a></span>;<br/><span class="kw">hence </span><a NAME="E3:91_4"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_4"><span class="lab"><font color="Green" title="E65">A27</font></span></a>, <a class="ref" href="semi_af1.html#T84" target="_self" title="SEMI_AF1:th.84">Th84</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:91"/>
 <a href="semi_af1.html#R3" title="SEMI_AF1:pred.3">congr</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="txt" href="#E3:91"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E4:91"><span class="lab"><font color="Green" title="E64">A25</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
