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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="firebrick">::  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="firebrick">::  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="E1:71"/><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="firebrick">::  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="71_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:71_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="firebrick">::  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="E2:71_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#T64" target="_self" title="SEMI_AF1:th.64">Th84</a></span>;<br/><a NAME="E3:71_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="#E1:71"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E1:71_1"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="ref" href="semi_af1.html#T62" target="_self" title="SEMI_AF1:th.62">Th81</a></span>;<br/><span class="kw">hence </span><a NAME="E4:71_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="#E1:71_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="firebrick">::  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="71_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71_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="E2:71_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="E3:71_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="firebrick">::  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="E4:71_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="#E3:71_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="E6:71_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="E7:71_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="#E1:71"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E1:71_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="E8:71_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#T1" target="_self" title="SEMI_AF1:th.1">Th12</a></span>;<br/><a NAME="E9:71_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="#E6:71_2"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="ref" href="semi_af1.html#T36" target="_self" title="SEMI_AF1:th.36">Th54</a>, <a class="ref" href="semi_af1.html#T38" target="_self" title="SEMI_AF1:th.38">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="E11:71_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="#E2:71_2"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E4:71_2"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E8:71_2"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="ref" href="semi_af1.html#T23" target="_self" title="SEMI_AF1:th.23">Th39</a>, <a class="ref" href="semi_af1.html#T44" target="_self" title="SEMI_AF1:th.44">Th62</a></span>;<br/><a NAME="E12:71_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="#E11:71_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="E13:71_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="#E7:71_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="E14:71_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="#E6:71_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#T36" target="_self" title="SEMI_AF1:th.36">Th54</a></span>;<br/><a NAME="E15:71_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="#E13:71_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="E16:71_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="#E1:71_2"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E3:71_2"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="semi_af1.html#T32" target="_self" title="SEMI_AF1:th.32">Th49</a></span>;<br/><a NAME="E17:71_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="#E2:71_2"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E12:71_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="E18:71_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="#E11:71_2"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="semi_af1.html#T43" target="_self" title="SEMI_AF1:th.43">Th61</a></span>;<br/><a NAME="E19:71_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="E20:71_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="#E6:71_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#T36" target="_self" title="SEMI_AF1:th.36">Th54</a></span>;<br/><a NAME="E21:71_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="#E19:71_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="E22:71_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="#E7:71_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="E23:71_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#T6" target="_self" title="SEMI_AF1:th.6">Th17</a></span>;<br/><a NAME="E24:71_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="#E6:71_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="E25:71_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#T6" target="_self" title="SEMI_AF1:th.6">Th17</a></span>;<br/><a NAME="E26:71_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="#E11:71_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="E27:71_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="#E3:71_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="E28:71_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="#E1:71_2"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E25:71_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="E29:71_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="#E2:71_2"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E26:71_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="E30:71_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#T7" target="_self" title="SEMI_AF1:th.7">Th18</a></span>;<br/><a NAME="E31:71_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="#E3:71_2"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="semi_af1.html#T22" target="_self" title="SEMI_AF1:th.22">Th37</a></span>;<br/><a NAME="E32:71_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="#E11:71_2"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="semi_af1.html#T39" target="_self" title="SEMI_AF1:th.39">Th57</a></span>;<br/><a NAME="E33:71_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="#E11:71_2"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="semi_af1.html#T43" target="_self" title="SEMI_AF1:th.43">Th61</a></span>;<br/><a NAME="E34:71_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="#E1:71_2"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E3:71_2"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="txt" href="#E15:71_2"><span class="lab"><font color="Green" title="E74">A15</font></span></a>, <a class="ref" href="semi_af1.html#T33" target="_self" title="SEMI_AF1:th.33">Th50</a></span>;<br/><a NAME="E35:71_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#T7" target="_self" title="SEMI_AF1:th.7">Th18</a></span>;<br/><a NAME="E36:71_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="#E30:71_2"><span class="lab"><font color="Green" title="E81">A22</font></span></a>, <a class="txt" href="#E21:71_2"><span class="lab"><font color="Green" title="E77">A18</font></span></a>, <a class="txt" href="#E23:71_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="E37:71_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="#E32:71_2"><span class="lab"><font color="Green" title="E82">A23</font></span></a>, <a class="txt" href="#E17:71_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="E38:71_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="#E33:71_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="firebrick">::  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="71_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71_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="E2:71_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="E3:71_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="firebrick">::  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="E4:71_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="#E1:71"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E3:71_3"><span class="lab"><font color="Green" title="E65">A26</font></span></a>, <a class="ref" href="semi_af1.html#T59" target="_self" title="SEMI_AF1:th.59">Th78</a></span>;<br/><a NAME="E5:71_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#T43" target="_self" title="SEMI_AF1:th.43">Th61</a></span>;<br/><span class="kw">hence </span><a NAME="E6:71_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#T60" target="_self" title="SEMI_AF1:th.60">Th79</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="71_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:71_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="firebrick">::  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="E2:71_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="#E1:71"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="semi_af1.html#T54" target="_self" title="SEMI_AF1:th.54">Th73</a></span>;<br/><span class="kw">hence </span><a NAME="E3:71_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="#E1:71_4"><span class="lab"><font color="Green" title="E65">A27</font></span></a>, <a class="ref" href="semi_af1.html#T64" target="_self" title="SEMI_AF1:th.64">Th84</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:71"/>
 <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="#E2:71"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E4:71"><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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
