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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V3" title="CONMETR1:attr.3">satisfying_Scherungssatz</a>  iff ( <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V1" title="CONMETR1:attr.1">satisfying_minor_Scherungssatz</a>  &amp; <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V2" title="CONMETR1:attr.2">satisfying_major_Scherungssatz</a>  ) )</span><br/>

<a NAME="E1:8"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
( <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V3" title="CONMETR1:attr.3">satisfying_Scherungssatz</a>  implies <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V1" title="CONMETR1:attr.1">satisfying_minor_Scherungssatz</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:8_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V3" title="CONMETR1:attr.3">satisfying_Scherungssatz</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V1" title="CONMETR1:attr.1">satisfying_minor_Scherungssatz</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</font>, <font color="Maroon" title="c4">a3</font>, <font color="Maroon" title="c5">a4</font>, <font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c7">b2</font>, <font color="Maroon" title="c8">b3</font>, <font color="Maroon" title="c9">b4</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <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">M</font>, <font color="Olive" title="b2">N</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; not <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; not <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; not <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; not <font color="Maroon" title="c9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; not <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; not <font color="Maroon" title="c4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; not <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; not <font color="Maroon" title="c8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c7">b2</font> &amp; <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c9">b4</font> holds <br/><font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font></span><br/><span class="kw">let </span><font color="Maroon" title="c10">M</font>, <font color="Maroon" title="c11">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; not <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; not <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; not <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; not <font color="Maroon" title="c9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; not <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; not <font color="Maroon" title="c4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; not <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; not <font color="Maroon" title="c8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c7">b2</font> &amp; <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c9">b4</font> implies <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:8_1_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c10">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E2:8_1_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E3:8_1_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E4:8_1_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E5:8_1_1"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E6:8_1_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E7:8_1_1"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E8:8_1_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E9:8_1_1"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E10:8_1_1"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
not <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E11:8_1_1"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
not <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E12:8_1_1"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
not <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E13:8_1_1"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
not <font color="Maroon" title="c9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E14:8_1_1"/><span class="lab"><font color="Green" title="E24">A16</font></span>: 
not <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E15:8_1_1"/><span class="lab"><font color="Green" title="E25">A17</font></span>: 
not <font color="Maroon" title="c4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E16:8_1_1"/><span class="lab"><font color="Green" title="E26">A18</font></span>: 
not <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E17:8_1_1"/><span class="lab"><font color="Green" title="E27">A19</font></span>: 
not <font color="Maroon" title="c8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E18:8_1_1"/><span class="lab"><font color="Green" title="E28">A20</font></span>: 
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c7">b2</font>
 <span class="kw">and </span><br/><a NAME="E19:8_1_1"/><span class="lab"><font color="Green" title="E29">A21</font></span>: 
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font>
 <span class="kw">and </span><br/><a NAME="E20:8_1_1"/><span class="lab"><font color="Green" title="E30">A22</font></span>: 
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c9">b4</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font></span><br/><a NAME="E21:8_1_1"/><span class="lab"><font color="Green" title="E31">A23</font></span>: 
<font color="Maroon" title="c11">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/><a NAME="E22:8_1_1"/>
<font color="Maroon" title="c10">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/><span class="kw">hence </span><a NAME="E23:8_1_1"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E2:8_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E3:8_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E4:8_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E6:8_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E7:8_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E8:8_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E9:8_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E10:8_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E11:8_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E12:8_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E13:8_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E14:8_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E15:8_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E16:8_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E17:8_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E18:8_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E19:8_1_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E20:8_1_1"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E21:8_1_1"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="ref" href="conmetr1.html#D3" target="_self" title="CONMETR1:def.3">Def3</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="E3:8_1"/>
<font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V1" title="CONMETR1:attr.1">satisfying_minor_Scherungssatz</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="conmetr1.html#D1" target="_self" title="CONMETR1:def.1">Def1</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>
<a NAME="E2:8"/><span class="lab"><font color="Green" title="E10">A24</font></span>: 
( <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V1" title="CONMETR1:attr.1">satisfying_minor_Scherungssatz</a>  &amp; <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V2" title="CONMETR1:attr.2">satisfying_major_Scherungssatz</a>  implies <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V3" title="CONMETR1:attr.3">satisfying_Scherungssatz</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:8_2"/><span class="lab"><font color="Green" title="E11">A25</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V1" title="CONMETR1:attr.1">satisfying_minor_Scherungssatz</a> 
 <span class="kw">and </span><br/><a NAME="E2:8_2"/><span class="lab"><font color="Green" title="E12">A26</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V2" title="CONMETR1:attr.2">satisfying_major_Scherungssatz</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V3" title="CONMETR1:attr.3">satisfying_Scherungssatz</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</font>, <font color="Maroon" title="c4">a3</font>, <font color="Maroon" title="c5">a4</font>, <font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c7">b2</font>, <font color="Maroon" title="c8">b3</font>, <font color="Maroon" title="c9">b4</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <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">M</font>, <font color="Olive" title="b2">N</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; not <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; not <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; not <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; not <font color="Maroon" title="c9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; not <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; not <font color="Maroon" title="c4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; not <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; not <font color="Maroon" title="c8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c7">b2</font> &amp; <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c9">b4</font> holds <br/><font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font></span><br/><span class="kw">let </span><font color="Maroon" title="c10">M</font>, <font color="Maroon" title="c11">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c11">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; not <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; not <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; not <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; not <font color="Maroon" title="c9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; not <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; not <font color="Maroon" title="c4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; not <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; not <font color="Maroon" title="c8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> &amp; <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c7">b2</font> &amp; <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c9">b4</font> implies <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:8_2_1"/><span class="lab"><font color="Green" title="E13">A27</font></span>: 
<font color="Maroon" title="c10">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E2:8_2_1"/><span class="lab"><font color="Green" title="E14">A28</font></span>: 
<font color="Maroon" title="c11">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:8_2_1"/><span class="lab"><font color="Green" title="E15">A29</font></span>: 
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E4:8_2_1"/><span class="lab"><font color="Green" title="E16">A30</font></span>: 
<font color="Maroon" title="c4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E5:8_2_1"/><span class="lab"><font color="Green" title="E17">A31</font></span>: 
<font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E6:8_2_1"/><span class="lab"><font color="Green" title="E18">A32</font></span>: 
<font color="Maroon" title="c8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E7:8_2_1"/><span class="lab"><font color="Green" title="E19">A33</font></span>: 
<font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E8:8_2_1"/><span class="lab"><font color="Green" title="E20">A34</font></span>: 
<font color="Maroon" title="c5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E9:8_2_1"/><span class="lab"><font color="Green" title="E21">A35</font></span>: 
<font color="Maroon" title="c7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E10:8_2_1"/><span class="lab"><font color="Green" title="E22">A36</font></span>: 
<font color="Maroon" title="c9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E11:8_2_1"/><span class="lab"><font color="Green" title="E23">A37</font></span>: 
not <font color="Maroon" title="c5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E12:8_2_1"/><span class="lab"><font color="Green" title="E24">A38</font></span>: 
not <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E13:8_2_1"/><span class="lab"><font color="Green" title="E25">A39</font></span>: 
not <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E14:8_2_1"/><span class="lab"><font color="Green" title="E26">A40</font></span>: 
not <font color="Maroon" title="c9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E15:8_2_1"/><span class="lab"><font color="Green" title="E27">A41</font></span>: 
not <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E16:8_2_1"/><span class="lab"><font color="Green" title="E28">A42</font></span>: 
not <font color="Maroon" title="c4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E17:8_2_1"/><span class="lab"><font color="Green" title="E29">A43</font></span>: 
not <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E18:8_2_1"/><span class="lab"><font color="Green" title="E30">A44</font></span>: 
not <font color="Maroon" title="c8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font>
 <span class="kw">and </span><br/><a NAME="E19:8_2_1"/><span class="lab"><font color="Green" title="E31">A45</font></span>: 
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c7">b2</font>
 <span class="kw">and </span><br/><a NAME="E20:8_2_1"/><span class="lab"><font color="Green" title="E32">A46</font></span>: 
<font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c6">b1</font>
 <span class="kw">and </span><br/><a NAME="E21:8_2_1"/><span class="lab"><font color="Green" title="E33">A47</font></span>: 
<font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c9">b4</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:8_2_1_1"/>
not <font color="Maroon" title="c10">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font></span><br/><a NAME="E2:8_2_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">o</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_2_1"><span class="lab"><font color="Green" title="E13">A27</font></span></a>, <a class="txt" href="#E2:8_2_1"><span class="lab"><font color="Green" title="E14">A28</font></span></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></span>;<br/><span class="kw">hence </span><a NAME="E3:8_2_1_1"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_2"><span class="lab"><font color="Green" title="E12">A26</font></span></a>, <a class="txt" href="#E1:8_2_1"><span class="lab"><font color="Green" title="E13">A27</font></span></a>, <a class="txt" href="#E2:8_2_1"><span class="lab"><font color="Green" title="E14">A28</font></span></a>, <a class="txt" href="#E3:8_2_1"><span class="lab"><font color="Green" title="E15">A29</font></span></a>, <a class="txt" href="#E4:8_2_1"><span class="lab"><font color="Green" title="E16">A30</font></span></a>, <a class="txt" href="#E5:8_2_1"><span class="lab"><font color="Green" title="E17">A31</font></span></a>, <a class="txt" href="#E6:8_2_1"><span class="lab"><font color="Green" title="E18">A32</font></span></a>, <a class="txt" href="#E7:8_2_1"><span class="lab"><font color="Green" title="E19">A33</font></span></a>, <a class="txt" href="#E8:8_2_1"><span class="lab"><font color="Green" title="E20">A34</font></span></a>, <a class="txt" href="#E9:8_2_1"><span class="lab"><font color="Green" title="E21">A35</font></span></a>, <a class="txt" href="#E10:8_2_1"><span class="lab"><font color="Green" title="E22">A36</font></span></a>, <a class="txt" href="#E11:8_2_1"><span class="lab"><font color="Green" title="E23">A37</font></span></a>, <a class="txt" href="#E12:8_2_1"><span class="lab"><font color="Green" title="E24">A38</font></span></a>, <a class="txt" href="#E13:8_2_1"><span class="lab"><font color="Green" title="E25">A39</font></span></a>, <a class="txt" href="#E14:8_2_1"><span class="lab"><font color="Green" title="E26">A40</font></span></a>, <a class="txt" href="#E15:8_2_1"><span class="lab"><font color="Green" title="E27">A41</font></span></a>, <a class="txt" href="#E16:8_2_1"><span class="lab"><font color="Green" title="E28">A42</font></span></a>, <a class="txt" href="#E17:8_2_1"><span class="lab"><font color="Green" title="E29">A43</font></span></a>, <a class="txt" href="#E18:8_2_1"><span class="lab"><font color="Green" title="E30">A44</font></span></a>, <a class="txt" href="#E19:8_2_1"><span class="lab"><font color="Green" title="E31">A45</font></span></a>, <a class="txt" href="#E20:8_2_1"><span class="lab"><font color="Green" title="E32">A46</font></span></a>, <a class="txt" href="#E21:8_2_1"><span class="lab"><font color="Green" title="E33">A47</font></span></a>, <a class="ref" href="conmetr1.html#D2" target="_self" title="CONMETR1:def.2">Def2</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="E23:8_2_1"/>
<font color="Maroon" title="c4">a3</font>,<font color="Maroon" title="c5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b3</font>,<font color="Maroon" title="c9">b4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_2"><span class="lab"><font color="Green" title="E11">A25</font></span></a>, <a class="txt" href="#E3:8_2_1"><span class="lab"><font color="Green" title="E15">A29</font></span></a>, <a class="txt" href="#E4:8_2_1"><span class="lab"><font color="Green" title="E16">A30</font></span></a>, <a class="txt" href="#E5:8_2_1"><span class="lab"><font color="Green" title="E17">A31</font></span></a>, <a class="txt" href="#E6:8_2_1"><span class="lab"><font color="Green" title="E18">A32</font></span></a>, <a class="txt" href="#E7:8_2_1"><span class="lab"><font color="Green" title="E19">A33</font></span></a>, <a class="txt" href="#E8:8_2_1"><span class="lab"><font color="Green" title="E20">A34</font></span></a>, <a class="txt" href="#E9:8_2_1"><span class="lab"><font color="Green" title="E21">A35</font></span></a>, <a class="txt" href="#E10:8_2_1"><span class="lab"><font color="Green" title="E22">A36</font></span></a>, <a class="txt" href="#E11:8_2_1"><span class="lab"><font color="Green" title="E23">A37</font></span></a>, <a class="txt" href="#E12:8_2_1"><span class="lab"><font color="Green" title="E24">A38</font></span></a>, <a class="txt" href="#E13:8_2_1"><span class="lab"><font color="Green" title="E25">A39</font></span></a>, <a class="txt" href="#E14:8_2_1"><span class="lab"><font color="Green" title="E26">A40</font></span></a>, <a class="txt" href="#E15:8_2_1"><span class="lab"><font color="Green" title="E27">A41</font></span></a>, <a class="txt" href="#E16:8_2_1"><span class="lab"><font color="Green" title="E28">A42</font></span></a>, <a class="txt" href="#E17:8_2_1"><span class="lab"><font color="Green" title="E29">A43</font></span></a>, <a class="txt" href="#E18:8_2_1"><span class="lab"><font color="Green" title="E30">A44</font></span></a>, <a class="txt" href="#E19:8_2_1"><span class="lab"><font color="Green" title="E31">A45</font></span></a>, <a class="txt" href="#E20:8_2_1"><span class="lab"><font color="Green" title="E32">A46</font></span></a>, <a class="txt" href="#E21:8_2_1"><span class="lab"><font color="Green" title="E33">A47</font></span></a>, <a class="ref" href="conmetr1.html#D1" target="_self" title="CONMETR1:def.1">Def1</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="E4:8_2"/>
<font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V3" title="CONMETR1:attr.3">satisfying_Scherungssatz</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="conmetr1.html#D3" target="_self" title="CONMETR1:def.3">Def3</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>
<a NAME="E3:8"/>
( <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V3" title="CONMETR1:attr.3">satisfying_Scherungssatz</a>  implies <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V2" title="CONMETR1:attr.2">satisfying_major_Scherungssatz</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:8_3"/>
<font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V3" title="CONMETR1:attr.3">satisfying_Scherungssatz</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V2" title="CONMETR1:attr.2">satisfying_major_Scherungssatz</a> </span><br/>

<a NAME="E2:8_3"/><span class="kw">then </span>
for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a1</font>, <font color="Olive" title="b3">a2</font>, <font color="Olive" title="b4">a3</font>, <font color="Olive" title="b5">a4</font>, <font color="Olive" title="b6">b1</font>, <font color="Olive" title="b7">b2</font>, <font color="Olive" title="b8">b3</font>, <font color="Olive" title="b9">b4</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b10">M</font>, <font color="Olive" title="b11">N</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b10">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b11">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp; <font color="Olive" title="b2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; <font color="Olive" title="b4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; <font color="Olive" title="b6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; <font color="Olive" title="b8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; <font color="Olive" title="b3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp; <font color="Olive" title="b5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp; <font color="Olive" title="b7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp; <font color="Olive" title="b9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp; not <font color="Olive" title="b5">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; not <font color="Olive" title="b3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; not <font color="Olive" title="b7">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; not <font color="Olive" title="b9">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">M</font> &amp; not <font color="Olive" title="b2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp; not <font color="Olive" title="b4">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp; not <font color="Olive" title="b6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp; not <font color="Olive" title="b8">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b11">N</font> &amp; <font color="Olive" title="b4">a3</font>,<font color="Olive" title="b3">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">b3</font>,<font color="Olive" title="b7">b2</font> &amp; <font color="Olive" title="b3">a2</font>,<font color="Olive" title="b2">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">b2</font>,<font color="Olive" title="b6">b1</font> &amp; <font color="Olive" title="b2">a1</font>,<font color="Olive" title="b5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">b1</font>,<font color="Olive" title="b9">b4</font> holds <br/><font color="Olive" title="b4">a3</font>,<font color="Olive" title="b5">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">b3</font>,<font color="Olive" title="b9">b4</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="conmetr1.html#D3" target="_self" title="CONMETR1:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:8_3"/>
<font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V2" title="CONMETR1:attr.2">satisfying_major_Scherungssatz</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="conmetr1.html#D2" target="_self" title="CONMETR1:def.2">Def2</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="E4:8"/>
( <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V3" title="CONMETR1:attr.3">satisfying_Scherungssatz</a>  iff ( <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V1" title="CONMETR1:attr.1">satisfying_minor_Scherungssatz</a>  &amp; <font color="Maroon" title="c1">X</font> is  <a href="conmetr1.html#V2" title="CONMETR1:attr.2">satisfying_major_Scherungssatz</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E10">A24</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>
