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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <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="conmetr.html#V6" title="CONMETR:attr.6">satisfying_SCH</a>  )</span><br/>

<a NAME="E1:19"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
( <font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V6" title="CONMETR:attr.6">satisfying_SCH</a>  implies  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) is  <a href="conmetr1.html#V3" title="CONMETR1:attr.3">satisfying_Scherungssatz</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:19_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V6" title="CONMETR:attr.6">satisfying_SCH</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font>, <font color="Olive" title="b3">a3</font>, <font color="Olive" title="b4">a4</font>, <font color="Olive" title="b5">b1</font>, <font color="Olive" title="b6">b2</font>, <font color="Olive" title="b7">b3</font>, <font color="Olive" title="b8">b4</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #)<br/>  for <font color="Olive" title="b9">M</font>, <font color="Olive" title="b10">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #)  st <font color="Olive" title="b9">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b10">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M</font> &amp; <font color="Olive" title="b3">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M</font> &amp; <font color="Olive" title="b5">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M</font> &amp; <font color="Olive" title="b7">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M</font> &amp; <font color="Olive" title="b2">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N</font> &amp; <font color="Olive" title="b4">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N</font> &amp; <font color="Olive" title="b6">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N</font> &amp; <font color="Olive" title="b8">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N</font> &amp;  not <font color="Olive" title="b4">a4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M</font> &amp;  not <font color="Olive" title="b2">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M</font> &amp;  not <font color="Olive" title="b6">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M</font> &amp;  not <font color="Olive" title="b8">b4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M</font> &amp;  not <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N</font> &amp;  not <font color="Olive" title="b3">a3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N</font> &amp;  not <font color="Olive" title="b5">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N</font> &amp;  not <font color="Olive" title="b7">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N</font> &amp; <font color="Olive" title="b3">a3</font>,<font color="Olive" title="b2">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">b3</font>,<font color="Olive" title="b6">b2</font> &amp; <font color="Olive" title="b2">a2</font>,<font color="Olive" title="b1">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">b2</font>,<font color="Olive" title="b5">b1</font> &amp; <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b4">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b1</font>,<font color="Olive" title="b8">b4</font> holds <br/><font color="Olive" title="b3">a3</font>,<font color="Olive" title="b4">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">b3</font>,<font color="Olive" title="b8">b4</font></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 <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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 <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</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 <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E1:19_1_1"/><span class="lab"><font color="Green" title="E11">A3</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="E2:19_1_1"/><span class="lab"><font color="Green" title="E12">A4</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="E3:19_1_1"/><span class="lab"><font color="Green" title="E13">A5</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="E4:19_1_1"/><span class="lab"><font color="Green" title="E14">A6</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="E5:19_1_1"/><span class="lab"><font color="Green" title="E15">A7</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="E6:19_1_1"/><span class="lab"><font color="Green" title="E16">A8</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="E7:19_1_1"/><span class="lab"><font color="Green" title="E17">A9</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="E8:19_1_1"/><span class="lab"><font color="Green" title="E18">A10</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="E9:19_1_1"/><span class="lab"><font color="Green" title="E19">A11</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="E10:19_1_1"/><span class="lab"><font color="Green" title="E20">A12</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="E11:19_1_1"/><span class="lab"><font color="Green" title="E21">A13</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="E12:19_1_1"/><span class="lab"><font color="Green" title="E22">A14</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="E13:19_1_1"/><span class="lab"><font color="Green" title="E23">A15</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="E14:19_1_1"/><span class="lab"><font color="Green" title="E24">A16</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="E15:19_1_1"/><span class="lab"><font color="Green" title="E25">A17</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="E16:19_1_1"/><span class="lab"><font color="Green" title="E26">A18</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="E17:19_1_1"/><span class="lab"><font color="Green" title="E27">A19</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="E18:19_1_1"/><span class="lab"><font color="Green" title="E28">A20</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="E19:19_1_1"/><span class="lab"><font color="Green" title="E29">A21</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="E20:19_1_1"/><span class="lab"><font color="Green" title="E30">A22</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="E21:19_1_1"/><span class="lab"><font color="Green" title="E31">A23</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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/><span class="kw">reconsider </span><font color="Maroon" title="c12">a19</font> = <font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c13">a29</font> = <font color="Maroon" title="c3">a2</font>, <font color="Maroon" title="c14">a39</font> = <font color="Maroon" title="c4">a3</font>, <font color="Maroon" title="c15">a49</font> = <font color="Maroon" title="c5">a4</font>, <font color="Maroon" title="c16">b19</font> = <font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c17">b29</font> = <font color="Maroon" title="c7">b2</font>, <font color="Maroon" title="c18">b39</font> = <font color="Maroon" title="c8">b3</font>, <font color="Maroon" title="c19">b49</font> = <font color="Maroon" title="c9">b4</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E23:19_1_1"/><span class="lab"><font color="Green" title="E32">A24</font></span>: 
<font color="Maroon" title="c14">a39</font>,<font color="Maroon" title="c13">a29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">b39</font>,<font color="Maroon" title="c17">b29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:19_1_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>;<br/><a NAME="E24:19_1_1"/><span class="lab"><font color="Green" title="E33">A25</font></span>: 
<font color="Maroon" title="c12">a19</font>,<font color="Maroon" title="c15">a49</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">b19</font>,<font color="Maroon" title="c19">b49</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:19_1_1"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>;<br/><a NAME="E25:19_1_1"/><span class="lab"><font color="Green" title="E34">A26</font></span>: 
<font color="Maroon" title="c13">a29</font>,<font color="Maroon" title="c12">a19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">b29</font>,<font color="Maroon" title="c16">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:19_1_1"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">M9</font> = <font color="Maroon" title="c10">M</font>, <font color="Maroon" title="c21">N9</font> = <font color="Maroon" title="c11">N</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E27:19_1_1"/><span class="lab"><font color="Green" title="E35">A27</font></span>: 
<font color="Maroon" title="c21">N9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="analmetr.html#T43" onmouseover="rs('analmetr/T43')" onmouseout="rh()">ANALMETR:43</a></span>;<br/><a NAME="E28:19_1_1"/>
<font color="Maroon" title="c20">M9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="analmetr.html#T43" onmouseover="rs('analmetr/T43')" onmouseout="rh()">ANALMETR:43</a></span>;<br/><a NAME="E29:19_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">a39</font>,<font color="Maroon" title="c15">a49</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">b39</font>,<font color="Maroon" title="c19">b49</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:19_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E4:19_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E5:19_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E6:19_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E7:19_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E8:19_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E9:19_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E10:19_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E11:19_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E12:19_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E13:19_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E14:19_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E15:19_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E16:19_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E17:19_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E18:19_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E27:19_1_1"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E23:19_1_1"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E25:19_1_1"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E24:19_1_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="ref" href="conmetr.html#D6" onmouseover="rs('conmetr/D6')" onmouseout="rh()">CONMETR:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E30:19_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="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:19_1"/>
 <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) is  <a href="conmetr1.html#V3" title="CONMETR1:attr.3">satisfying_Scherungssatz</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E2:19"/>
(  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <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="conmetr.html#V6" title="CONMETR:attr.6">satisfying_SCH</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:19_2"/><span class="lab"><font color="Green" title="E10">A28</font></span>: 
 <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) is  <a href="conmetr1.html#V3" title="CONMETR1:attr.3">satisfying_Scherungssatz</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V6" title="CONMETR:attr.6">satisfying_SCH</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a19</font>, <font color="Olive" title="b2">a29</font>, <font color="Olive" title="b3">a39</font>, <font color="Olive" title="b4">a49</font>, <font color="Olive" title="b5">b19</font>, <font color="Olive" title="b6">b29</font>, <font color="Olive" title="b7">b39</font>, <font color="Olive" title="b8">b49</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="b9">M9</font>, <font color="Olive" title="b10">N9</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="b9">M9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a>  &amp; <font color="Olive" title="b10">N9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a>  &amp; <font color="Olive" title="b1">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M9</font> &amp; <font color="Olive" title="b3">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M9</font> &amp; <font color="Olive" title="b5">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M9</font> &amp; <font color="Olive" title="b7">b39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M9</font> &amp; <font color="Olive" title="b2">a29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N9</font> &amp; <font color="Olive" title="b4">a49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N9</font> &amp; <font color="Olive" title="b6">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N9</font> &amp; <font color="Olive" title="b8">b49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N9</font> &amp;  not <font color="Olive" title="b4">a49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M9</font> &amp;  not <font color="Olive" title="b2">a29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M9</font> &amp;  not <font color="Olive" title="b6">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M9</font> &amp;  not <font color="Olive" title="b8">b49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b9">M9</font> &amp;  not <font color="Olive" title="b1">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N9</font> &amp;  not <font color="Olive" title="b3">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N9</font> &amp;  not <font color="Olive" title="b5">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N9</font> &amp;  not <font color="Olive" title="b7">b39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b10">N9</font> &amp; <font color="Olive" title="b3">a39</font>,<font color="Olive" title="b2">a29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">b39</font>,<font color="Olive" title="b6">b29</font> &amp; <font color="Olive" title="b2">a29</font>,<font color="Olive" title="b1">a19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">b29</font>,<font color="Olive" title="b5">b19</font> &amp; <font color="Olive" title="b1">a19</font>,<font color="Olive" title="b4">a49</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b19</font>,<font color="Olive" title="b8">b49</font> holds <br/><font color="Olive" title="b3">a39</font>,<font color="Olive" title="b4">a49</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">b39</font>,<font color="Olive" title="b8">b49</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">a19</font>, <font color="Maroon" title="c3">a29</font>, <font color="Maroon" title="c4">a39</font>, <font color="Maroon" title="c5">a49</font>, <font color="Maroon" title="c6">b19</font>, <font color="Maroon" title="c7">b29</font>, <font color="Maroon" title="c8">b39</font>, <font color="Maroon" title="c9">b49</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M9</font>, <font color="Olive" title="b2">N9</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">M9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a>  &amp; <font color="Olive" title="b2">N9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a>  &amp; <font color="Maroon" title="c2">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M9</font> &amp; <font color="Maroon" title="c4">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M9</font> &amp; <font color="Maroon" title="c6">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M9</font> &amp; <font color="Maroon" title="c8">b39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M9</font> &amp; <font color="Maroon" title="c3">a29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N9</font> &amp; <font color="Maroon" title="c5">a49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N9</font> &amp; <font color="Maroon" title="c7">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N9</font> &amp; <font color="Maroon" title="c9">b49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N9</font> &amp;  not <font color="Maroon" title="c5">a49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M9</font> &amp;  not <font color="Maroon" title="c3">a29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M9</font> &amp;  not <font color="Maroon" title="c7">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M9</font> &amp;  not <font color="Maroon" title="c9">b49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M9</font> &amp;  not <font color="Maroon" title="c2">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N9</font> &amp;  not <font color="Maroon" title="c4">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N9</font> &amp;  not <font color="Maroon" title="c6">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N9</font> &amp;  not <font color="Maroon" title="c8">b39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N9</font> &amp; <font color="Maroon" title="c4">a39</font>,<font color="Maroon" title="c3">a29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b39</font>,<font color="Maroon" title="c7">b29</font> &amp; <font color="Maroon" title="c3">a29</font>,<font color="Maroon" title="c2">a19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b29</font>,<font color="Maroon" title="c6">b19</font> &amp; <font color="Maroon" title="c2">a19</font>,<font color="Maroon" title="c5">a49</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b19</font>,<font color="Maroon" title="c9">b49</font> holds <br/><font color="Maroon" title="c4">a39</font>,<font color="Maroon" title="c5">a49</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b39</font>,<font color="Maroon" title="c9">b49</font></span><br/><span class="kw">let </span><font color="Maroon" title="c10">M9</font>, <font color="Maroon" title="c11">N9</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">M9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a>  &amp; <font color="Maroon" title="c11">N9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a>  &amp; <font color="Maroon" title="c2">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font> &amp; <font color="Maroon" title="c4">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font> &amp; <font color="Maroon" title="c6">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font> &amp; <font color="Maroon" title="c8">b39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font> &amp; <font color="Maroon" title="c3">a29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font> &amp; <font color="Maroon" title="c5">a49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font> &amp; <font color="Maroon" title="c7">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font> &amp; <font color="Maroon" title="c9">b49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font> &amp;  not <font color="Maroon" title="c5">a49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font> &amp;  not <font color="Maroon" title="c3">a29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font> &amp;  not <font color="Maroon" title="c7">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font> &amp;  not <font color="Maroon" title="c9">b49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font> &amp;  not <font color="Maroon" title="c2">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font> &amp;  not <font color="Maroon" title="c4">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font> &amp;  not <font color="Maroon" title="c6">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font> &amp;  not <font color="Maroon" title="c8">b39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font> &amp; <font color="Maroon" title="c4">a39</font>,<font color="Maroon" title="c3">a29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b39</font>,<font color="Maroon" title="c7">b29</font> &amp; <font color="Maroon" title="c3">a29</font>,<font color="Maroon" title="c2">a19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b29</font>,<font color="Maroon" title="c6">b19</font> &amp; <font color="Maroon" title="c2">a19</font>,<font color="Maroon" title="c5">a49</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b19</font>,<font color="Maroon" title="c9">b49</font> implies <font color="Maroon" title="c4">a39</font>,<font color="Maroon" title="c5">a49</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b39</font>,<font color="Maroon" title="c9">b49</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:19_2_1"/><span class="lab"><font color="Green" title="E11">A29</font></span>: 
<font color="Maroon" title="c10">M9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E2:19_2_1"/><span class="lab"><font color="Green" title="E12">A30</font></span>: 
<font color="Maroon" title="c11">N9</font> is  <a href="analmetr.html#V5" title="ANALMETR:attr.5">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:19_2_1"/><span class="lab"><font color="Green" title="E13">A31</font></span>: 
<font color="Maroon" title="c2">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font>
 <span class="kw">and </span><br/><a NAME="E4:19_2_1"/><span class="lab"><font color="Green" title="E14">A32</font></span>: 
<font color="Maroon" title="c4">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font>
 <span class="kw">and </span><br/><a NAME="E5:19_2_1"/><span class="lab"><font color="Green" title="E15">A33</font></span>: 
<font color="Maroon" title="c6">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font>
 <span class="kw">and </span><br/><a NAME="E6:19_2_1"/><span class="lab"><font color="Green" title="E16">A34</font></span>: 
<font color="Maroon" title="c8">b39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font>
 <span class="kw">and </span><br/><a NAME="E7:19_2_1"/><span class="lab"><font color="Green" title="E17">A35</font></span>: 
<font color="Maroon" title="c3">a29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font>
 <span class="kw">and </span><br/><a NAME="E8:19_2_1"/><span class="lab"><font color="Green" title="E18">A36</font></span>: 
<font color="Maroon" title="c5">a49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font>
 <span class="kw">and </span><br/><a NAME="E9:19_2_1"/><span class="lab"><font color="Green" title="E19">A37</font></span>: 
<font color="Maroon" title="c7">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font>
 <span class="kw">and </span><br/><a NAME="E10:19_2_1"/><span class="lab"><font color="Green" title="E20">A38</font></span>: 
<font color="Maroon" title="c9">b49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font>
 <span class="kw">and </span><br/><a NAME="E11:19_2_1"/><span class="lab"><font color="Green" title="E21">A39</font></span>: 
 not <font color="Maroon" title="c5">a49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font>
 <span class="kw">and </span><br/><a NAME="E12:19_2_1"/><span class="lab"><font color="Green" title="E22">A40</font></span>: 
 not <font color="Maroon" title="c3">a29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font>
 <span class="kw">and </span><br/><a NAME="E13:19_2_1"/><span class="lab"><font color="Green" title="E23">A41</font></span>: 
 not <font color="Maroon" title="c7">b29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font>
 <span class="kw">and </span><br/><a NAME="E14:19_2_1"/><span class="lab"><font color="Green" title="E24">A42</font></span>: 
 not <font color="Maroon" title="c9">b49</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M9</font>
 <span class="kw">and </span><br/><a NAME="E15:19_2_1"/><span class="lab"><font color="Green" title="E25">A43</font></span>: 
 not <font color="Maroon" title="c2">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font>
 <span class="kw">and </span><br/><a NAME="E16:19_2_1"/><span class="lab"><font color="Green" title="E26">A44</font></span>: 
 not <font color="Maroon" title="c4">a39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font>
 <span class="kw">and </span><br/><a NAME="E17:19_2_1"/><span class="lab"><font color="Green" title="E27">A45</font></span>: 
 not <font color="Maroon" title="c6">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font>
 <span class="kw">and </span><br/><a NAME="E18:19_2_1"/><span class="lab"><font color="Green" title="E28">A46</font></span>: 
 not <font color="Maroon" title="c8">b39</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">N9</font>
 <span class="kw">and </span><br/><a NAME="E19:19_2_1"/><span class="lab"><font color="Green" title="E29">A47</font></span>: 
<font color="Maroon" title="c4">a39</font>,<font color="Maroon" title="c3">a29</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b39</font>,<font color="Maroon" title="c7">b29</font>
 <span class="kw">and </span><br/><a NAME="E20:19_2_1"/><span class="lab"><font color="Green" title="E30">A48</font></span>: 
<font color="Maroon" title="c3">a29</font>,<font color="Maroon" title="c2">a19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b29</font>,<font color="Maroon" title="c6">b19</font>
 <span class="kw">and </span><br/><a NAME="E21:19_2_1"/><span class="lab"><font color="Green" title="E31">A49</font></span>: 
<font color="Maroon" title="c2">a19</font>,<font color="Maroon" title="c5">a49</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b19</font>,<font color="Maroon" title="c9">b49</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a39</font>,<font color="Maroon" title="c5">a49</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b39</font>,<font color="Maroon" title="c9">b49</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c12">a1</font> = <font color="Maroon" title="c2">a19</font>, <font color="Maroon" title="c13">a2</font> = <font color="Maroon" title="c3">a29</font>, <font color="Maroon" title="c14">a3</font> = <font color="Maroon" title="c4">a39</font>, <font color="Maroon" title="c15">a4</font> = <font color="Maroon" title="c5">a49</font>, <font color="Maroon" title="c16">b1</font> = <font color="Maroon" title="c6">b19</font>, <font color="Maroon" title="c17">b2</font> = <font color="Maroon" title="c7">b29</font>, <font color="Maroon" title="c18">b3</font> = <font color="Maroon" title="c8">b39</font>, <font color="Maroon" title="c19">b4</font> = <font color="Maroon" title="c9">b49</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) ;<br/><a NAME="E23:19_2_1"/><span class="lab"><font color="Green" title="E32">A50</font></span>: 
<font color="Maroon" title="c14">a3</font>,<font color="Maroon" title="c13">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">b3</font>,<font color="Maroon" title="c17">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:19_2_1"><span class="lab"><font color="Green" title="E29">A47</font></span></a>, <a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>;<br/><a NAME="E24:19_2_1"/><span class="lab"><font color="Green" title="E33">A51</font></span>: 
<font color="Maroon" title="c12">a1</font>,<font color="Maroon" title="c15">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">b1</font>,<font color="Maroon" title="c19">b4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:19_2_1"><span class="lab"><font color="Green" title="E31">A49</font></span></a>, <a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>;<br/><a NAME="E25:19_2_1"/><span class="lab"><font color="Green" title="E34">A52</font></span>: 
<font color="Maroon" title="c13">a2</font>,<font color="Maroon" title="c12">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">b2</font>,<font color="Maroon" title="c16">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:19_2_1"><span class="lab"><font color="Green" title="E30">A48</font></span></a>, <a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">M</font> = <font color="Maroon" title="c10">M9</font>, <font color="Maroon" title="c21">N</font> = <font color="Maroon" title="c11">N9</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) ;<br/><a NAME="E27:19_2_1"/><span class="lab"><font color="Green" title="E35">A53</font></span>: 
<font color="Maroon" title="c21">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="#E2:19_2_1"><span class="lab"><font color="Green" title="E12">A30</font></span></a>, <a class="ref" href="analmetr.html#T43" onmouseover="rs('analmetr/T43')" onmouseout="rh()">ANALMETR:43</a></span>;<br/><a NAME="E28:19_2_1"/>
<font color="Maroon" title="c20">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="#E1:19_2_1"><span class="lab"><font color="Green" title="E11">A29</font></span></a>, <a class="ref" href="analmetr.html#T43" onmouseover="rs('analmetr/T43')" onmouseout="rh()">ANALMETR:43</a></span>;<br/><a NAME="E29:19_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">a3</font>,<font color="Maroon" title="c15">a4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">b3</font>,<font color="Maroon" title="c19">b4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_2"><span class="lab"><font color="Green" title="E10">A28</font></span></a>, <a class="txt" href="#E3:19_2_1"><span class="lab"><font color="Green" title="E13">A31</font></span></a>, <a class="txt" href="#E4:19_2_1"><span class="lab"><font color="Green" title="E14">A32</font></span></a>, <a class="txt" href="#E5:19_2_1"><span class="lab"><font color="Green" title="E15">A33</font></span></a>, <a class="txt" href="#E6:19_2_1"><span class="lab"><font color="Green" title="E16">A34</font></span></a>, <a class="txt" href="#E7:19_2_1"><span class="lab"><font color="Green" title="E17">A35</font></span></a>, <a class="txt" href="#E8:19_2_1"><span class="lab"><font color="Green" title="E18">A36</font></span></a>, <a class="txt" href="#E9:19_2_1"><span class="lab"><font color="Green" title="E19">A37</font></span></a>, <a class="txt" href="#E10:19_2_1"><span class="lab"><font color="Green" title="E20">A38</font></span></a>, <a class="txt" href="#E11:19_2_1"><span class="lab"><font color="Green" title="E21">A39</font></span></a>, <a class="txt" href="#E12:19_2_1"><span class="lab"><font color="Green" title="E22">A40</font></span></a>, <a class="txt" href="#E13:19_2_1"><span class="lab"><font color="Green" title="E23">A41</font></span></a>, <a class="txt" href="#E14:19_2_1"><span class="lab"><font color="Green" title="E24">A42</font></span></a>, <a class="txt" href="#E15:19_2_1"><span class="lab"><font color="Green" title="E25">A43</font></span></a>, <a class="txt" href="#E16:19_2_1"><span class="lab"><font color="Green" title="E26">A44</font></span></a>, <a class="txt" href="#E17:19_2_1"><span class="lab"><font color="Green" title="E27">A45</font></span></a>, <a class="txt" href="#E18:19_2_1"><span class="lab"><font color="Green" title="E28">A46</font></span></a>, <a class="txt" href="#E27:19_2_1"><span class="lab"><font color="Green" title="E35">A53</font></span></a>, <a class="txt" href="#E23:19_2_1"><span class="lab"><font color="Green" title="E32">A50</font></span></a>, <a class="txt" href="#E25:19_2_1"><span class="lab"><font color="Green" title="E34">A52</font></span></a>, <a class="txt" href="#E24:19_2_1"><span class="lab"><font color="Green" title="E33">A51</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E30:19_2_1"/>
<font color="Maroon" title="c4">a39</font>,<font color="Maroon" title="c5">a49</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b39</font>,<font color="Maroon" title="c9">b49</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:19_2"/>
<font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V6" title="CONMETR:attr.6">satisfying_SCH</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="conmetr.html#D6" onmouseover="rs('conmetr/D6')" onmouseout="rh()">CONMETR:def 6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:19"/>
(  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <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="conmetr.html#V6" title="CONMETR:attr.6">satisfying_SCH</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
