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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   <a href="midsp_1.html#NM1" title="MIDSP_1:NM.1">MidSp</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font>, <font color="Olive" title="b4">w9</font> being    <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font>  st  ex <font color="Olive" title="b5">p</font>, <font color="Olive" title="b6">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">p</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a>  &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">q</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a>  &amp; <font color="Olive" title="b5">p</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">q</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a>  &amp; <font color="Olive" title="b3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">p</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b6">q</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a>  ) &amp;  ex <font color="Olive" title="b5">p</font>, <font color="Olive" title="b6">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">p</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a>  &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">q</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a>  &amp; <font color="Olive" title="b5">p</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">q</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a>  &amp; <font color="Olive" title="b4">w9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">p</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b6">q</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a>  ) holds <br/><font color="Olive" title="b3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font>, <font color="Maroon" title="c5">w9</font> be    <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a>  &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a>  &amp; <font color="Olive" title="b1">p</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a>  &amp; <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">q</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a>  ) &amp;  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a>  &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a>  &amp; <font color="Olive" title="b1">p</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a>  &amp; <font color="Maroon" title="c5">w9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">q</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a>  ) implies <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w9</font> )</span><br/>









<span class="kw">given </span><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><span class="kw"> such that </span><a NAME="E2:42"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a> 
 <span class="kw">and </span><br/><a NAME="E3:42"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a> 
 <span class="kw">and </span><br/><a NAME="E4:42"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a> 
 <span class="kw">and </span><br/><a NAME="E5:42"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  holds <br/>(  not <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a>  or  not <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a>  or  not <font color="Olive" title="b1">p</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a>  or  not <font color="Maroon" title="c5">w9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">q</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a>  ) or <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w9</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c8">p9</font>, <font color="Maroon" title="c9">q9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><span class="kw"> such that </span><a NAME="E7:42"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p9</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a> 
 <span class="kw">and </span><br/><a NAME="E8:42"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q9</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a> 
 <span class="kw">and </span><br/><a NAME="E9:42"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<font color="Maroon" title="c8">p9</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q9</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a> 
 <span class="kw">and </span><br/><a NAME="E10:42"/><span class="lab"><font color="Green" title="E37">A8</font></span>: 
<font color="Maroon" title="c5">w9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p9</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">q9</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w9</font></span><br/>

<a NAME="E11:42"/>
<font color="Maroon" title="c7">q</font> <a href="midsp_1.html#R2" title="MIDSP_1:pred.2">##</a> <font color="Maroon" title="c9">q9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="midsp_1.html#T28" target="_self" title="MIDSP_1:th.28">Th43</a></span>;<br/>
<a NAME="E12:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<font color="Maroon" title="c7">q</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a> ,<font color="Maroon" title="c7">q</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a>  <a href="midsp_1.html#R1" title="MIDSP_1:pred.1">@@</a> <font color="Maroon" title="c9">q9</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a> ,<font color="Maroon" title="c9">q9</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_1.html#D5" target="_self" title="MIDSP_1:def.5">Def6</a></span>;<br/>
<a NAME="E13:42"/>
<font color="Maroon" title="c6">p</font> <a href="midsp_1.html#R2" title="MIDSP_1:pred.2">##</a> <font color="Maroon" title="c8">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="midsp_1.html#T28" target="_self" title="MIDSP_1:th.28">Th43</a></span>;<br/>
<a NAME="E14:42"/><span class="kw">then </span>
<font color="Maroon" title="c6">p</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a> ,<font color="Maroon" title="c6">p</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a>  <a href="midsp_1.html#R1" title="MIDSP_1:pred.1">@@</a> <font color="Maroon" title="c8">p9</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a> ,<font color="Maroon" title="c8">p9</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_1.html#D5" target="_self" title="MIDSP_1:def.5">Def6</a></span>;<br/>
<a NAME="E15:42"/><span class="kw">then </span>
<font color="Maroon" title="c6">p</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a> ,<font color="Maroon" title="c7">q</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a>  <a href="midsp_1.html#R1" title="MIDSP_1:pred.1">@@</a> <font color="Maroon" title="c8">p9</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a> ,<font color="Maroon" title="c9">q9</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="ref" href="midsp_1.html#T18" target="_self" title="MIDSP_1:th.18">Th29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:42"/>
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="midsp_1.html#T19" target="_self" title="MIDSP_1:th.19">Th31</a>, <a class="ref" href="midsp_1.html#T27" target="_self" title="MIDSP_1:th.27">Th42</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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