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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">w'</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">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>  ) holds <br/><font color="Olive" title="b3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w'</font></span><br/><b>let </b><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">w'</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:()"><i><font color="Red">::  thesis: </font></i></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">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>  ) implies <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w'</font> )</span><br/>









<b>given </b><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><b> such that </b><a NAME="E2:43"/><i><font color="Green" title="E31">A1</font></i>: 
<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> 
 <b>and </b><br/><a NAME="E3:43"/><i><font color="Green" title="E32">A2</font></i>: 
<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> 
 <b>and </b><br/><a NAME="E4:43"/><i><font color="Green" title="E33">A3</font></i>: 
<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> 
 <b>and </b><br/><a NAME="E5:43"/><i><font color="Green" title="E34">A4</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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">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>  ) or <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w'</font> )</span><br/>

<b>given </b><font color="Maroon" title="c8">p'</font>, <font color="Maroon" title="c9">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><b> such that </b><a NAME="E7:43"/><i><font color="Green" title="E35">A5</font></i>: 
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p'</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a> 
 <b>and </b><br/><a NAME="E8:43"/><i><font color="Green" title="E36">A6</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q'</font> <a href="midsp_1.html#K7" title="MIDSP_1:func.7">~</a> 
 <b>and </b><br/><a NAME="E9:43"/><i><font color="Green" title="E37">A7</font></i>: 
<font color="Maroon" title="c8">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="c9">q'</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a> 
 <b>and </b><br/><a NAME="E10:43"/><i><font color="Green" title="E38">A8</font></i>: 
<font color="Maroon" title="c5">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="c8">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="c9">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:()"><i><font color="Red">::  thesis: </font></i></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">w'</font></span><br/>

<a NAME="E11:43"/>
<font color="Maroon" title="c6">p</font> <a href="midsp_1.html#R2" title="MIDSP_1:pred.2">##</a> <font color="Maroon" title="c8">p'</font>
 
<b>by </b><i><a class="txt" href="midsp_1.html#E2:43"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="midsp_1.html#E7:43"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="midsp_1.html#T43" target="_self" title="MIDSP_1:th.43">Th43</a></i>;<br/>
<a NAME="E12:43"/><b>then </b><i><font color="Green" title="E39">A9</font></i>: 
<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">p'</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a> ,<font color="Maroon" title="c8">p'</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a> 
 
<b>by </b><i><a class="ref" href="midsp_1.html#D6" target="_self" title="MIDSP_1:def.6">Def6</a></i>;<br/>
<a NAME="E13:43"/>
<font color="Maroon" title="c7">q</font> <a href="midsp_1.html#R2" title="MIDSP_1:pred.2">##</a> <font color="Maroon" title="c9">q'</font>
 
<b>by </b><i><a class="txt" href="midsp_1.html#E3:43"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="midsp_1.html#E8:43"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="midsp_1.html#T43" target="_self" title="MIDSP_1:th.43">Th43</a></i>;<br/>
<a NAME="E14:43"/><b>then </b>
<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">q'</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a> ,<font color="Maroon" title="c9">q'</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a> 
 
<b>by </b><i><a class="ref" href="midsp_1.html#D6" target="_self" title="MIDSP_1:def.6">Def6</a></i>;<br/>
<a NAME="E15:43"/><b>then </b>
<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">p'</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">`1</a> ,<font color="Maroon" title="c9">q'</font> <a href="midsp_1.html#K5" title="MIDSP_1:func.5">`2</a> 
 
<b>by </b><i><a class="txt" href="midsp_1.html#E4:43"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="midsp_1.html#E9:43"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="midsp_1.html#E12:43"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="midsp_1.html#T29" target="_self" title="MIDSP_1:th.29">Th29</a></i>;<br/>
<b>hence </b><a NAME="E16:43"/>
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w'</font>
 <b>by </b><i><a class="txt" href="midsp_1.html#E5:43"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="midsp_1.html#E10:43"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="midsp_1.html#T31" target="_self" title="MIDSP_1:th.31">Th31</a>, <a class="ref" href="midsp_1.html#T42" target="_self" title="MIDSP_1:th.42">Th42</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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