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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">W</font> being   <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">d</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b6">x</font> being   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Olive" title="b1">W</font> holds <br/> ( <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Olive" title="b1">W</font> &amp; ( <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Olive" title="b1">W</font> implies <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> ) &amp; <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b3">b</font>,<font color="Olive" title="b2">a</font>)</span>)</span> &amp; ( <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">d</font>) implies <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b3">b</font>,<font color="Olive" title="b2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b5">d</font>,<font color="Olive" title="b4">c</font>) ) &amp; (  for <font color="Olive" title="b7">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b8">x</font> being   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Olive" title="b1">W</font>  ex <font color="Olive" title="b9">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b9">a</font>,<font color="Olive" title="b7">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">x</font> ) &amp; ( <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b3">b</font>,<font color="Olive" title="b2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b4">c</font>,<font color="Olive" title="b2">a</font>) implies <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c</font> ) &amp; ( <font color="Olive" title="b2">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c</font> implies <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b4">c</font>,<font color="Olive" title="b3">b</font>) ) &amp; ( <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b4">c</font>,<font color="Olive" title="b3">b</font>) implies <font color="Olive" title="b2">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c</font> ) &amp; ( <font color="Olive" title="b2">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b5">d</font> implies <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b4">c</font>,<font color="Olive" title="b3">b</font>) ) &amp; ( <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b4">c</font>,<font color="Olive" title="b3">b</font>) implies <font color="Olive" title="b2">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b5">d</font> ) &amp; ( <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">x</font> implies (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> ) &amp; ( (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b6">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> implies <font color="Olive" title="b1">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">x</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">W</font> be   <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</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">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b5">x</font> being   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</font> holds <br/> ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c2">W</font> &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c2">W</font> implies <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) &amp; <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>)</span>)</span> &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>) implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b4">d</font>,<font color="Olive" title="b3">c</font>) ) &amp; (  for <font color="Olive" title="b6">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b7">x</font> being   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</font>  ex <font color="Olive" title="b8">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b8">a</font>,<font color="Olive" title="b6">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">x</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b3">c</font>,<font color="Olive" title="b1">a</font>) implies <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> ) &amp; ( <font color="Olive" title="b1">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font>) implies <font color="Olive" title="b1">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> ) &amp; ( <font color="Olive" title="b1">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b4">d</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font>) implies <font color="Olive" title="b1">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b4">d</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">x</font> implies (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) &amp; ( (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">x</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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">  for <font color="Olive" title="b1">x</font> being   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</font> holds <br/> ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c2">W</font> &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c2">W</font> implies <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) &amp; <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>)</span>)</span> &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>) implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c5">c</font>) ) &amp; (  for <font color="Olive" title="b2">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b3">x</font> being   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</font>  ex <font color="Olive" title="b4">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b4">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>) implies <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> implies (<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) &amp; ( (<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">x</font> be   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</font>; <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="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c2">W</font> &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c2">W</font> implies <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) &amp; <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>)</span>)</span> &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>) implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c5">c</font>) ) &amp; (  for <font color="Olive" title="b1">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b2">x</font> being   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</font>  ex <font color="Olive" title="b3">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>) implies <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> implies (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) &amp; ( (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> ) )</span><br/>













<span class="kw">set </span><font color="Maroon" title="c8">w</font> =  the <a href="midsp_2.html#U2" title="MIDSP_2:sel.2">function</a> of <font color="Maroon" title="c2">W</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">G</font> =  the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font>;<br/>
<a NAME="E1:58"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 the <a href="midsp_2.html#U2" title="MIDSP_2:sel.2">function</a> of <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#R2" title="MIDSP_2:pred.2">is_atlas_of</a>  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="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#D12" target="_self" title="MIDSP_2:def.12">Def12</a></span>;<br/>
<a NAME="E2:58"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
(  the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> is  <a href="midsp_2.html#V1" title="MIDSP_2:attr.1">midpoint_operator</a>  &amp;  the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#D12" target="_self" title="MIDSP_2:def.12">Def12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:58"/>
<font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="midsp_2.html#T2" target="_self" title="MIDSP_2:th.2">Th4</a></span>; <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="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c2">W</font> implies <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) &amp; <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>)</span>)</span> &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>) implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c5">c</font>) ) &amp; (  for <font color="Olive" title="b1">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b2">x</font> being   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</font>  ex <font color="Olive" title="b3">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>) implies <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> implies (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) &amp; ( (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:58"/>
( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c2">W</font> implies <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="midsp_2.html#T3" target="_self" title="MIDSP_2:th.3">Th5</a></span>; <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="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>)</span>)</span> &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>) implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c5">c</font>) ) &amp; (  for <font color="Olive" title="b1">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b2">x</font> being   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</font>  ex <font color="Olive" title="b3">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>) implies <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> implies (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) &amp; ( (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:58"/>
<font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="midsp_2.html#T4" target="_self" title="MIDSP_2:th.4">Th6</a></span>; <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="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>) implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c5">c</font>) ) &amp; (  for <font color="Olive" title="b1">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b2">x</font> being   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</font>  ex <font color="Olive" title="b3">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>) implies <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> implies (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) &amp; ( (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E6:58"/>
( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>) implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c5">c</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="midsp_2.html#T5" target="_self" title="MIDSP_2:th.5">Th7</a></span>; <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">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b2">x</font> being   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</font>  ex <font color="Olive" title="b3">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>) implies <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> implies (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) &amp; ( (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E7:58"/>
 for <font color="Olive" title="b1">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/>  for <font color="Olive" title="b2">x</font> being   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</font>  ex <font color="Olive" title="b3">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font>
  <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="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>) implies <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> implies (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) &amp; ( (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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">  for <font color="Olive" title="b1">x</font> being   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</font>  ex <font color="Olive" title="b2">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b2">a</font>,<font color="Maroon" title="c10">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">x</font> be   <a href="midsp_2.html#NM2" title="MIDSP_2:NM.2">Vector</a> of <font color="Maroon" title="c2">W</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">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c10">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font></span><br/>



<span class="kw">consider </span><font color="Maroon" title="c12">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><br/><a NAME="E2:58_1"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
 the <a href="midsp_2.html#U2" title="MIDSP_2:sel.2">function</a> of <font color="Maroon" title="c2">W</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c10">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="midsp_2.html#T6" target="_self" title="MIDSP_2:th.6">Th8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">a</font>
; <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="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c10">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font></span><br/>

<span class="kw">thus </span><a NAME="E3:58_1"/>
<font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c10">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E8:58"/>
( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>) implies <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="midsp_2.html#T7" target="_self" title="MIDSP_2:th.7">Th9</a></span>; <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="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> ) &amp; ( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> implies (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) &amp; ( (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> ) )</span><br/>

<a NAME="E9:58"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
<font color="Maroon" title="c1">M</font>, the <a href="midsp_2.html#U1" title="MIDSP_2:sel.1">algebra</a> of <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a>  the <a href="midsp_2.html#U2" title="MIDSP_2:sel.2">function</a> of <font color="Maroon" title="c2">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#D12" target="_self" title="MIDSP_2:def.12">Def12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:58"/>
( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> iff <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#D2" target="_self" title="MIDSP_2:def.2">Def2</a></span>; <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="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) implies <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> ) &amp; ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> implies (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) &amp; ( (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E11:58"/>
( <font color="Maroon" title="c3">a</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c6">d</font> iff <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E9:58"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="midsp_2.html#T13" target="_self" title="MIDSP_2:th.13">Th16</a></span>; <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="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> implies (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) &amp; ( (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E12:58"/>
( ( <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> implies (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) &amp; ( (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">x</font>) <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c2">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="midsp_2.html#D4" target="_self" title="MIDSP_2:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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