<?xml version="1.0"?>
<div class="add">

<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">  <a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</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>, <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">w</font> =  <a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font>;<br/>
<a NAME="E1:33"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
 for <font color="Olive" title="b1">p</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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span>)</span>  ex <font color="Olive" title="b3">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">p</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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span>)</span>  ex <font color="Olive" title="b2">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b2">q</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="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span>)</span>; <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">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c5">xx</font> = <font color="Maroon" title="c4">x</font> as    <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T15" target="_self" title="MIDSP_2:th.15">Th18</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">q</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="E3:33_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c5">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">q</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_1.html#T35" title="MIDSP_1:th.35">MIDSP_1:35</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">q</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font></span><br/>

<span class="kw">thus </span><a NAME="E4:33_1"/>
<span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="midsp_2.html#D8" target="_self" title="MIDSP_2:def.8">Def8</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>
<a NAME="E2:33"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
 for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>  st <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">r</font>) holds <br/><font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c5">r</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"> ( <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>) implies <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:33_2"/>
<span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</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="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font></span><br/>

<span class="kw">then </span> <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>) = 
<span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#D8" target="_self" title="MIDSP_2:def.8">Def8</a></span>
<br/>.= 
 <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#D8" target="_self" title="MIDSP_2:def.8">Def8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:33_2"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_1.html#T39" title="MIDSP_1:th.39">MIDSP_1:39</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>
<a NAME="E3:33"/>
 for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">r</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c5">r</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>)</span><br/>





<a NAME="E1:33_3"/>
( <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>) &amp; <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> (<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#D8" target="_self" title="MIDSP_2:def.8">Def8</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>)</span>)</span> <a href="midsp_1.html#K9" title="MIDSP_1:func.9">+</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> (<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T15" target="_self" title="MIDSP_2:th.15">Th18</a></span>
<br/>.= 
 <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_1.html#T40" title="MIDSP_1:th.40">MIDSP_1:40</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#D8" target="_self" title="MIDSP_2:def.8">Def8</a></span>
;<br/> <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">hence </span><a NAME="E4:33"/>
 <a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</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>, <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="midsp_2.html#D3" target="_self" title="MIDSP_2:def.3">Def3</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>
