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<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="midsp_1.html#NM1" title="MIDSP_1:NM.1">MidSp</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> is   <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> iff <font color="Olive" title="b1">a</font> is    <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font> ) ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_1.html#K8" title="MIDSP_1:func.8">ID</a> <font color="Maroon" title="c1">M</font> &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y</font> holds <br/><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_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">x</font> <a href="midsp_1.html#K9" title="MIDSP_1:func.9">+</a> <font color="Olive" title="b4">y</font> ) )</span><br/>

<b>set </b><font color="Maroon" title="c2">G</font> =  <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font>;<br/>
<b>thus </b><a NAME="E1:19"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> is   <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> iff <font color="Olive" title="b1">a</font> is    <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_1.html#K8" title="MIDSP_1:func.8">ID</a> <font color="Maroon" title="c1">M</font> &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y</font> holds <br/><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_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">x</font> <a href="midsp_1.html#K9" title="MIDSP_1:func.9">+</a> <font color="Olive" title="b4">y</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> is   <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> iff <font color="Maroon" title="c3">a</font> is    <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font> )</span><br/>

<a NAME="E1:19_1"/>
( <font color="Maroon" title="c3">a</font> is   <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> iff <font color="Maroon" title="c3">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="midsp_1.html#K12" title="MIDSP_1:func.12">setvect</a> <font color="Maroon" title="c1">M</font> )
 
<b>by </b><i><a class="ref" href="midsp_1.html#T82" title="MIDSP_1:th.82">MIDSP_1:82</a></i>;<br/>
<b>hence </b><a NAME="E2:19_1"/>
( <font color="Maroon" title="c3">a</font> is   <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> iff <font color="Maroon" title="c3">a</font> is    <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font> )
 <b>by </b><i><a class="ref" href="midsp_1.html#T71" title="MIDSP_1:th.71">MIDSP_1:71</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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 href="midsp_1.html#K16" title="MIDSP_1:func.16">zerovect</a> <font color="Maroon" title="c1">M</font>
<b>by </b><i><a class="ref" href="midsp_1.html#T85" title="MIDSP_1:th.85">MIDSP_1:85</a></i>
<br/>.= 
 <a href="midsp_1.html#K8" title="MIDSP_1:func.8">ID</a> <font color="Maroon" title="c1">M</font>
<b>by </b><i><a class="ref" href="midsp_1.html#D17" title="MIDSP_1:def.17">MIDSP_1:def 17</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y</font> holds <br/><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_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">x</font> <a href="midsp_1.html#K9" title="MIDSP_1:func.9">+</a> <font color="Olive" title="b4">y</font></span><br/>

<b>thus </b><a NAME="E3:19"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y</font> holds <br/><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_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">x</font> <a href="midsp_1.html#K9" title="MIDSP_1:func.9">+</a> <font color="Olive" title="b4">y</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font>  st <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="midsp_1.html#K9" title="MIDSP_1:func.9">+</a> <font color="Olive" title="b2">y</font></span><br/><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> implies <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K3" title="RLVECT_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">x</font> <a href="midsp_1.html#K9" title="MIDSP_1:func.9">+</a> <font color="Maroon" title="c6">y</font> )</span><br/>



<b>assume </b><a NAME="E1:19_3"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K3" title="RLVECT_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">x</font> <a href="midsp_1.html#K9" title="MIDSP_1:func.9">+</a> <font color="Maroon" title="c6">y</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">xx</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">yy</font> = <font color="Maroon" title="c6">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="midsp_1.html#K12" title="MIDSP_1:func.12">setvect</a> <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="ref" href="midsp_1.html#T71" title="MIDSP_1:th.71">MIDSP_1:71</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">b</font> = 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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 href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="midsp_1.html#K14" title="MIDSP_1:func.14">addvect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">xx</font>,<font color="Maroon" title="c8">yy</font>
<b>by </b><i><a class="txt" href="#E1:19_3"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="midsp_1.html#T83" title="MIDSP_1:th.83">MIDSP_1:83</a></i>
<br/>.= 
<font color="Maroon" title="c7">xx</font> <a href="midsp_1.html#K13" title="MIDSP_1:func.13">+</a> <font color="Maroon" title="c8">yy</font>
<b>by </b><i><a class="ref" href="midsp_1.html#D15" title="MIDSP_1:def.15">MIDSP_1:def 15</a></i>
<br/>.= 
<font color="Maroon" title="c5">x</font> <a href="midsp_1.html#K9" title="MIDSP_1:func.9">+</a> <font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="ref" href="midsp_1.html#D14" title="MIDSP_1:def.14">MIDSP_1:def 14</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div>
