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<span class="kw">let </span><font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_4.html#K1" title="PRVECT_4:func.1">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K1" title="PRVECT_4:func.1">:]</a></span> iff  ex <font color="Olive" title="b2">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b3">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ex <font color="Olive" title="b4">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Olive" title="b2">x1</font>,<font color="Olive" title="b3">x2</font>,<font color="Olive" title="b4">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> ) ) &amp; (  for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">y2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b5">x3</font>, <font color="Olive" title="b6">y3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds  <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b3">x2</font>,<font color="Olive" title="b5">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Olive" title="b2">y1</font>,<font color="Olive" title="b4">y2</font>,<font color="Olive" title="b6">y3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">x2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b4">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">x3</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b6">y3</font></span>)</span></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1"><a href="prvect_4.html#K1" title="PRVECT_4:func.1">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K1" title="PRVECT_4:func.1">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">G</font></span>)</span></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E1:10"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_4.html#K1" title="PRVECT_4:func.1">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K1" title="PRVECT_4:func.1">:]</a></span> iff  ex <font color="Olive" title="b2">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b3">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ex <font color="Olive" title="b4">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Olive" title="b2">x1</font>,<font color="Olive" title="b3">x2</font>,<font color="Olive" title="b4">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">y2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b5">x3</font>, <font color="Olive" title="b6">y3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds  <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b3">x2</font>,<font color="Olive" title="b5">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Olive" title="b2">y1</font>,<font color="Olive" title="b4">y2</font>,<font color="Olive" title="b6">y3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">x2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b4">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">x3</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b6">y3</font></span>)</span></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1"><a href="prvect_4.html#K1" title="PRVECT_4:func.1">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K1" title="PRVECT_4:func.1">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">G</font></span>)</span></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_4.html#K1" title="PRVECT_4:func.1">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K1" title="PRVECT_4:func.1">:]</a></span> iff  ex <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ex <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ex <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> implies <font color="Maroon" title="c4">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_4.html#K1" title="PRVECT_4:func.1">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K1" title="PRVECT_4:func.1">:]</a></span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:10_1_1"/>
<font color="Maroon" title="c4">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_4.html#K1" title="PRVECT_4:func.1">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K1" title="PRVECT_4:func.1">:]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ex <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x1x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K4" title="PRVECT_3:func.4">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K4" title="PRVECT_3:func.4">:]</a></span>, <font color="Maroon" title="c6">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font><span class="kw"> such that </span><br/><a NAME="E3:10_1_1"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_3.html#K5" title="PRVECT_3:func.5">[</a><span class="default"><font color="Maroon" title="c5">x1x2</font>,<font color="Maroon" title="c6">x3</font></span><a href="prvect_3.html#K5" title="PRVECT_3:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#T7" onmouseover="rs('prvect_3/T7')" onmouseout="rh()">PRVECT_3:7</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c8">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E5:10_1_1"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c5">x1x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_3.html#K5" title="PRVECT_3:func.5">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font></span><a href="prvect_3.html#K5" title="PRVECT_3:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#T7" onmouseover="rs('prvect_3/T7')" onmouseout="rh()">PRVECT_3:7</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c9">x1</font> = <font color="Maroon" title="c7">x1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ex <font color="Olive" title="b2">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span></span><br/><span class="kw">take </span><font color="Maroon" title="c10">x2</font> = <font color="Maroon" title="c8">x2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>,<font color="Olive" title="b1">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span></span><br/><span class="kw">take </span><font color="Maroon" title="c11">x3</font> = <font color="Maroon" title="c6">x3</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span></span><br/><span class="kw">thus </span><a NAME="E6:10_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E5:10_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:10_1"/>
(  ex <font color="Olive" title="b1">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ex <font color="Olive" title="b3">x3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> implies <font color="Maroon" title="c4">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_4.html#K1" title="PRVECT_4:func.1">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K1" title="PRVECT_4:func.1">:]</a></span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1"><a href="prvect_4.html#K1" title="PRVECT_4:func.1">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K1" title="PRVECT_4:func.1">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">G</font></span>)</span></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">y1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x2</font>, <font color="Olive" title="b2">y2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x3</font>, <font color="Olive" title="b4">y3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds  <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Olive" title="b2">y2</font>,<font color="Olive" title="b4">y3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">x2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">x3</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b4">y3</font></span>)</span></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">x2</font>, <font color="Maroon" title="c7">y2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x3</font>, <font color="Olive" title="b2">y3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds  <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Olive" title="b1">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c7">y2</font>,<font color="Olive" title="b2">y3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">x3</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y3</font></span>)</span></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">x3</font>, <font color="Maroon" title="c9">y3</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c8">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c9">y3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">x3</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">y3</font></span>)</span></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span></span><br/><a NAME="E1:10_2"/>
<span class="p1"><a href="prvect_3.html#K5" title="PRVECT_3:func.5">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">x2</font></span><a href="prvect_3.html#K5" title="PRVECT_3:func.5">]</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="prvect_3.html#K5" title="PRVECT_3:func.5">[</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c7">y2</font></span><a href="prvect_3.html#K5" title="PRVECT_3:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_3.html#K5" title="PRVECT_3:func.5">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y2</font></span>)</span></span><a href="prvect_3.html#K5" title="PRVECT_3:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#T7" onmouseover="rs('prvect_3/T7')" onmouseout="rh()">PRVECT_3:7</a></span>;<br/><span class="kw">hence </span><a NAME="E2:10_2"/>
<span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c8">x3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c9">y3</font></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">x3</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">y3</font></span>)</span></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#T7" onmouseover="rs('prvect_3/T7')" onmouseout="rh()">PRVECT_3:7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:10"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1"><a href="prvect_4.html#K1" title="PRVECT_4:func.1">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K1" title="PRVECT_4:func.1">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="prvect_4.html#K2" title="PRVECT_4:func.2">[</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">E</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">G</font></span>)</span></span><a href="prvect_4.html#K2" title="PRVECT_4:func.2">]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
