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

<b>thus </b><a NAME="E1:26_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> &amp; not  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><br/>

<b>thus </b><a NAME="E2:26_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> &amp; not  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><br/>

<b>thus </b><a NAME="E3:26_1_1"/>
 <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> &amp; not  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> is <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c3">x'</font> = <font color="Maroon" title="c2">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">y</font> =  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">x'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  ;<br/>
<b>take </b><font color="Maroon" title="c5">y'</font> = <font color="Maroon" title="c4">y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> </span><br/>

<b>thus </b><a NAME="E3:26_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> 
 ; <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 NAME="E4:26_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">x</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> &amp; not  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><br/>

<b>thus </b><a NAME="E5:26_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> &amp; not  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><br/>

<b>thus </b><a NAME="E6:26_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> &amp; not  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><br/>

<b>thus </b><a NAME="E7:26_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  holds  <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D19" target="_self" title="VECTSP_1:def.19">VECTSP_1:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> &amp; not  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><br/>

<b>thus </b><a NAME="E8:26_1_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D13" target="_self" title="VECTSP_1:def.13">VECTSP_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp;  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> &amp; not  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><br/>

<b>thus </b><a NAME="E9:26_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font></span>)</span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D18" target="_self" title="VECTSP_1:def.18">VECTSP_1:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> &amp; not  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> )</span><br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D20" target="_self" title="VECTSP_1:def.20">VECTSP_1:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  is <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a></span>
<div class="add"><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  implies  ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  st <font color="Olive" title="b1">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  )</span><br/><b>reconsider </b><font color="Maroon" title="c3">x'</font> = <font color="Maroon" title="c2">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><b>assume </b><a NAME="E2:26_1_1_2"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  st <font color="Olive" title="b1">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> </span><br/><b>reconsider </b><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c3">x'</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  ;<br/><b>take </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">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="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> </span><br/><b>thus </b><a NAME="E4:26_1_1_2"/>
<font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> 
 <b>by </b><i><a class="txt" href="vectsp_1.html#E2:26_1_1_2"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</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 NAME="E11:26_1_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="struct_0.html#D8" title="STRUCT_0:def.8">STRUCT_0:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
