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<b>thus </b><a NAME="E1:59_1_2"/>
for <font color="Olive">G1</font>, <font color="Olive">G2</font> being <a href="rlvect_1.html#V1">strict</a> <a href="vectsp_1.html#NM1">AbGroup</a>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">F</font> &amp; ( for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">F</font> holds  the <a href="rlvect_1.html#U1">add</a> of <font color="Olive">G1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">A</font>,<font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Olive">B</font> ) &amp; the <a href="struct_0.html#U2">Zero</a> of <font color="Olive">G1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K11" target="_self">0.</a> <font color="Maroon">F</font>,<font color="Maroon">n</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">F</font> &amp; ( for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">F</font> holds  the <a href="rlvect_1.html#U1">add</a> of <font color="Olive">G2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">A</font>,<font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Olive">B</font> ) &amp; the <a href="struct_0.html#U2">Zero</a> of <font color="Olive">G2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K11" target="_self">0.</a> <font color="Maroon">F</font>,<font color="Maroon">n</font> holds <br/><font color="Olive">G1</font> <a href="hidden.html#R1">=</a> <font color="Olive">G2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G1</font> be  <a href="rlvect_1.html#V1">strict</a> <a href="vectsp_1.html#NM1">AbGroup</a>, <font color="Maroon">G2</font> be  <a href="rlvect_1.html#V1">strict</a> <a href="vectsp_1.html#NM1">AbGroup</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:59_1_2_1"/><i><font color="Green">E64</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E2:59_1_2_1"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">F</font> holds  the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">G1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">A</font>,<font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Olive">B</font>
 <b>and </b><br/><a NAME="E3:59_1_2_1"/><i><font color="Green">E79</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K11" target="_self">0.</a> <font color="Maroon">F</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:59_1_2_1"/><i><font color="Green">E84</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E5:59_1_2_1"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">F</font> holds  the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">G2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">A</font>,<font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Olive">B</font>
 <b>and </b><br/><a NAME="E6:59_1_2_1"/><i><font color="Green">E87</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K11" target="_self">0.</a> <font color="Maroon">F</font>,<font color="Maroon">n</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>;<br/><b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</font>, <font color="Maroon">B</font> = <font color="Maroon">b</font> as   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">F</font> <b>by </b><i>, </i>;<br/><b>thus </b>the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">G1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<font color="Maroon">A</font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Maroon">B</font>
<b>by </b><i/>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">G2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:59_1_2_1"/>
<font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font>
 <b>by </b><i>, , , , <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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


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