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<b>thus </b><a NAME="E1:58_1_2"/>
for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font> being <a href="algstr_0.html#V8" title="ALGSTR_0:attr.8">strict</a> <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font> &amp; ( for <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b1">G1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Olive" title="b4">B</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font> &amp; ( for <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b2">G2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Olive" title="b4">B</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b2">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font> holds <br/><font color="Olive" title="b1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G2</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="58_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">G1</font>, <font color="Maroon" title="c4">G2</font> be  <a href="algstr_0.html#V8" title="ALGSTR_0:attr.8">strict</a> <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font> &amp; ( for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">G1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Olive" title="b2">B</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font> &amp; ( for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c4">G2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Olive" title="b2">B</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font> implies <font color="Maroon" title="c3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:58_1_2_1"/><i><font color="Green" title="E36">A7</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font>
 <b>and </b><br/><a NAME="E2:58_1_2_1"/><i><font color="Green" title="E37">A8</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">G1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Olive" title="b2">B</font>
 <b>and </b><br/><a NAME="E3:58_1_2_1"/><i><font color="Green" title="E38">A9</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E4:58_1_2_1"/><i><font color="Green" title="E39">A10</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font>
 <b>and </b><br/><a NAME="E5:58_1_2_1"/><i><font color="Green" title="E40">A11</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c4">G2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Olive" title="b2">B</font>
 <b>and </b><br/><a NAME="E6:58_1_2_1"/><i><font color="Green" title="E41">A12</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</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">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">G1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c4">G2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><br/><b>reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">B</font> = <font color="Maroon" title="c6">b</font> as   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> <b>by </b><i><a class="txt" href="matrix_1.html#E1:58_1_2_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="matrix_1.html#T31" target="_self" title="MATRIX_1:th.31">Th31</a></i>;<br/><b>thus </b>the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">G1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> = 
<font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c8">B</font>
<b>by </b><i><a class="txt" href="matrix_1.html#E2:58_1_2_1"><i><font color="Green" title="E37">A8</font></i></a></i>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c4">G2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="txt" href="matrix_1.html#E5:58_1_2_1"><i><font color="Green" title="E40">A11</font></i></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>hence </b><a NAME="E8:58_1_2_1"/>
<font color="Maroon" title="c3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G2</font>
 <b>by </b><i><a class="txt" href="matrix_1.html#E1:58_1_2_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="matrix_1.html#E3:58_1_2_1"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="matrix_1.html#E4:58_1_2_1"><i><font color="Green" title="E39">A10</font></i></a>, <a class="txt" href="matrix_1.html#E6:58_1_2_1"><i><font color="Green" title="E41">A12</font></i></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</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>
