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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L1">LoopStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green">E4</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V1">commutative</a>
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green">E5</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V2">associative</a>
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green">E6</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:4"/><i><font color="Green">E7</font></i>: 
 <a href="vectsp_1.html#K7">comp</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseqop.html#R1">is_an_inverseOp_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="prvect_1.html#E1:4"><i><font color="Green">E4</font></i></a>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="prvect_1.html#E2:4"><i><font color="Green">E5</font></i></a>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
;<br/><b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>1</sub></font>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="prvect_1.html#E3:4"><i><font color="Green">E6</font></i></a>, <a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>
;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
 <a href="binop_1.html#K3">the_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="prvect_1.html#E4:4"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finseqop.html#D1">FINSEQOP:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="prvect_1.html#E3:4"><i><font color="Green">E6</font></i></a>, <a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:4"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="vectsp_1.html#NM1">AbGroup</a>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T7">VECTSP_1:7</a></i>;<br/>


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