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<b>thus </b><a NAME="E1:36_1_1"/>
( <font color="Maroon" title="c1">IT</font> is <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">well-unital</a> implies the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <font color="Maroon" title="c1">IT</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">IT</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1_1_1"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c1">IT</font> is <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">well-unital</a>
 ;<br/>

<b>thus </b><a NAME="E2:36_1_1_1"/>
the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <font color="Maroon" title="c1">IT</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">IT</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D7" title="BINOP_1:def.7">BINOP_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">IT</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D16" title="BINOP_1:def.16">BINOP_1:def 16</a><br/>

<b>thus </b>the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">IT</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <font color="Maroon" title="c1">IT</font>,<font color="Maroon" title="c2">a</font> = 
<span class="p1">(<span class="default"><a href="vectsp_1.html#K2" title="VECTSP_1:func.2">1_</a> <font color="Maroon" title="c1">IT</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D9" title="VECTSP_1:def.9">VECTSP_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:36_1_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="vectsp_2.html#D2" title="VECTSP_2:def.2">VECTSP_2:def 2</a></i>
;<br/>


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

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">IT</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D17" title="BINOP_1:def.17">BINOP_1:def 17</a><br/>

<b>thus </b>the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">IT</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">a</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <font color="Maroon" title="c1">IT</font> = 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="vectsp_1.html#K2" title="VECTSP_1:func.2">1_</a> <font color="Maroon" title="c1">IT</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D9" title="VECTSP_1:def.9">VECTSP_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:36_1_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="vectsp_2.html#D2" title="VECTSP_2:def.2">VECTSP_2:def 2</a></i>
;<br/>


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

<b>assume </b><a NAME="E2:36_1_1"/><i><font color="Green" title="E23">A2</font></i>: 
the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <font color="Maroon" title="c1">IT</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">IT</font>
 ;<br/>

<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 <font color="Maroon" title="c1">IT</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_2.html#D2" title="VECTSP_2:def.2">VECTSP_2:def 2</a><br/>

<a NAME="E3:36_1_1"/><i><font color="Green" title="E24">A3</font></i>: 
the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <font color="Maroon" title="c1">IT</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">IT</font>
 
<b>by </b><i><a class="txt" href="monoid_0.html#E2:36_1_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="binop_1.html#D7" title="BINOP_1:def.7">BINOP_1:def 7</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="vectsp_1.html#K2" title="VECTSP_1:func.2">1_</a> <font color="Maroon" title="c1">IT</font></span>)</span> = 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">IT</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">x</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <font color="Maroon" title="c1">IT</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D9" title="VECTSP_1:def.9">VECTSP_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E3:36_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="binop_1.html#D6" title="BINOP_1:def.6">BINOP_1:def 6</a></i>
;<br/>

<a NAME="E5:36_1_1"/><i><font color="Green" title="E25">A4</font></i>: 
the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <font color="Maroon" title="c1">IT</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">IT</font>
 
<b>by </b><i><a class="txt" href="monoid_0.html#E2:36_1_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="binop_1.html#D7" title="BINOP_1:def.7">BINOP_1:def 7</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="vectsp_1.html#K2" title="VECTSP_1:func.2">1_</a> <font color="Maroon" title="c1">IT</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font> = 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">IT</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <font color="Maroon" title="c1">IT</font>,<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D9" title="VECTSP_1:def.9">VECTSP_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E5:36_1_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="binop_1.html#D5" title="BINOP_1:def.5">BINOP_1:def 5</a></i>
;<br/>


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