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<b>let </b><font color="Maroon">F</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">V</font>;<br/>







<b>assume </b><a NAME="E1:69"/><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a>  <a href="vectsp10.html#K8" target="_self">ker</a> <font color="Maroon">f</font>
 ;<br/>

<b>set </b><font color="Maroon">qf</font> =  <a href="vectsp10.html#K10" target="_self">QFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">W</font>;<br/>
<b>set </b><font color="Maroon">vq</font> =  <a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="vectsp_1.html#NM4">Scalar</a> of ;<br/><b>set </b><font color="Maroon">C</font> =  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/><a NAME="E1:69_1"/><i><font color="Green">E35</font></i>: 
 <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E2:69_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">V</font>,<font color="Maroon">W</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">aa</font> being    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E4:69_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">aa</font>
 ;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:69_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">aa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">A</font> = 
the <a href="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" target="_self">VectQuot</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">r</font>,<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">V</font>,<font color="Maroon">W</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">A</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">W</font>
<b>by </b><i>, , , </i>
;<br/><b>hence </b><span class="p1">(<span class="default"><a href="vectsp10.html#K10" target="_self">QFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">W</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">A</font></span>)</span> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp10.html#T2" target="_self">Th2</a>, <a class="txt" href="vectsp10.html#E1:69_1"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D12">HAHNBAN1:def 12</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp10.html#K10" target="_self">QFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">W</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp10.html#T2" target="_self">Th2</a>, , , <a class="txt" href="vectsp10.html#E1:69_1"><i><font color="Green">E35</font></i></a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:69"/>
 <a href="vectsp10.html#K10" target="_self">QFunctional</a> <font color="Maroon">f</font>,<font color="Maroon">W</font> is <a href="hahnban1.html#V2">homogeneous</a>
 <b>by </b><i><a class="ref" href="hahnban1.html#D12">HAHNBAN1:def 12</a></i>;<br/>


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