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<b>let </b><font color="Maroon" title="c1">F</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font><br/> for <font color="Olive" title="b2">W</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">V</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Olive" title="b1">V</font>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Olive" title="b3">f</font> holds <br/> <a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b2">W</font> is <a href="hahnban1.html#V2" title="HAHNBAN1:attr.2">homogeneous</a></span><br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c2">V</font>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Olive" title="b2">f</font> holds <br/> <a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">W</font> is <a href="hahnban1.html#V2" title="HAHNBAN1:attr.2">homogeneous</a></span><br/><b>let </b><font color="Maroon" title="c3">W</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c2">V</font>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Olive" title="b1">f</font> holds <br/> <a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">W</font> is <a href="hahnban1.html#V2" title="HAHNBAN1:attr.2">homogeneous</a></span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c2">V</font>; <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">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c4">f</font> implies  <a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">W</font> is <a href="hahnban1.html#V2" title="HAHNBAN1:attr.2">homogeneous</a> )</span><br/>







<b>assume </b><a NAME="E1:69"/><i><font color="Green" title="E37">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">W</font> is <a href="hahnban1.html#V2" title="HAHNBAN1:attr.2">homogeneous</a></span><br/>

<b>set </b><font color="Maroon" title="c5">qf</font> =  <a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">W</font>;<br/>
<b>set </b><font color="Maroon" title="c6">vq</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">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" title="c7">A</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Scalar</a> of  holds  <span class="p1">(<span class="default"><a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c8">r</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Scalar</a> of ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font></span>)</span></span><br/><b>set </b><font color="Maroon" title="c9">C</font> =  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>;<br/><a NAME="E1:69_1"/><i><font color="Green" title="E38">A2</font></i>: 
 <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#D6" target="_self" title="VECTSP10:def.6">Def6</a></i>;<br/><a NAME="E2:69_1"/><b>then </b>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">aa</font> being    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of <font color="Maroon" title="c3">W</font><b> such that </b><br/><a NAME="E4:69_1"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">aa</font>
 ;<br/><b>consider </b><font color="Maroon" title="c11">a</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E6:69_1"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c10">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6" title="VECTSP_4:def.6">VECTSP_4:def 6</a></i>;<br/><font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">A</font> = 
the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c7">A</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D24" title="VECTSP_1:def.24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c7">A</font>
<b>by </b><i><a class="ref" href="vectsp10.html#D6" target="_self" title="VECTSP10:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <font color="Maroon" title="c3">W</font>
<b>by </b><i><a class="txt" href="vectsp10.html#E1:69_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="vectsp10.html#E4:69_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="vectsp10.html#E6:69_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="vectsp10.html#D5" target="_self" title="VECTSP10:def.5">Def5</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">A</font></span>)</span> = 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp10.html#E1:69"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="vectsp10.html#D12" target="_self" title="VECTSP10:def.12">Def12</a></i>
<br/>.= 
<font color="Maroon" title="c8">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D12" title="HAHNBAN1:def.12">HAHNBAN1:def 12</a></i>
<br/>.= 
<font color="Maroon" title="c8">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">A</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp10.html#E1:69"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="vectsp10.html#E4:69_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="vectsp10.html#E6:69_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="vectsp10.html#D12" target="_self" title="VECTSP10:def.12">Def12</a></i>
;<br/> <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="E3:69"/>
 <a href="vectsp10.html#K10" title="VECTSP10:func.10">QFunctional</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">W</font> is <a href="hahnban1.html#V2" title="HAHNBAN1:attr.2">homogeneous</a>
 <b>by </b><i><a class="ref" href="hahnban1.html#D12" title="HAHNBAN1:def.12">HAHNBAN1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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