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<b>let </b><font color="Maroon" title="c3">V1</font>, <font color="Maroon" title="c4">V2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_1.html#L4" title="VECTSP_1:struct.4">VectSpStr</a> of <font color="Maroon" title="c1">K</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:43_1_2"/><i><font color="Green" title="E26">A11</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V1</font> iff <font color="Olive" title="b1">x</font> is   <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c2">V</font> )
 <b>and </b><br/><a NAME="E2:43_1_2"/><i><font color="Green" title="E27">A12</font></i>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c2">V</font> holds  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c3">V1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="hahnban1.html#K3" title="HAHNBAN1:func.3">+</a> <font color="Olive" title="b2">g</font>
 <b>and </b><br/><a NAME="E3:43_1_2"/><i><font color="Green" title="E28">A13</font></i>: 
the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hahnban1.html#K7" title="HAHNBAN1:func.7">0Functional</a> <font color="Maroon" title="c2">V</font>
 <b>and </b><br/><a NAME="E4:43_1_2"/><i><font color="Green" title="E29">A14</font></i>: 
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><br/> for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> holds  the <a href="vectsp_1.html#U2" title="VECTSP_1:sel.2">lmult</a> of <font color="Maroon" title="c3">V1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Olive" title="b1">f</font>
 <b>and </b><br/><a NAME="E5:43_1_2"/><i><font color="Green" title="E30">A15</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">V2</font> iff <font color="Olive" title="b1">x</font> is   <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c2">V</font> )
 <b>and </b><br/><a NAME="E6:43_1_2"/><i><font color="Green" title="E31">A16</font></i>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c2">V</font> holds  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c4">V2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="hahnban1.html#K3" title="HAHNBAN1:func.3">+</a> <font color="Olive" title="b2">g</font>
 <b>and </b><br/><a NAME="E7:43_1_2"/><i><font color="Green" title="E32">A17</font></i>: 
the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c4">V2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hahnban1.html#K7" title="HAHNBAN1:func.7">0Functional</a> <font color="Maroon" title="c2">V</font>
 <b>and </b><br/><a NAME="E8:43_1_2"/><i><font color="Green" title="E33">A18</font></i>: 
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><br/> for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> holds  the <a href="vectsp_1.html#U2" title="VECTSP_1:sel.2">lmult</a> of <font color="Maroon" title="c4">V2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Olive" title="b1">f</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:43_1_2_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V1</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">V2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_2_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V1</font>
 ;<br/>

<a NAME="E2:43_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> is   <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="txt" href="hahnban1.html#E1:43_1_2"><i><font color="Green" title="E26">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_2_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">V2</font>
 <b>by </b><i><a class="txt" href="hahnban1.html#E5:43_1_2"><i><font color="Green" title="E30">A15</font></i></a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:43_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">V2</font>
 ;<br/><a NAME="E3:43_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> is   <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="hahnban1.html#E5:43_1_2"><i><font color="Green" title="E30">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E4:43_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V1</font>
 <b>by </b><i><a class="txt" href="hahnban1.html#E1:43_1_2"><i><font color="Green" title="E26">A11</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:43_1_2"/><b>then </b><i><font color="Green" title="E34">A19</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V1</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 <font color="Maroon" title="c4">V2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V1</font>;<br/><b>reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c6">y</font> as   <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="hahnban1.html#E1:43_1_2"><i><font color="Green" title="E26">A11</font></i></a></i>;<br/><b>thus </b>the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c3">V1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> = 
<font color="Maroon" title="c7">f</font> <a href="hahnban1.html#K3" title="HAHNBAN1:func.3">+</a> <font color="Maroon" title="c8">g</font>
<b>by </b><i><a class="txt" href="hahnban1.html#E2:43_1_2"><i><font color="Green" title="E27">A12</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c4">V2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="txt" href="hahnban1.html#E6:43_1_2"><i><font color="Green" title="E31">A16</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E12:43_1_2"/><b>then </b><i><font color="Green" title="E35">A20</font></i>: 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c3">V1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c4">V2</font>
 
<b>by </b><i><a class="txt" href="hahnban1.html#E10:43_1_2"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c6">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V1</font>;<br/><b>reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">x</font> as   <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="hahnban1.html#E1:43_1_2"><i><font color="Green" title="E26">A11</font></i></a></i>;<br/><b>thus </b>the <a href="vectsp_1.html#U2" title="VECTSP_1:sel.2">lmult</a> of <font color="Maroon" title="c3">V1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c5">r</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="hahnban1.html#E4:43_1_2"><i><font color="Green" title="E29">A14</font></i></a></i>
<br/>.= 
the <a href="vectsp_1.html#U2" title="VECTSP_1:sel.2">lmult</a> of <font color="Maroon" title="c4">V2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="hahnban1.html#E8:43_1_2"><i><font color="Green" title="E33">A18</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:43_1_2"/>
<font color="Maroon" title="c3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">V2</font>
 <b>by </b><i><a class="txt" href="hahnban1.html#E3:43_1_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="hahnban1.html#E7:43_1_2"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="hahnban1.html#E10:43_1_2"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="hahnban1.html#E12:43_1_2"><i><font color="Green" title="E35">A20</font></i></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>


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