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<span class="kw">let </span><font color="Maroon" title="c1">GF</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="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="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="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   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="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>   <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>   <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>   <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</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="vectsp_1.html#L1" title="VECTSP_1:struct.1">ModuleStr</a> over <font color="Maroon" title="c1">GF</font><br/>  for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">GF</font><br/>  for <font color="Olive" title="b3">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b4">L</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b4">L</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b3">A</font> holds <br/><font color="Olive" title="b2">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Olive" title="b4">L</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b3">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</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="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>   <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>   <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>   <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</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="vectsp_1.html#L1" title="VECTSP_1:struct.1">ModuleStr</a> over <font color="Maroon" title="c1">GF</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">GF</font><br/>  for <font color="Olive" title="b2">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b3">L</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b3">L</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b2">A</font> holds <br/><font color="Olive" title="b1">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Olive" title="b3">L</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b2">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">GF</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b2">L</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b2">L</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Olive" title="b2">L</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">L</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font> holds <br/><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Olive" title="b1">L</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">L</font> be    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">L</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font> implies <font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">L</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:37"/>
<font color="Maroon" title="c5">L</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">L</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font></span><br/>

<a NAME="E2:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#D4" target="_self" onmouseover="rs('vectsp_6/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E3:37"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c5">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#T28" target="_self" onmouseover="rs('vectsp_6/T28')" onmouseout="rh()">Th28</a></span>;<br/>
<a NAME="E4:37"/><span class="kw">then </span>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">L</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E24">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:37"/>
<font color="Maroon" title="c3">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c5">L</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#D4" target="_self" onmouseover="rs('vectsp_6/D4')" onmouseout="rh()">Def4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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