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<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</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">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font><br/>  for <font color="Olive" title="b2">K</font> being  <font color="Olive">b<sub>1</sub></font> <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a>  <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font><br/>  for <font color="Olive" title="b3">BE</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Olive" title="b2">K</font>,<font color="Olive" title="b1">E</font>)</span>)</span><br/>  for <font color="Olive" title="b4">BF</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Olive" title="b2">K</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>  st <font color="Olive" title="b4">BF</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">BE</font> holds <br/> for <font color="Olive" title="b5">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b4">BF</font> holds  <font color="Olive" title="b5">l</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b3">BE</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</font> be   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</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">K</font> being  <font color="Maroon" title="c2">E</font> <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a>  <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font><br/>  for <font color="Olive" title="b2">BE</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Olive" title="b1">K</font>,<font color="Maroon" title="c2">E</font>)</span>)</span><br/>  for <font color="Olive" title="b3">BF</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Olive" title="b1">K</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>  st <font color="Olive" title="b3">BF</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">BE</font> holds <br/> for <font color="Olive" title="b4">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b3">BF</font> holds  <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="b2">BE</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">K</font> be  <font color="Maroon" title="c2">E</font> <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a>  <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</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">BE</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">E</font>)</span>)</span><br/>  for <font color="Olive" title="b2">BF</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>  st <font color="Olive" title="b2">BF</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">BE</font> holds <br/> for <font color="Olive" title="b3">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b2">BF</font> holds  <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="b1">BE</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">BE</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">E</font>)</span>)</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">  for <font color="Olive" title="b1">BF</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>  st <font color="Olive" title="b1">BF</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">BE</font> holds <br/> for <font color="Olive" title="b2">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b1">BF</font> holds  <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="Maroon" title="c4">BE</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">BF</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">F</font>)</span>)</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"> ( <font color="Maroon" title="c5">BF</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">BE</font> implies  for <font color="Olive" title="b1">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c5">BF</font> holds  <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">BE</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:45"/><span class="lab"><font color="Green" title="E21">AS</font></span>: 
<font color="Maroon" title="c5">BF</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">BE</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#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c5">BF</font> holds  <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">BE</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">l</font> be    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c5">BF</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="c6">l</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">BE</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">l1</font> = <font color="Maroon" title="c6">l</font> as    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of  <a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">E</font>) <span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#T16" target="_self" onmouseover="rs('field_7/T16')" onmouseout="rh()">sp0</a></span>;<br/>
<span class="lab"><font color="Green" title="E22">H0</font></span>:  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> = 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D6" onmouseover="rs('field_4/D6')" onmouseout="rh()">FIELD_4:def 6</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">E</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D6" onmouseover="rs('field_4/D6')" onmouseout="rh()">FIELD_4:def 6</a></span>
;<br/>
<a NAME="E4:45"/><span class="lab"><font color="Green" title="E23">H1</font></span>: 
( <font color="Maroon" title="c1">F</font> is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of <font color="Maroon" title="c2">E</font> &amp; <font color="Maroon" title="c2">E</font> is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of <font color="Maroon" title="c3">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D1" onmouseover="rs('field_4/D1')" onmouseout="rh()">FIELD_4:def 1</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2"><span class="kw">now </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">  for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c7">l1</font> holds <br/><font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">BE</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c7">l1</font> implies <font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">BE</font> )</span><br/><span class="kw">assume </span><a NAME="E1:45_2"/>
<font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c7">l1</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="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">BE</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">v1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c2">E</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:45_2"/><span class="lab"><font color="Green" title="E24">A</font></span>: 
( <font color="Maroon" title="c8">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c7">l1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#T1" onmouseover="rs('vectsp_6/T1')" onmouseout="rh()">VECTSP_6:1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">v</font> = <font color="Maroon" title="c9">v1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E22">H0</font></span></a></span>;<br/><a NAME="E5:45_2"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E23">H1</font></span></a>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>;<br/><a NAME="E6:45_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">B</font></span>: 
<font color="Maroon" title="c10">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c6">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_2"><span class="lab"><font color="Green" title="E24">A</font></span></a>, <a class="ref" href="vectsp_6.html#T1" onmouseover="rs('vectsp_6/T1')" onmouseout="rh()">VECTSP_6:1</a></span>;<br/><a NAME="E7:45_2"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c6">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">BF</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#D4" onmouseover="rs('vectsp_6/D4')" onmouseout="rh()">VECTSP_6:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:45_2"/>
<font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">BE</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_2"><span class="lab"><font color="Green" title="E25">B</font></span></a>, <a class="txt" href="#E3:45_2"><span class="lab"><font color="Green" title="E24">A</font></span></a>, <a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E21">AS</font></span></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/></div><span class="kw">end;</span></div>
<a NAME="E6:45"/><span class="kw">then </span>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c7">l1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">BE</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E7:45"/>
<font color="Maroon" title="c6">l</font> is    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c4">BE</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#D4" onmouseover="rs('vectsp_6/D4')" onmouseout="rh()">VECTSP_6:def 4</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/>


</div>
