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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="Maroon" title="c1">F</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   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <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">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span><br/>  for <font color="Olive" title="b4">BK</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <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="b5">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of  <a href="field_7.html#K7" title="FIELD_7:func.7">Base</a> (<font color="Olive" title="b3">BE</font>,<font color="Olive" title="b4">BK</font>)  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Olive" title="b5">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Olive" title="b5">l</font></span>)</span></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="c1">F</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   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <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="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span><br/>  for <font color="Olive" title="b3">BK</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <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="b4">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of  <a href="field_7.html#K7" title="FIELD_7:func.7">Base</a> (<font color="Olive" title="b2">BE</font>,<font color="Olive" title="b3">BK</font>)  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Olive" title="b4">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Olive" title="b4">l</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">K</font> be  <font color="Maroon" title="c1">F</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   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <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="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span><br/>  for <font color="Olive" title="b2">BK</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <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="b3">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of  <a href="field_7.html#K7" title="FIELD_7:func.7">Base</a> (<font color="Olive" title="b1">BE</font>,<font color="Olive" title="b2">BK</font>)  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Olive" title="b3">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Olive" title="b3">l</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">BE</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <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="c2">E</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">  for <font color="Olive" title="b1">BK</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <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">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of  <a href="field_7.html#K7" title="FIELD_7:func.7">Base</a> (<font color="Maroon" title="c4">BE</font>,<font color="Olive" title="b1">BK</font>)  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Olive" title="b2">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Olive" title="b2">l</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">BK</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <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">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of  <a href="field_7.html#K7" title="FIELD_7:func.7">Base</a> (<font color="Maroon" title="c4">BE</font>,<font color="Maroon" title="c5">BK</font>)  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Olive" title="b1">l</font></span>)</span></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  <a href="field_7.html#K7" title="FIELD_7:func.7">Base</a> (<font color="Maroon" title="c4">BE</font>,<font color="Maroon" title="c5">BK</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"> (  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> )</span><br/>











<a NAME="E1:62"/><span class="lab"><font color="Green" title="E33">H1</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="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <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="c2">E</font> &amp;  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> <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="c3">K</font> &amp;  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> <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="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/>
<a NAME="E2:62"/><span class="lab"><font color="Green" title="E34">H2</font></span>: 
( <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> &amp; <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> )
 
<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/>
<a NAME="E3:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">H3</font></span>: 
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:62"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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">  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:62"/><span class="lab"><font color="Green" title="E36">D1</font></span>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T42" onmouseover="rs('card_2/T42')" onmouseout="rh()">CARD_2:42</a></span>;<br/>
<a NAME="E7:62"/><span class="lab"><font color="Green" title="E37">D2</font></span>: 
<font color="Maroon" title="c7">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="c6">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E36">D1</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">v</font> = <font color="Maroon" title="c7">o</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> ;<br/>
<a NAME="E9:62"/>
 <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>  <a href="field_7.html#K7" title="FIELD_7:func.7">Base</a> (<font color="Maroon" title="c4">BE</font>,<font color="Maroon" title="c5">BK</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/>
<a NAME="E10:62"/><span class="kw">then </span>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="field_7.html#K7" title="FIELD_7:func.7">Base</a> (<font color="Maroon" title="c4">BE</font>,<font color="Maroon" title="c5">BK</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E37">D2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font><span class="kw"> such that </span><br/><a NAME="E12:62"/><span class="lab"><font color="Green" title="E38">D3</font></span>: 
( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font> &amp; <font color="Maroon" title="c9">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">BE</font> &amp; <font color="Maroon" title="c10">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">BK</font> )
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">vab</font> = <font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</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="#E7:62"><span class="lab"><font color="Green" title="E37">D2</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E38">D3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">vb</font> = <font color="Maroon" title="c10">b</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="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/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">va</font> = <font color="Maroon" title="c9">a</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="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E38">D3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">aE</font> = <font color="Maroon" title="c13">va</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</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/>
<a NAME="E17:62"/>
<font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font> is   <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="ref" href="field_4.html#D6" onmouseover="rs('field_4/D6')" onmouseout="rh()">FIELD_4:def 6</a></span>;<br/>
<a NAME="E18:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">B0</font></span>: 
<font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">labE</font> = <font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E35">H3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">lab</font> = <font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:62"><span class="lab"><font color="Green" title="E39">B0</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E35">H3</font></span></a></span>;<br/>
<a NAME="E21:62"/>
<span class="p1"><a href="field_7.html#K2" title="FIELD_7:func.2">{</a><span class="default"><font color="Maroon" title="c13">va</font></span><a href="field_7.html#K2" title="FIELD_7:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">BE</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E38">D3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E22:62"/><span class="kw">then </span>
<font color="Maroon" title="c9">a</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> <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#T3" onmouseover="rs('vectsp_7/T3')" onmouseout="rh()">VECTSP_7:3</a>, <a class="ref" href="vectsp_7.html#T1" onmouseover="rs('vectsp_7/T1')" onmouseout="rh()">VECTSP_7:1</a></span>;<br/>
<a NAME="E23:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">D7a</font></span>: 
<font color="Maroon" title="c9">a</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="field_4.html#D6" onmouseover="rs('field_4/D6')" onmouseout="rh()">FIELD_4:def 6</a></span>;<br/>
<a NAME="E24:62"/>
<span class="p1"><a href="field_7.html#K2" title="FIELD_7:func.2">{</a><span class="default"><font color="Maroon" title="c12">vb</font></span><a href="field_7.html#K2" title="FIELD_7:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">BK</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E38">D3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E25:62"/><span class="kw">then </span>
<font color="Maroon" title="c10">b</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> <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="vectsp_7.html#T3" onmouseover="rs('vectsp_7/T3')" onmouseout="rh()">VECTSP_7:3</a>, <a class="ref" href="vectsp_7.html#T1" onmouseover="rs('vectsp_7/T1')" onmouseout="rh()">VECTSP_7:1</a></span>;<br/>
<a NAME="E26:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">D6a</font></span>: 
<font color="Maroon" title="c10">b</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="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/>
<a NAME="E27:62"/><span class="lab"><font color="Green" title="E42">D6</font></span>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_1"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E43">C1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_1_1"><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">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> holds <br/><font color="Olive" title="b1">u</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> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">u</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="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> implies <font color="Maroon" title="c17">u</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> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:62_1_1"/>
<font color="Maroon" title="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</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"> <font color="Maroon" title="c17">u</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> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span><br/><a NAME="E2:62_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">C2</font></span>: 
<font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E3:62_1_1"/><span class="lab"><font color="Green" title="E45">C3</font></span>: 
<span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E38">D3</font></span></a>, <a class="ref" href="field_7.html#D8" target="_self" onmouseover="rs('field_7/D8')" onmouseout="rh()">down1</a></span>;<br/><a NAME="E4:62_1_1"/>
<span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">va</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="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1"><span class="lab"><font color="Green" title="E45">C3</font></span></a>, <a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E37">D2</font></span></a>, <a class="ref" href="vectsp_6.html#T2" onmouseover="rs('vectsp_6/T2')" onmouseout="rh()">VECTSP_6:2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:62_1_1"/>
<font color="Maroon" title="c17">u</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> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1_1"><span class="lab"><font color="Green" title="E44">C2</font></span></a>, <a class="ref" href="vectsp_6.html#T2" onmouseover="rs('vectsp_6/T2')" onmouseout="rh()">VECTSP_6:2</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>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_1_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">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">u</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> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">u</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="c17">u</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> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> implies <font color="Maroon" title="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:62_1_2"/><span class="lab"><font color="Green" title="E44">C2</font></span>: 
<font color="Maroon" title="c17">u</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> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</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="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">aE</font> = <font color="Maroon" title="c17">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</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/><span class="kw">reconsider </span><font color="Maroon" title="c19">aK</font> = <font color="Maroon" title="c18">aE</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E35">H3</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">abV</font> = <font color="Maroon" title="c19">aK</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</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="ref" href="field_4.html#D6" onmouseover="rs('field_4/D6')" onmouseout="rh()">FIELD_4:def 6</a></span>;<br/><a NAME="E5:62_1_2"/><span class="lab"><font color="Green" title="E45">C3</font></span>: 
<span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">aE</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="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_2"><span class="lab"><font color="Green" title="E44">C2</font></span></a>, <a class="ref" href="vectsp_6.html#T2" onmouseover="rs('vectsp_6/T2')" onmouseout="rh()">VECTSP_6:2</a></span>;<br/><a NAME="E6:62_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c18">aE</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">BE</font> &amp; <font color="Maroon" title="c10">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">BK</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#D8" target="_self" onmouseover="rs('field_7/D8')" onmouseout="rh()">down1</a></span>;<br/><a NAME="E7:62_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">aK</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">aK</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#D8" target="_self" onmouseover="rs('field_7/D8')" onmouseout="rh()">down1</a></span>;<br/><a NAME="E8:62_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">abV</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="#E5:62_1_2"><span class="lab"><font color="Green" title="E45">C3</font></span></a>, <a class="ref" href="vectsp_6.html#T2" onmouseover="rs('vectsp_6/T2')" onmouseout="rh()">VECTSP_6:2</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c19">aK</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font> = 
<font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E36">D1</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E38">D3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D12" onmouseover="rs('group_1/D12')" onmouseout="rh()">GROUP_1:def 12</a></span>
;<br/><a NAME="E10:62_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">aK</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D12" onmouseover="rs('group_1/D12')" onmouseout="rh()">GROUP_1:def 12</a></span>;<br/><a NAME="E11:62_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">aK</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E41">D6a</font></span></a>, <a class="ref" href="vectsp_2.html#T8" onmouseover="rs('vectsp_2/T8')" onmouseout="rh()">VECTSP_2:8</a></span>;<br/><span class="kw">hence </span><a NAME="E12:62_1_2"/>
<font color="Maroon" title="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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>
<span class="kw">hence </span><a NAME="E3:62_1"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1"><span class="lab"><font color="Green" title="E43">C1</font></span></a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</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="E28:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">D7</font></span>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">va</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c13">va</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#T20" onmouseover="rs('vectsp_6/T20')" onmouseout="rh()">VECTSP_6:20</a></span>;<br/>
<a NAME="E29:62"/><span class="lab"><font color="Green" title="E44">D9</font></span>: 
( <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span> &amp; <font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">v</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E38">D3</font></span></a>, <a class="ref" href="field_7.html#D8" target="_self" onmouseover="rs('field_7/D8')" onmouseout="rh()">down1</a></span>;<br/>
<a NAME="E30:62"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">va</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="c1">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E37">D2</font></span></a>, <a class="ref" href="vectsp_6.html#T2" onmouseover="rs('vectsp_6/T2')" onmouseout="rh()">VECTSP_6:2</a></span>;<br/>
<a NAME="E31:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">D8</font></span>: 
<font color="Maroon" title="c15">labE</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="txt" href="#E29:62"><span class="lab"><font color="Green" title="E44">D9</font></span></a>, <a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E34">H2</font></span></a>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>;<br/>
<a NAME="E32:62"/><span class="lab"><font color="Green" title="E46">D10a</font></span>: 
 for <font color="Olive" title="b1">b1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font>  st <font color="Olive" title="b1">b1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">BK</font> &amp; <font color="Olive" title="b1">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b</font> holds <br/> <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Olive" title="b1">b1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">b1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</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="c17">b1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">BK</font> &amp; <font color="Maroon" title="c17">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b</font> implies  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c17">b1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:62_2"/><span class="lab"><font color="Green" title="E47">E0</font></span>: 
( <font color="Maroon" title="c17">b1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">BK</font> &amp; <font color="Maroon" title="c17">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b</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">  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c17">b1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_2_1"><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>   holds  not <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> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c17">b1</font>)</span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">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">  not <font color="Maroon" title="c18">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> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c17">b1</font>)</span>)</span></span><br/><span class="kw">assume </span><a NAME="E1:62_2_1"/><span class="lab"><font color="Green" title="E48">E1</font></span>: 
<font color="Maroon" title="c18">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> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c17">b1</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"> contradiction</span><br/><a NAME="E2:62_2_1"/><span class="lab"><font color="Green" title="E49">E2</font></span>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c17">b1</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">v</font> = <font color="Maroon" title="c18">o</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="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_1"><span class="lab"><font color="Green" title="E48">E1</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">vE</font> = <font color="Maroon" title="c19">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</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/><span class="kw">reconsider </span><font color="Maroon" title="c21">vb1</font> = <span class="p1">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c20">vE</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c17">b1</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="ref" href="field_4.html#D6" onmouseover="rs('field_4/D6')" onmouseout="rh()">FIELD_4:def 6</a></span>;<br/><a NAME="E6:62_2_1"/><span class="lab"><font color="Green" title="E50">E3</font></span>: 
<span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c17">b1</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c20">vE</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c17">b1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_1"><span class="lab"><font color="Green" title="E48">E1</font></span></a>, <a class="txt" href="#E2:62_2_1"><span class="lab"><font color="Green" title="E49">E2</font></span></a>, <a class="txt" href="#E1:62_2"><span class="lab"><font color="Green" title="E47">E0</font></span></a>, <a class="ref" href="field_7.html#D8" target="_self" onmouseover="rs('field_7/D8')" onmouseout="rh()">down1</a></span>;<br/><a NAME="E7:62_2_1"/><span class="lab"><font color="Green" title="E51">E4</font></span>: 
<span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c17">b1</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">v</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="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_1"><span class="lab"><font color="Green" title="E48">E1</font></span></a>, <a class="ref" href="vectsp_6.html#T2" onmouseover="rs('vectsp_6/T2')" onmouseout="rh()">VECTSP_6:2</a></span>;<br/><a NAME="E8:62_2_1"/>
<font color="Maroon" title="c21">vb1</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="#E6:62_2_1"><span class="lab"><font color="Green" title="E50">E3</font></span></a>, <a class="txt" href="#E7:62_2_1"><span class="lab"><font color="Green" title="E51">E4</font></span></a>, <a class="ref" href="vectsp_6.html#T2" onmouseover="rs('vectsp_6/T2')" onmouseout="rh()">VECTSP_6:2</a></span>;<br/><a NAME="E9:62_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="field_7.html#K5" title="FIELD_7:func.5">@</a> (<font color="Maroon" title="c20">vE</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c17">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E36">D1</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E38">D3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E10:62_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E38">D3</font></span></a>, <a class="txt" href="#E1:62_2"><span class="lab"><font color="Green" title="E47">E0</font></span></a>, <a class="txt" href="#E1:62_2_1"><span class="lab"><font color="Green" title="E48">E1</font></span></a>, <a class="txt" href="#E2:62_2_1"><span class="lab"><font color="Green" title="E49">E2</font></span></a>, <a class="txt" href="field_7.html#E29"><span class="lab"><font color="Green" title="E22">BE0</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>
<span class="kw">hence </span><a NAME="E3:62_2"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c17">b1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</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="E33:62"/><span class="lab"><font color="Green" title="E47">D10</font></span>: 
 for <font color="Olive" title="b1">b1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font>  st <font color="Olive" title="b1">b1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">BK</font> &amp; <font color="Olive" title="b1">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b</font> holds <br/><span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b1</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="c2">E</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">b1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</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="c17">b1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">BK</font> &amp; <font color="Maroon" title="c17">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b</font> implies <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">b1</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="c2">E</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:62_3"/><span class="lab"><font color="Green" title="E48">E1</font></span>: 
( <font color="Maroon" title="c17">b1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">BK</font> &amp; <font color="Maroon" title="c17">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b</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"> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">b1</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="c2">E</font></span><br/>

<a NAME="E2:62_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">E2</font></span>: 
<span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c17">b1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#D9" target="_self" onmouseover="rs('field_7/D9')" onmouseout="rh()">down2</a></span>;<br/>
<a NAME="E3:62_3"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c17">b1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_3"><span class="lab"><font color="Green" title="E48">E1</font></span></a>, <a class="txt" href="#E32:62"><span class="lab"><font color="Green" title="E46">D10a</font></span></a></span>;<br/>
<a NAME="E4:62_3"/><span class="kw">then </span>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c17">b1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="field_4.html#K2" title="FIELD_4:func.2">VecSp</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#T19" onmouseover="rs('vectsp_6/T19')" onmouseout="rh()">VECTSP_6:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:62_3"/>
<span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">b1</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="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_3"><span class="lab"><font color="Green" title="E49">E2</font></span></a>, <a class="ref" href="field_4.html#D6" onmouseover="rs('field_4/D6')" onmouseout="rh()">FIELD_4:def 6</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="E34:62"/><span class="lab"><font color="Green" title="E48">D5</font></span>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c10">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:62_4"/><span class="lab"><font color="Green" title="E49">C0</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">labE</font>,<font color="Maroon" title="c14">aE</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:62"><span class="lab"><font color="Green" title="E39">B0</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E50">C1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_4_1"><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">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c10">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> holds <br/><font color="Olive" title="b1">u</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> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">u</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="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c10">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> implies <font color="Maroon" title="c17">u</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> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:62_4_1"/><span class="lab"><font color="Green" title="E51">C2</font></span>: 
<font color="Maroon" title="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c10">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</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"> <font color="Maroon" title="c17">u</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> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span></span><br/><span class="lab"><font color="Green" title="E52">C3</font></span>: <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</font> = 
 the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</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="c2">E</font>,<font color="Maroon" title="c1">F</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">lab</font>,<font color="Maroon" title="c9">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E44">D9</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E38">D3</font></span></a>, <a class="txt" href="#E28:62"><span class="lab"><font color="Green" title="E43">D7</font></span></a>, <a class="ref" href="field_7.html#D9" target="_self" onmouseover="rs('field_7/D9')" onmouseout="rh()">down2</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">E</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">labE</font>,<font color="Maroon" title="c14">aE</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></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/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">labE</font>,<font color="Maroon" title="c14">aE</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_4"><span class="lab"><font color="Green" title="E49">C0</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/><a NAME="E3:62_4_1"/>
<font color="Maroon" title="c15">labE</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">aE</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="txt" href="#E23:62"><span class="lab"><font color="Green" title="E40">D7a</font></span></a>, <a class="txt" href="#E31:62"><span class="lab"><font color="Green" title="E45">D8</font></span></a>, <a class="ref" href="vectsp_2.html#D1" onmouseover="rs('vectsp_2/D1')" onmouseout="rh()">VECTSP_2:def 1</a></span>;<br/><a NAME="E4:62_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">vb</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> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_4_1"><span class="lab"><font color="Green" title="E52">C3</font></span></a>, <a class="ref" href="vectsp_6.html#T2" onmouseover="rs('vectsp_6/T2')" onmouseout="rh()">VECTSP_6:2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:62_4_1"/>
<font color="Maroon" title="c17">u</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> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_4_1"><span class="lab"><font color="Green" title="E51">C2</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_4_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">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">u</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> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c10">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">u</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="c17">u</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> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> implies <font color="Maroon" title="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c10">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:62_4_2"/><span class="lab"><font color="Green" title="E51">C2</font></span>: 
<font color="Maroon" title="c17">u</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> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</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="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c10">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span></span><br/><a NAME="E2:62_4_2"/><span class="lab"><font color="Green" title="E52">C3</font></span>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">BK</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">reconsider </span><font color="Maroon" title="c18">bK</font> = <font color="Maroon" title="c17">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_4_2"><span class="lab"><font color="Green" title="E51">C2</font></span></a>, <a class="ref" href="field_4.html#D6" onmouseover="rs('field_4/D6')" onmouseout="rh()">FIELD_4:def 6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">bV</font> = <font color="Maroon" title="c18">bK</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="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="E5:62_4_2"/>
<span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">bK</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="txt" href="#E1:62_4_2"><span class="lab"><font color="Green" title="E51">C2</font></span></a>, <a class="ref" href="vectsp_6.html#T2" onmouseover="rs('vectsp_6/T2')" onmouseout="rh()">VECTSP_6:2</a></span>;<br/><a NAME="E6:62_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">bK</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_4_2"><span class="lab"><font color="Green" title="E52">C3</font></span></a>, <a class="txt" href="#E1:62_4_2"><span class="lab"><font color="Green" title="E51">C2</font></span></a>, <a class="txt" href="#E33:62"><span class="lab"><font color="Green" title="E47">D10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:62_4_2"/>
<font color="Maroon" title="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c10">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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>
<span class="kw">hence </span><a NAME="E4:62_4"/>
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c10">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_4"><span class="lab"><font color="Green" title="E50">C1</font></span></a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</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="E35:62"/><span class="lab"><font color="Green" title="E49">B2</font></span>: 
<span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E38">D3</font></span></a>, <a class="ref" href="field_7.html#D8" target="_self" onmouseover="rs('field_7/D8')" onmouseout="rh()">down1</a></span>;<br/>
<a NAME="E36:62"/><span class="lab"><font color="Green" title="E50">B3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c9">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E33">H1</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E38">D3</font></span></a>, <a class="txt" href="#E18:62"><span class="lab"><font color="Green" title="E39">B0</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E37:62"/>
<font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:62"><span class="lab"><font color="Green" title="E39">B0</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E35">H3</font></span></a></span>;<br/>
<a NAME="E38:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">B4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c9">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E33">H1</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E38">D3</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E52">B6</font></span>: <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">vb</font> = 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E38">D3</font></span></a>, <a class="ref" href="field_7.html#D9" target="_self" onmouseover="rs('field_7/D9')" onmouseout="rh()">down2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">va</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c13">va</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:62"><span class="lab"><font color="Green" title="E42">D6</font></span></a>, <a class="ref" href="vectsp_6.html#T20" onmouseover="rs('vectsp_6/T20')" onmouseout="rh()">VECTSP_6:20</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">E</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c9">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:62"><span class="lab"><font color="Green" title="E49">B2</font></span></a>, <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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">E</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c9">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:62"><span class="lab"><font color="Green" title="E50">B3</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">K</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>,<font color="Maroon" title="c9">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E34">H2</font></span></a>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c16">lab</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:62"><span class="lab"><font color="Green" title="E51">B4</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<a NAME="E40:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">B7</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">lab</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">a</font></span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E54">B</font></span>:  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">vb</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c12">vb</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:62"><span class="lab"><font color="Green" title="E48">D5</font></span></a>, <a class="ref" href="vectsp_6.html#T20" onmouseover="rs('vectsp_6/T20')" onmouseout="rh()">VECTSP_6:20</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">K</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">E</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c16">lab</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">a</font></span>)</span>,<font color="Maroon" title="c10">b</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:62"><span class="lab"><font color="Green" title="E52">B6</font></span></a>, <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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c16">lab</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">a</font></span>)</span>,<font color="Maroon" title="c10">b</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:62"><span class="lab"><font color="Green" title="E53">B7</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<a NAME="E42:62"/><span class="lab"><font color="Green" title="E55">B1</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:62"><span class="lab"><font color="Green" title="E39">B0</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c6">l</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">vab</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c11">vab</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E36">D1</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E38">D3</font></span></a>, <a class="ref" href="vectsp_6.html#T20" onmouseover="rs('vectsp_6/T20')" onmouseout="rh()">VECTSP_6:20</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">K</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</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/>.= 
<font color="Maroon" title="c16">lab</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:62"><span class="lab"><font color="Green" title="E55">B1</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">lab</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E44:62"/>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:62"><span class="lab"><font color="Green" title="E54">B</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>
