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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>, <font color="Olive" title="b6">l1</font>, <font color="Olive" title="b7">l2</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 <font color="Olive" title="b5">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Olive" title="b6">l1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Olive" title="b7">l2</font> holds <br/> <a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Olive" title="b5">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Olive" title="b6">l1</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Olive" title="b7">l2</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>, <font color="Olive" title="b5">l1</font>, <font color="Olive" title="b6">l2</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 <font color="Olive" title="b4">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Olive" title="b5">l1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Olive" title="b6">l2</font> holds <br/> <a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Olive" title="b4">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</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">l1</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Olive" title="b6">l2</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>, <font color="Olive" title="b4">l1</font>, <font color="Olive" title="b5">l2</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 <font color="Olive" title="b3">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Olive" title="b4">l1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Olive" title="b5">l2</font> holds <br/> <a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Olive" title="b3">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</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">l1</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</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">l2</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>, <font color="Olive" title="b3">l1</font>, <font color="Olive" title="b4">l2</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 <font color="Olive" title="b2">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Olive" title="b3">l1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Olive" title="b4">l2</font> holds <br/> <a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Olive" title="b2">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</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">l1</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</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">l2</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>, <font color="Olive" title="b2">l1</font>, <font color="Olive" title="b3">l2</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 <font color="Olive" title="b1">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Olive" title="b2">l1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Olive" title="b3">l2</font> holds <br/> <a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Olive" title="b1">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</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">l1</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</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">l2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">l</font>, <font color="Maroon" title="c7">l1</font>, <font color="Maroon" title="c8">l2</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"> ( <font color="Maroon" title="c6">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c7">l1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c8">l2</font> implies  <a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c7">l1</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c8">l2</font></span>)</span> )</span><br/>











<span class="kw">assume </span><a NAME="E1:61"/><span class="lab"><font color="Green" title="E32">AS</font></span>: 
<font color="Maroon" title="c6">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c7">l1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c8">l2</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="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c7">l1</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c8">l2</font></span>)</span></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_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>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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> 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">o</font> <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#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c7">l1</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p2">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c8">l2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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> 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="Olive">b<sub>1</sub></font> <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#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c7">l1</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p2">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c8">l2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:61_1"/><span class="lab"><font color="Green" title="E33">A0</font></span>: 
<font color="Maroon" title="c9">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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"> <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">b<sub>1</sub></font> <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#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c7">l1</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p2">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c8">l2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c9">o</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="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="c11">bV</font> = <font color="Maroon" title="c9">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="c2">E</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E33">A0</font></span></a></span>;<br/><a NAME="E4:61_1"/><span class="lab"><font color="Green" title="E34">H1</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> &amp;  <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="c7">l1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">BK</font> &amp;  <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="c8">l2</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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:61_1_1"/>
( <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> or  not <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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_1"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<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>
 ; <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="Olive">b<sub>1</sub></font> <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#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c7">l1</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p2">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c8">l2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:61_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A2</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="c7">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</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="c7">l1</font>,<font color="Maroon" title="c10">b</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/><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> = 
 <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="#E1:61_1_1_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="field_7.html#D9" target="_self" onmouseover="rs('field_7/D9')" onmouseout="rh()">down2</a></span>
<br/>.= 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</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="c7">l1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p2">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c8">l2</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E32">AS</font></span></a>, <a class="ref" href="field_7.html#T24" target="_self" onmouseover="rs('field_7/T24')" onmouseout="rh()">LSum1a</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p2">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c7">l1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p2">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c8">l2</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#T44" onmouseover="rs('vectsp_6/T44')" onmouseout="rh()">VECTSP_6:44</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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"><span class="p2">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p3">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c7">l1</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p3">(<span class="default"><a href="field_7.html#K8" title="FIELD_7:func.8">down</a> (<font color="Maroon" title="c8">l2</font>,<font color="Maroon" title="c10">b</font>)</span>)</span></span>)</span></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/>.= 
<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="c7">l1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">bV</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <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="c8">l2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">bV</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E2:61_1_1_1"><span class="lab"><font color="Green" title="E36">A2</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="kw">hence </span><a NAME="E4:61_1_1_1"/>
<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="c9">o</font> <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#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c7">l1</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p2">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c8">l2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#T22" onmouseover="rs('vectsp_6/T22')" onmouseout="rh()">VECTSP_6:22</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" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_2"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
 not <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>
 ; <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="Olive">b<sub>1</sub></font> <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#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c7">l1</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p2">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c8">l2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">then </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="c11">bV</font> = 
 <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="#E4:61_1"><span class="lab"><font color="Green" title="E34">H1</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="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="c7">l1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">bV</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_2"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E4:61_1"><span class="lab"><font color="Green" title="E34">H1</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="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="c7">l1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">bV</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <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="c8">l2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">bV</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_2"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E4:61_1"><span class="lab"><font color="Green" title="E34">H1</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="E3:61_1_1_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="c9">o</font> <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#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c7">l1</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p2">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c8">l2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#T22" onmouseover="rs('vectsp_6/T22')" onmouseout="rh()">VECTSP_6:22</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:61"/>
 <a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c6">l</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c7">l1</font></span>)</span> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <span class="p1">(<span class="default"><a href="field_7.html#K9" title="FIELD_7:func.9">down</a> <font color="Maroon" title="c8">l2</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"> verum</span><br/>


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