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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">A</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="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">A</font> holds <br/> for <font color="Olive" title="b2">E</font> being    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Olive" title="b1">A</font>  st <font color="Olive" title="b2">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> holds <br/> for <font color="Olive" title="b3">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">k</font></span>)</span><br/>  for <font color="Olive" title="b4">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span>  st <font color="Olive" title="b4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b5">v</font> where <font color="Olive" title="b5">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b5">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Olive" title="b2">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">P</font> </span> } </span>  holds <br/>( <font color="Olive" title="b3">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Olive" title="b4">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">TRn</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">A</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="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</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="c2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">A</font> implies  for <font color="Olive" title="b1">E</font> being    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c4">A</font>  st <font color="Olive" title="b1">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> holds <br/> for <font color="Olive" title="b2">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">k</font></span>)</span><br/>  for <font color="Olive" title="b3">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span>  st <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b4">v</font> where <font color="Olive" title="b4">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b4">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Olive" title="b1">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">P</font> </span> } </span>  holds <br/>( <font color="Olive" title="b2">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Olive" title="b3">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:39"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font> being    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c4">A</font>  st <font color="Olive" title="b1">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> holds <br/> for <font color="Olive" title="b2">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">k</font></span>)</span><br/>  for <font color="Olive" title="b3">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span>  st <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b4">v</font> where <font color="Olive" title="b4">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b4">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Olive" title="b1">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">P</font> </span> } </span>  holds <br/>( <font color="Olive" title="b2">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Olive" title="b3">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">c1</font> = <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="nat_1.html#T21" onmouseover="rs('nat_1/T21')" onmouseout="rh()">NAT_1:21</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">AA</font> =  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">E</font> be    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c4">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> implies  for <font color="Olive" title="b1">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">k</font></span>)</span><br/>  for <font color="Olive" title="b2">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">v</font> where <font color="Olive" title="b3">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b3">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">P</font> </span> } </span>  holds <br/>( <font color="Olive" title="b1">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Olive" title="b2">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:39"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c7">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</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">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">k</font></span>)</span><br/>  for <font color="Olive" title="b2">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">v</font> where <font color="Olive" title="b3">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b3">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">P</font> </span> } </span>  holds <br/>( <font color="Olive" title="b1">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Olive" title="b2">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<a NAME="E4:39"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#D1" target="_self" onmouseover="rs('rlaffin3/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E5:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T62" onmouseover="rs('finseq_4/T62')" onmouseout="rh()">FINSEQ_4:62</a></span>;<br/>
<a NAME="E6:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">E</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/>
<a NAME="E7:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E8:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A7</font></span>: 
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E9:39"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T66" onmouseover="rs('euclid/T66')" onmouseout="rh()">EUCLID:66</a></span>;<br/>
<a NAME="E10:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A9</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E11:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A10</font></span>: 
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T46" onmouseover="rs('rlaffin1/T46')" onmouseout="rh()">RLAFFIN1:46</a></span>;<br/>
<a NAME="E12:39"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#T5" target="_self" onmouseover="rs('rlaffin3/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E13:39"/><span class="kw">then </span>
<font color="Maroon" title="c5">c1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#T4" target="_self" onmouseover="rs('rlaffin3/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E14:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A11</font></span>: 
<font color="Maroon" title="c5">c1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">nc</font> = <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">k</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">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">v</font> where <font color="Olive" title="b2">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font> </span> } </span>  holds <br/>( <font color="Maroon" title="c9">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Olive" title="b1">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c10">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></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="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font> </span> } </span>  implies ( <font color="Maroon" title="c9">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Maroon" title="c10">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E15:39"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
<font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</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="c9">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Maroon" title="c10">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<a NAME="E16:39"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">F</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E18:39"/><span class="lab"><font color="Green" title="E44">A14</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">and </span><br/><a NAME="E19:39"/><span class="lab"><font color="Green" title="E45">A15</font></span>: 
<font color="Maroon" title="c11">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T58" onmouseover="rs('finseq_4/T58')" onmouseout="rh()">FINSEQ_4:58</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">V</font> = <font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">Bn</font> =  <a href="matrix13.html#K11" title="MATRIX13:func.11">MX2FinS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K3" title="MATRIX_1:func.3">1.</a> (<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>,<font color="Maroon" title="c1">n</font>)</span>)</span> as    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T45" onmouseover="rs('matrlin2/T45')" onmouseout="rh()">MATRLIN2:45</a></span>;<br/>
<a NAME="E21:39"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">Bn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop1.html#T19" onmouseover="rs('matrtop1/T19')" onmouseout="rh()">MATRTOP1:19</a></span>;<br/>
<a NAME="E22:39"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">FinS2MX</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Bn</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">c1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:39"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">BnC</font> =  <a href="matrix13.html#K10" title="MATRIX13:func.10">FinS2MX</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Bn</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">c1</font></span>)</span> as   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c5">c1</font>,<font color="Maroon" title="c1">n</font>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">MBC</font> =  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E25:39"/><span class="lab"><font color="Green" title="E46">A16</font></span>: 
<font color="Maroon" title="c5">c1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">A</font>
 
;<br/>
<span class="lab"><font color="Green" title="E47">A17</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">F</font> = 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E19:39"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="ref" href="finseq_4.html#T62" onmouseover="rs('finseq_4/T62')" onmouseout="rh()">FINSEQ_4:62</a></span>
<br/>.= 
<font color="Maroon" title="c5">c1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:39"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E25:39"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="ref" href="stirl2_1.html#T55" onmouseover="rs('stirl2_1/T55')" onmouseout="rh()">STIRL2_1:55</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">MBc</font> =  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">TRc</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font>;<br/>
<a NAME="E27:39"/><span class="lab"><font color="Green" title="E48">A18</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span></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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="matrtop1.html#T29" onmouseover="rs('matrtop1/T29')" onmouseout="rh()">MATRTOP1:29</a></span>;<br/>
<a NAME="E28:39"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">Bn</font> is   <a href="vectsp_7.html#NM1" title="VECTSP_7:NM.1">Basis</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D2" onmouseover="rs('matrlin/D2')" onmouseout="rh()">MATRLIN:def 2</a></span>;<br/>
<a NAME="E29:39"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">Bn</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#D3" onmouseover="rs('vectsp_7/D3')" onmouseout="rh()">VECTSP_7:def 3</a></span>;<br/>
<a NAME="E30:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A19</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Bn</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">c1</font></span>)</span> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T70" onmouseover="rs('relat_1/T70')" onmouseout="rh()">RELAT_1:70</a>, <a class="ref" href="vectsp_7.html#T1" onmouseover="rs('vectsp_7/T1')" onmouseout="rh()">VECTSP_7:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">F</font> = <font color="Maroon" title="c11">F</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E32:39"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">Bn</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span></span>)</span></span>)</span></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D2" onmouseover="rs('vectsp_4/D2')" onmouseout="rh()">VECTSP_4:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">BF</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">Bn</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="rlaffin3.html#E10"><span class="lab"><font color="Green" title="E10">Lm3</font></span></a></span>;<br/>
<a NAME="E34:39"/><span class="lab"><font color="Green" title="E50">A20</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c15">MBC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">BF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:39"><span class="lab"><font color="Green" title="E47">A17</font></span></a>, <a class="ref" href="matrtop2.html#T18" onmouseover="rs('matrtop2/T18')" onmouseout="rh()">MATRTOP2:18</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">M</font> being   <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of <font color="Maroon" title="c1">n</font>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a><span class="kw"> such that </span><br/><a NAME="E36:39"/><span class="lab"><font color="Green" title="E51">A21</font></span>: 
<font color="Maroon" title="c20">M</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> 
 <span class="kw">and </span><br/><a NAME="E37:39"/><span class="lab"><font color="Green" title="E52">A22</font></span>: 
<font color="Maroon" title="c20">M</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E18:39"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E19:39"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="ref" href="matrtop2.html#T19" onmouseover="rs('matrtop2/T19')" onmouseout="rh()">MATRTOP2:19</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">MTI</font> =  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span>;<br/>
<a NAME="E38:39"/><span class="lab"><font color="Green" title="E53">A23</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">BF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">BF</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E39:39"/>
<font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a>  is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:39"><span class="lab"><font color="Green" title="E51">A21</font></span></a>, <a class="ref" href="matrix_6.html#T16" onmouseover="rs('matrix_6/T16')" onmouseout="rh()">MATRIX_6:16</a></span>;<br/>
<a NAME="E40:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A24</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span> <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> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="laplace.html#T34" onmouseover="rs('laplace/T34')" onmouseout="rh()">LAPLACE:34</a></span>;<br/>
<a NAME="E41:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A25</font></span>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T83" onmouseover="rs('matrix13/T83')" onmouseout="rh()">MATRIX13:83</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">MTe</font> = <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c7">E</font> as    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#T15" target="_self" onmouseover="rs('rlaffin3/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<a NAME="E43:39"/><span class="lab"><font color="Green" title="E56">A26</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c22">MTe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#D1" target="_self" onmouseover="rs('rlaffin3/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E44:39"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:39"><span class="lab"><font color="Green" title="E54">A24</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="matrtop1.html#T40" onmouseover="rs('matrtop1/T40')" onmouseout="rh()">MATRTOP1:40</a></span>;<br/>
<a NAME="E45:39"/><span class="kw">then </span>
<font color="Maroon" title="c4">A</font>,<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">A</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T33" onmouseover="rs('card_1/T33')" onmouseout="rh()">CARD_1:33</a></span>;<br/>
<a NAME="E46:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A27</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T5" onmouseover="rs('card_1/T5')" onmouseout="rh()">CARD_1:5</a></span>;<br/>
<a NAME="E47:39"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">MTe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E43:39"><span class="lab"><font color="Green" title="E56">A26</font></span></a>, <a class="ref" href="finseq_4.html#T62" onmouseover="rs('finseq_4/T62')" onmouseout="rh()">FINSEQ_4:62</a></span>;<br/>
<a NAME="E48:39"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">MTe</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E58">A28</font></span>: <font color="Maroon" title="c22">MTe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">E</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop1.html#T29" onmouseover="rs('matrtop1/T29')" onmouseout="rh()">MATRTOP1:29</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">MT</font> =  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c20">M</font>;<br/>
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E8:39"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="rlaffin3.html#T11" target="_self" onmouseover="rs('rlaffin3/T11')" onmouseout="rh()">Th11</a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c18">F</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="rlaffin3.html#E4"><span class="lab"><font color="Green" title="E4">Lm2</font></span></a></span>
;<br/>
<a NAME="E51:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A29</font></span>: 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c20">M</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c19">BF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E18:39"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E19:39"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="txt" href="#E36:39"><span class="lab"><font color="Green" title="E51">A21</font></span></a>, <a class="txt" href="#E37:39"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="ref" href="matrtop2.html#T21" onmouseover="rs('matrtop2/T21')" onmouseout="rh()">MATRTOP2:21</a></span>;<br/>
<a NAME="E52:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A30</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c20">M</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c19">BF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T115" onmouseover="rs('relat_1/T115')" onmouseout="rh()">RELAT_1:115</a></span>;<br/>
<a NAME="E53:39"/><span class="lab"><font color="Green" title="E61">A31</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c20">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E54:39"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c20">M</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c19">BF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">BF</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">MT1</font> = <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c20">M</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c19">BF</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">BF</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:39"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E38:39"><span class="lab"><font color="Green" title="E53">A23</font></span></a>, <a class="txt" href="#E52:39"><span class="lab"><font color="Green" title="E60">A30</font></span></a>, <a class="ref" href="funct_2.html#T1" onmouseover="rs('funct_2/T1')" onmouseout="rh()">FUNCT_2:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c25">MT</font> =  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c20">M</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E57:39"/><span class="lab"><font color="Green" title="E62">A32</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c20">M</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> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:39"><span class="lab"><font color="Green" title="E51">A21</font></span></a>, <a class="ref" href="laplace.html#T34" onmouseover="rs('laplace/T34')" onmouseout="rh()">LAPLACE:34</a></span>;<br/>
<a NAME="E58:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A33</font></span>: 
<font color="Maroon" title="c25">MT</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop1.html#T43" onmouseover="rs('matrtop1/T43')" onmouseout="rh()">MATRTOP1:43</a></span>;<br/>
<a NAME="E59:39"/>
<font color="Maroon" title="c24">MT1</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:39"><span class="lab"><font color="Green" title="E51">A21</font></span></a>, <a class="txt" href="#E51:39"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="ref" href="metrizts.html#T2" onmouseover="rs('metrizts/T2')" onmouseout="rh()">METRIZTS:2</a></span>;<br/>
<a NAME="E60:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A34</font></span>: 
( <font color="Maroon" title="c24">MT1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c10">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Maroon" title="c10">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topgrp_1.html#T26" onmouseover="rs('topgrp_1/T26')" onmouseout="rh()">TOPGRP_1:26</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c26">nc0</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c27">Vc1</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> : verum </span> } </span> ;<br/>
<a NAME="E61:39"/><span class="lab"><font color="Green" title="E65">A35</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:39"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="xreal_1.html#T233" onmouseover="rs('xreal_1/T233')" onmouseout="rh()">XREAL_1:233</a></span>;<br/>
<a NAME="E62:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A36</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span>
 
;<br/>
<a NAME="E63:39"/><span class="lab"><font color="Green" title="E67">A37</font></span>: 
<font color="Maroon" title="c25">MT</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:39"><span class="lab"><font color="Green" title="E62">A32</font></span></a>, <a class="ref" href="matrtop1.html#T40" onmouseover="rs('matrtop1/T40')" onmouseout="rh()">MATRTOP1:40</a></span>;<br/>
<a NAME="E64:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A38</font></span>: 
<font color="Maroon" title="c25">MT</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">BF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:39"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="txt" href="#E53:39"><span class="lab"><font color="Green" title="E61">A31</font></span></a>, <a class="ref" href="funct_1.html#T94" onmouseover="rs('funct_1/T94')" onmouseout="rh()">FUNCT_1:94</a></span>;<br/>
<a NAME="E65:39"/><span class="lab"><font color="Green" title="E69">A39</font></span>: 
<font color="Maroon" title="c25">MT</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">MT</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T143" onmouseover="rs('relat_1/T143')" onmouseout="rh()">RELAT_1:143</a>, <a class="ref" href="rlaffin1.html#T49" onmouseover="rs('rlaffin1/T49')" onmouseout="rh()">RLAFFIN1:49</a></span>;<br/>
<a NAME="E66:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A40</font></span>: 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">BF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:39"><span class="lab"><font color="Green" title="E63">A33</font></span></a>, <a class="txt" href="#E63:39"><span class="lab"><font color="Green" title="E67">A37</font></span></a>, <a class="txt" href="#E64:39"><span class="lab"><font color="Green" title="E68">A38</font></span></a>, <a class="ref" href="funct_1.html#T85" onmouseover="rs('funct_1/T85')" onmouseout="rh()">FUNCT_1:85</a></span>;<br/>
<a NAME="E67:39"/>
<font color="Maroon" title="c13">Bn</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D2" onmouseover="rs('matrlin/D2')" onmouseout="rh()">MATRLIN:def 2</a></span>;<br/>
<a NAME="E68:39"/><span class="kw">then </span>
<font color="Maroon" title="c13">Bn</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">c1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T52" onmouseover="rs('funct_1/T52')" onmouseout="rh()">FUNCT_1:52</a></span>;<br/>
<a NAME="E69:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A41</font></span>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c14">BnC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:39"><span class="lab"><font color="Green" title="E49">A19</font></span></a>, <a class="ref" href="matrix13.html#T121" onmouseover="rs('matrix13/T121')" onmouseout="rh()">MATRIX13:121</a></span>;<br/>
<a NAME="E70:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A42</font></span>: 
<font color="Maroon" title="c15">MBC</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop1.html#T39" onmouseover="rs('matrtop1/T39')" onmouseout="rh()">MATRTOP1:39</a></span>;<br/>
<a NAME="E71:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A43</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">MBC</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c15">MBC</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T33" onmouseover="rs('funct_1/T33')" onmouseout="rh()">FUNCT_1:33</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">MBCe</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c15">MBC</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c22">MTe</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:39"><span class="lab"><font color="Green" title="E50">A20</font></span></a>, <a class="txt" href="#E43:39"><span class="lab"><font color="Green" title="E56">A26</font></span></a>, <a class="txt" href="#E66:39"><span class="lab"><font color="Green" title="E70">A40</font></span></a>, <a class="ref" href="finseq_1.html#T16" onmouseover="rs('finseq_1/T16')" onmouseout="rh()">FINSEQ_1:16</a></span>;<br/>
<a NAME="E73:39"/><span class="lab"><font color="Green" title="E74">A44</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c28">MBCe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">MBC</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:39"><span class="lab"><font color="Green" title="E56">A26</font></span></a>, <a class="ref" href="relat_1.html#T127" onmouseover="rs('relat_1/T127')" onmouseout="rh()">RELAT_1:127</a></span>;<br/>
<a NAME="E74:39"/>
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">A</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:39"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="ref" href="matrtop2.html#T24" onmouseover="rs('matrtop2/T24')" onmouseout="rh()">MATRTOP2:24</a></span>;<br/>
<a NAME="E75:39"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">A</font></span>)</span> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:39"><span class="lab"><font color="Green" title="E71">A41</font></span></a>, <a class="ref" href="matrtop2.html#T27" onmouseover="rs('matrtop2/T27')" onmouseout="rh()">MATRTOP2:27</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">R</font> =  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c28">MBCe</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:39"><span class="lab"><font color="Green" title="E72">A42</font></span></a>, <a class="txt" href="#E73:39"><span class="lab"><font color="Green" title="E74">A44</font></span></a>, <a class="ref" href="funct_1.html#T85" onmouseover="rs('funct_1/T85')" onmouseout="rh()">FUNCT_1:85</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c30">MBCe</font> = <font color="Maroon" title="c28">MBCe</font> as    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c29">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:39"><span class="lab"><font color="Green" title="E72">A42</font></span></a>, <a class="ref" href="rlaffin3.html#D1" target="_self" onmouseover="rs('rlaffin3/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c31">MBCeE</font> = <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c30">MBCe</font> as    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c29">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:39"><span class="lab"><font color="Green" title="E71">A41</font></span></a>, <a class="ref" href="rlaffin3.html#T15" target="_self" onmouseover="rs('rlaffin3/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<a NAME="E79:39"/>
<font color="Maroon" title="c15">MBC</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">MBC</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c15">MBC</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:39"><span class="lab"><font color="Green" title="E72">A42</font></span></a>, <a class="ref" href="funct_1.html#T39" onmouseover="rs('funct_1/T39')" onmouseout="rh()">FUNCT_1:39</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E75">A45</font></span>: <font color="Maroon" title="c31">MBCeE</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c15">MBC</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">MTe</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" onmouseover="rs('relat_1/T36')" onmouseout="rh()">RELAT_1:36</a></span>
<br/>.= 
<font color="Maroon" title="c22">MTe</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:39"><span class="lab"><font color="Green" title="E50">A20</font></span></a>, <a class="txt" href="#E43:39"><span class="lab"><font color="Green" title="E56">A26</font></span></a>, <a class="txt" href="#E66:39"><span class="lab"><font color="Green" title="E70">A40</font></span></a>, <a class="ref" href="relat_1.html#T53" onmouseover="rs('relat_1/T53')" onmouseout="rh()">RELAT_1:53</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c32">PPP</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c30">MBCe</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font> </span> } </span> ;<br/>
<a NAME="E81:39"/><span class="lab"><font color="Green" title="E76">A46</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c30">MBCe</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c33">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c30">MBCe</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font> </span> } </span>  or <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_5"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c30">MBCe</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</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="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span></span><br/>

<a NAME="E2:39_5"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> st <br/>( <font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c30">MBCe</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:39_5"/>
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</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/>


</div><span class="kw">end;</span></div>
<a NAME="E82:39"/><span class="lab"><font color="Green" title="E77">A47</font></span>: 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">MT</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:39"><span class="lab"><font color="Green" title="E63">A33</font></span></a>, <a class="txt" href="#E63:39"><span class="lab"><font color="Green" title="E67">A37</font></span></a>, <a class="ref" href="funct_1.html#T85" onmouseover="rs('funct_1/T85')" onmouseout="rh()">FUNCT_1:85</a></span>;<br/>
<a NAME="E83:39"/><span class="lab"><font color="Green" title="E78">A48</font></span>: 
<font color="Maroon" title="c25">MT</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">MT</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:39"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="ref" href="relat_1.html#T143" onmouseover="rs('relat_1/T143')" onmouseout="rh()">RELAT_1:143</a></span>;<br/>
<a NAME="E84:39"/><span class="lab"><font color="Green" title="E79">A49</font></span>: 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c29">R</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:39"><span class="lab"><font color="Green" title="E50">A20</font></span></a>, <a class="txt" href="#E70:39"><span class="lab"><font color="Green" title="E72">A42</font></span></a>, <a class="txt" href="#E71:39"><span class="lab"><font color="Green" title="E73">A43</font></span></a>, <a class="txt" href="#E66:39"><span class="lab"><font color="Green" title="E70">A40</font></span></a>, <a class="txt" href="#E73:39"><span class="lab"><font color="Green" title="E74">A44</font></span></a>, <a class="ref" href="card_1.html#T33" onmouseover="rs('card_1/T33')" onmouseout="rh()">CARD_1:33</a></span>;<br/>
<a NAME="E85:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A50</font></span>: 
<font color="Maroon" title="c5">c1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c29">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:39"><span class="lab"><font color="Green" title="E57">A27</font></span></a>, <a class="ref" href="card_1.html#T5" onmouseover="rs('card_1/T5')" onmouseout="rh()">CARD_1:5</a></span>;<br/>
<a NAME="E86:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A51</font></span>: 
( <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">c1</font> &amp;  not <font color="Maroon" title="c29">R</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c33">PPP</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c30">MBCe</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font> </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:39"><span class="lab"><font color="Green" title="E76">A46</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c34">nPP</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c33">PPP</font> &amp; <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c19">BF</font> ) </span> } </span> ;<br/>
<a NAME="E88:39"/>
 <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#T4" target="_self" onmouseover="rs('rlaffin3/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E89:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A52</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c29">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:39"><span class="lab"><font color="Green" title="E80">A50</font></span></a>, <a class="ref" href="rlaffin3.html#T6" target="_self" onmouseover="rs('rlaffin3/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<span class="lab"><font color="Green" title="E83">A53</font></span>: <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c29">R</font> = 
<font color="Maroon" title="c15">MBC</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">MBC</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:39"><span class="lab"><font color="Green" title="E56">A26</font></span></a>, <a class="ref" href="relat_1.html#T127" onmouseover="rs('relat_1/T127')" onmouseout="rh()">RELAT_1:127</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">MBC</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">MBC</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T126" onmouseover="rs('relat_1/T126')" onmouseout="rh()">RELAT_1:126</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c15">MBC</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:39"><span class="lab"><font color="Green" title="E72">A42</font></span></a>, <a class="ref" href="funct_1.html#T39" onmouseover="rs('funct_1/T39')" onmouseout="rh()">FUNCT_1:39</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E82:39"><span class="lab"><font color="Green" title="E77">A47</font></span></a>, <a class="txt" href="#E64:39"><span class="lab"><font color="Green" title="E68">A38</font></span></a>, <a class="txt" href="#E65:39"><span class="lab"><font color="Green" title="E69">A39</font></span></a>, <a class="txt" href="#E34:39"><span class="lab"><font color="Green" title="E50">A20</font></span></a>, <a class="ref" href="funct_1.html#T92" onmouseover="rs('funct_1/T92')" onmouseout="rh()">FUNCT_1:92</a></span>
;<br/>
<a NAME="E91:39"/><span class="lab"><font color="Green" title="E84">A54</font></span>: 
<font color="Maroon" title="c25">MT</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c33">PPP</font> &amp; <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c19">BF</font> ) </span> } </span> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c35">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">MT</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <font color="Maroon" title="c10">B</font> or <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c33">PPP</font> &amp; <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c19">BF</font> ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_7"/><span class="lab"><font color="Green" title="E85">A55</font></span>: 
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">MT</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</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"> <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c33">PPP</font> &amp; <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c19">BF</font> ) </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c36">w</font> = <font color="Maroon" title="c35">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">BF</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:39"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="txt" href="#E38:39"><span class="lab"><font color="Green" title="E53">A23</font></span></a>, <a class="txt" href="#E53:39"><span class="lab"><font color="Green" title="E61">A31</font></span></a>, <a class="txt" href="#E63:39"><span class="lab"><font color="Green" title="E67">A37</font></span></a>, <a class="txt" href="#E83:39"><span class="lab"><font color="Green" title="E78">A48</font></span></a>, <a class="ref" href="funct_1.html#T94" onmouseover="rs('funct_1/T94')" onmouseout="rh()">FUNCT_1:94</a></span>;<br/>
<a NAME="E3:39_7"/>
<font color="Maroon" title="c25">MT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_7"><span class="lab"><font color="Green" title="E85">A55</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c37">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:39_7"/><span class="lab"><font color="Green" title="E86">A56</font></span>: 
<font color="Maroon" title="c25">MT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">v</font>
 <span class="kw">and </span><br/><a NAME="E6:39_7"/><span class="lab"><font color="Green" title="E87">A57</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c37">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c7">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E42">A12</font></span></a></span>;<br/>
<a NAME="E7:39_7"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c25">MT</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_7"><span class="lab"><font color="Green" title="E85">A55</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E8:39_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A58</font></span>: 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:39"><span class="lab"><font color="Green" title="E63">A33</font></span></a>, <a class="txt" href="#E63:39"><span class="lab"><font color="Green" title="E67">A37</font></span></a>, <a class="txt" href="#E5:39_7"><span class="lab"><font color="Green" title="E86">A56</font></span></a>, <a class="ref" href="funct_1.html#T34" onmouseover="rs('funct_1/T34')" onmouseout="rh()">FUNCT_1:34</a></span>;<br/>
<a NAME="E9:39_7"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">BF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:39"><span class="lab"><font color="Green" title="E68">A38</font></span></a>, <a class="txt" href="#E83:39"><span class="lab"><font color="Green" title="E78">A48</font></span></a>, <a class="txt" href="#E1:39_7"><span class="lab"><font color="Green" title="E85">A55</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c38">W</font> = <font color="Maroon" title="c35">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<a NAME="E11:39_7"/><span class="lab"><font color="Green" title="E89">A59</font></span>: 
<font color="Maroon" title="c37">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:39"><span class="lab"><font color="Green" title="E43">A13</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c39">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E13:39_7"/><span class="lab"><font color="Green" title="E90">A60</font></span>: 
<font color="Maroon" title="c39">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:39_7"/><span class="lab"><font color="Green" title="E91">A61</font></span>: 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:39"><span class="lab"><font color="Green" title="E68">A38</font></span></a>, <a class="txt" href="#E83:39"><span class="lab"><font color="Green" title="E78">A48</font></span></a>, <a class="txt" href="#E34:39"><span class="lab"><font color="Green" title="E50">A20</font></span></a>, <a class="txt" href="#E1:39_7"><span class="lab"><font color="Green" title="E85">A55</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E15:39_7"/><span class="lab"><font color="Green" title="E92">A62</font></span>: 
<font color="Maroon" title="c38">W</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39_7"><span class="lab"><font color="Green" title="E90">A60</font></span></a>, <a class="txt" href="#E14:39_7"><span class="lab"><font color="Green" title="E91">A61</font></span></a>, <a class="ref" href="matrtop1.html#T38" onmouseover="rs('matrtop1/T38')" onmouseout="rh()">MATRTOP1:38</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c40">u</font> = <font color="Maroon" title="c39">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39_7"><span class="lab"><font color="Green" title="E90">A60</font></span></a></span>;<br/>
<font color="Maroon" title="c40">u</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c30">MBCe</font> = 
<font color="Maroon" title="c36">w</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c22">MTe</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:39_7"><span class="lab"><font color="Green" title="E91">A61</font></span></a>, <a class="txt" href="#E90:39"><span class="lab"><font color="Green" title="E83">A53</font></span></a>, <a class="txt" href="#E80:39"><span class="lab"><font color="Green" title="E75">A45</font></span></a>, <a class="txt" href="#E69:39"><span class="lab"><font color="Green" title="E71">A41</font></span></a>, <a class="txt" href="#E89:39"><span class="lab"><font color="Green" title="E82">A52</font></span></a>, <a class="ref" href="rlaffin3.html#T19" target="_self" onmouseover="rs('rlaffin3/T19')" onmouseout="rh()">Th19</a></span>
<br/>.= 
<font color="Maroon" title="c37">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c7">E</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:39"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="txt" href="#E8:39_7"><span class="lab"><font color="Green" title="E88">A58</font></span></a>, <a class="txt" href="#E11:39_7"><span class="lab"><font color="Green" title="E89">A59</font></span></a>, <a class="ref" href="rlaffin3.html#T19" target="_self" onmouseover="rs('rlaffin3/T19')" onmouseout="rh()">Th19</a></span>
;<br/>
<a NAME="E18:39_7"/><span class="kw">then </span>
<font color="Maroon" title="c40">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c33">PPP</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_7"><span class="lab"><font color="Green" title="E87">A57</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:39_7"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c33">PPP</font> &amp; <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c19">BF</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:39"><span class="lab"><font color="Green" title="E68">A38</font></span></a>, <a class="txt" href="#E83:39"><span class="lab"><font color="Green" title="E78">A48</font></span></a>, <a class="txt" href="#E1:39_7"><span class="lab"><font color="Green" title="E85">A55</font></span></a>, <a class="txt" href="#E15:39_7"><span class="lab"><font color="Green" title="E92">A62</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E92:39"/><span class="lab"><font color="Green" title="E85">A63</font></span>: 
<font color="Maroon" title="c19">BF</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> : verum </span> } </span> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c35">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">BF</font> or <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> : verum </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_8"/><span class="lab"><font color="Green" title="E86">A64</font></span>: 
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">BF</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> : verum </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c36">f</font> = <font color="Maroon" title="c35">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<a NAME="E3:39_8"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E4:39_8"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:39"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/>
<a NAME="E5:39_8"/><span class="kw">then </span>
<font color="Maroon" title="c36">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font> is <font color="Maroon" title="c5">c1</font> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E6:39_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A65</font></span>: 
<font color="Maroon" title="c36">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="rlaffin3.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E7:39_8"/>
<font color="Maroon" title="c36">f</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Bn</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">c1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:39"><span class="lab"><font color="Green" title="E47">A17</font></span></a>, <a class="txt" href="#E1:39_8"><span class="lab"><font color="Green" title="E86">A64</font></span></a></span>;<br/>
<a NAME="E8:39_8"/><span class="kw">then </span>
<font color="Maroon" title="c36">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop2.html#T20" onmouseover="rs('matrtop2/T20')" onmouseout="rh()">MATRTOP2:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:39_8"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_8"><span class="lab"><font color="Green" title="E87">A65</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E93:39"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> : verum </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">BF</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c35">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> : verum </span> } </span>  or <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">BF</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_9"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> : verum </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="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">BF</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c36">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:39_9"/><span class="lab"><font color="Green" title="E86">A66</font></span>: 
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:39_9"/>
verum
 ;<br/>
<a NAME="E5:39_9"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c36">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c36">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c36">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c36">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T21" onmouseover="rs('finseq_1/T21')" onmouseout="rh()">FINSEQ_1:21</a></span>
;<br/>
<a NAME="E7:39_9"/><span class="kw">then </span>
<font color="Maroon" title="c36">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">Bn</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">c1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E61:39"><span class="lab"><font color="Green" title="E65">A35</font></span></a>, <a class="ref" href="matrtop2.html#T20" onmouseover="rs('matrtop2/T20')" onmouseout="rh()">MATRTOP2:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:39_9"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">BF</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:39"><span class="lab"><font color="Green" title="E47">A17</font></span></a>, <a class="txt" href="#E3:39_9"><span class="lab"><font color="Green" title="E86">A66</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E94:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A67</font></span>: 
<font color="Maroon" title="c19">BF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E92:39"><span class="lab"><font color="Green" title="E85">A63</font></span></a></span>;<br/>
<a NAME="E95:39"/><span class="lab"><font color="Green" title="E87">A68</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c33">PPP</font> &amp; <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c19">BF</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c25">MT</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <font color="Maroon" title="c10">B</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c35">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c33">PPP</font> &amp; <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c19">BF</font> ) </span> } </span>  or <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">MT</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <font color="Maroon" title="c10">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_10"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c33">PPP</font> &amp; <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c19">BF</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="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">MT</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <font color="Maroon" title="c10">B</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c36">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:39_10"/><span class="lab"><font color="Green" title="E88">A69</font></span>: 
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">v</font>
 <span class="kw">and </span><br/><a NAME="E4:39_10"/><span class="lab"><font color="Green" title="E89">A70</font></span>: 
<font color="Maroon" title="c36">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c33">PPP</font>
 <span class="kw">and </span><br/><a NAME="E5:39_10"/><span class="lab"><font color="Green" title="E90">A71</font></span>: 
<font color="Maroon" title="c36">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c19">BF</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c37">v1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:39_10"/><span class="lab"><font color="Green" title="E91">A72</font></span>: 
<font color="Maroon" title="c36">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">v1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E94:39"><span class="lab"><font color="Green" title="E86">A67</font></span></a>, <a class="txt" href="#E5:39_10"><span class="lab"><font color="Green" title="E90">A71</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c38">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E9:39_10"/><span class="lab"><font color="Green" title="E92">A73</font></span>: 
<font color="Maroon" title="c38">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font>
 <span class="kw">and </span><br/><a NAME="E10:39_10"/><span class="lab"><font color="Green" title="E93">A74</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c38">u</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c30">MBCe</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_10"><span class="lab"><font color="Green" title="E89">A70</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c39">w</font> = <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">u</font>;<br/>
<a NAME="E11:39_10"/><span class="lab"><font color="Green" title="E94">A75</font></span>: 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">MT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">u</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:39"><span class="lab"><font color="Green" title="E61">A31</font></span></a>, <a class="txt" href="#E58:39"><span class="lab"><font color="Green" title="E63">A33</font></span></a>, <a class="txt" href="#E63:39"><span class="lab"><font color="Green" title="E67">A37</font></span></a>, <a class="ref" href="funct_1.html#T34" onmouseover="rs('funct_1/T34')" onmouseout="rh()">FUNCT_1:34</a></span>;<br/>
<a NAME="E12:39_10"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E13:39_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A76</font></span>: 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c19">BF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:39"><span class="lab"><font color="Green" title="E50">A20</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E14:39_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A77</font></span>: 
<font color="Maroon" title="c25">MT</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="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">u</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:39"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="txt" href="#E53:39"><span class="lab"><font color="Green" title="E61">A31</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<font color="Maroon" title="c38">u</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c30">MBCe</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">u</font></span>)</span> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c31">MBCeE</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:39"><span class="lab"><font color="Green" title="E71">A41</font></span></a>, <a class="txt" href="#E89:39"><span class="lab"><font color="Green" title="E82">A52</font></span></a>, <a class="ref" href="rlaffin3.html#T19" target="_self" onmouseover="rs('rlaffin3/T19')" onmouseout="rh()">Th19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c25">MT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">u</font></span>)</span></span>)</span> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c7">E</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:39"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="txt" href="#E11:39_10"><span class="lab"><font color="Green" title="E94">A75</font></span></a>, <a class="txt" href="#E14:39_10"><span class="lab"><font color="Green" title="E96">A77</font></span></a>, <a class="ref" href="rlaffin3.html#T19" target="_self" onmouseover="rs('rlaffin3/T19')" onmouseout="rh()">Th19</a>, <a class="txt" href="#E90:39"><span class="lab"><font color="Green" title="E83">A53</font></span></a>, <a class="txt" href="#E80:39"><span class="lab"><font color="Green" title="E75">A45</font></span></a></span>
;<br/>
<a NAME="E16:39_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A78</font></span>: 
<font color="Maroon" title="c25">MT</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="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">u</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="txt" href="#E16:39"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E10:39_10"><span class="lab"><font color="Green" title="E93">A74</font></span></a>, <a class="txt" href="#E14:39_10"><span class="lab"><font color="Green" title="E96">A77</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c40">w1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E18:39_10"/><span class="lab"><font color="Green" title="E98">A79</font></span>: 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">w1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">c1</font></span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E94:39"><span class="lab"><font color="Green" title="E86">A67</font></span></a>, <a class="txt" href="#E13:39_10"><span class="lab"><font color="Green" title="E95">A76</font></span></a></span>;<br/>
<font color="Maroon" title="c40">w1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">u</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c40">w1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:39_10"><span class="lab"><font color="Green" title="E98">A79</font></span></a>, <a class="ref" href="finseq_1.html#T21" onmouseover="rs('finseq_1/T21')" onmouseout="rh()">FINSEQ_1:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">u</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">w1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c14">BnC</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">u</font></span>)</span> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c36">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_10"><span class="lab"><font color="Green" title="E92">A73</font></span></a>, <a class="ref" href="matrtop1.html#T38" onmouseover="rs('matrtop1/T38')" onmouseout="rh()">MATRTOP1:38</a></span>
<br/>.= 
<font color="Maroon" title="c36">v</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">v1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c36">v</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c37">v1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c37">v1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_10"><span class="lab"><font color="Green" title="E91">A72</font></span></a>, <a class="ref" href="finseq_1.html#T21" onmouseover="rs('finseq_1/T21')" onmouseout="rh()">FINSEQ_1:21</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E20:39_10"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">MT</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <font color="Maroon" title="c10">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:39"><span class="lab"><font color="Green" title="E61">A31</font></span></a>, <a class="txt" href="#E3:39_10"><span class="lab"><font color="Green" title="E88">A69</font></span></a>, <a class="txt" href="#E18:39_10"><span class="lab"><font color="Green" title="E98">A79</font></span></a>, <a class="txt" href="#E7:39_10"><span class="lab"><font color="Green" title="E91">A72</font></span></a>, <a class="txt" href="#E16:39_10"><span class="lab"><font color="Green" title="E97">A78</font></span></a>, <a class="ref" href="funct_1.html#D7" onmouseover="rs('funct_1/D7')" onmouseout="rh()">FUNCT_1:def 7</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="E96:39"/><span class="lab"><font color="Green" title="E88">A80</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">MBCe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E85:39"><span class="lab"><font color="Green" title="E80">A50</font></span></a>, <a class="ref" href="finseq_4.html#T62" onmouseover="rs('finseq_4/T62')" onmouseout="rh()">FINSEQ_4:62</a></span>;<br/>
<a NAME="E97:39"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c30">MBCe</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c30">MBCe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">E</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">MBC</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:39"><span class="lab"><font color="Green" title="E58">A28</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c5">c1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:39"><span class="lab"><font color="Green" title="E72">A42</font></span></a>, <a class="txt" href="#E27:39"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="ref" href="funct_1.html#T34" onmouseover="rs('funct_1/T34')" onmouseout="rh()">FUNCT_1:34</a></span>
;<br/>
<a NAME="E99:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A81</font></span>: 
<font color="Maroon" title="c30">MBCe</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">MBCe</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c5">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E96:39"><span class="lab"><font color="Green" title="E88">A80</font></span></a>, <a class="ref" href="euclid.html#T70" onmouseover="rs('euclid/T70')" onmouseout="rh()">EUCLID:70</a></span>;<br/>
<font color="Maroon" title="c24">MT1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c10">B</font> = 
<font color="Maroon" title="c19">BF</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">MT</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <font color="Maroon" title="c10">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T70" onmouseover="rs('funct_1/T70')" onmouseout="rh()">FUNCT_1:70</a></span>
<br/>.= 
<font color="Maroon" title="c25">MT</font> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <font color="Maroon" title="c10">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:39"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E64:39"><span class="lab"><font color="Green" title="E68">A38</font></span></a>, <a class="ref" href="relat_1.html#T143" onmouseover="rs('relat_1/T143')" onmouseout="rh()">RELAT_1:143</a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E101:39"/><span class="kw">then </span>
<font color="Maroon" title="c24">MT1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c5">c1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c33">PPP</font> &amp; <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c19">BF</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E91:39"><span class="lab"><font color="Green" title="E84">A54</font></span></a>, <a class="txt" href="#E95:39"><span class="lab"><font color="Green" title="E87">A68</font></span></a></span>;<br/>
<a NAME="E102:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A82</font></span>: 
( <font color="Maroon" title="c33">PPP</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Maroon" title="c10">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:39"><span class="lab"><font color="Green" title="E64">A34</font></span></a>, <a class="txt" href="#E94:39"><span class="lab"><font color="Green" title="E86">A67</font></span></a>, <a class="txt" href="#E62:39"><span class="lab"><font color="Green" title="E66">A36</font></span></a>, <a class="ref" href="rlaffin3.html#T8" target="_self" onmouseover="rs('rlaffin3/T8')" onmouseout="rh()">Th8</a></span>;<br/>
<a NAME="E103:39"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c29">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:39"><span class="lab"><font color="Green" title="E57">A27</font></span></a>, <a class="txt" href="#E84:39"><span class="lab"><font color="Green" title="E79">A49</font></span></a>, <a class="ref" href="card_1.html#T5" onmouseover="rs('card_1/T5')" onmouseout="rh()">CARD_1:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E104:39"/>
( <font color="Maroon" title="c9">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Maroon" title="c10">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E99:39"><span class="lab"><font color="Green" title="E89">A81</font></span></a>, <a class="txt" href="#E102:39"><span class="lab"><font color="Green" title="E90">A82</font></span></a>, <a class="txt" href="#E86:39"><span class="lab"><font color="Green" title="E81">A51</font></span></a>, <a class="txt" href="rlaffin3.html#E33"><span class="lab"><font color="Green" title="E29">Lm7</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>
