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<span class="kw">let </span><font color="Maroon" title="c1">D</font>, <font color="Maroon" title="c2">C</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="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">A</font> being  <a href="rearran1.html#NM1" title="REARRAN1:NM.1">RearrangmentGen</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">F</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">C</font> holds <br/>(  <a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b1">F</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp;  <a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b1">F</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">F</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">F</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="rearran1.html#NM1" title="REARRAN1:NM.1">RearrangmentGen</a> of <font color="Maroon" title="c2">C</font>  st <font color="Maroon" title="c3">F</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">C</font> holds <br/>(  <a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c3">F</font>,<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c3">F</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp;  <a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c3">F</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c3">F</font>,<font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">F</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">F</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="rearran1.html#NM1" title="REARRAN1:NM.1">RearrangmentGen</a> of <font color="Maroon" title="c2">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">C</font> implies (  <a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">F</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp;  <a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">F</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">F</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">F</font> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 <span class="kw">and </span><br/><a NAME="E2:51"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">F</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp;  <a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">F</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">F</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">F</font> )</span><br/>

<a NAME="E3:51"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">|</a> <font color="Maroon" title="c1">D</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">D</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
<font color="Maroon" title="c1">D</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E37">A3</font></span></a></span>
;<br/>
<a NAME="E5:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A4</font></span>: 
 <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c3">F</font> <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">|</a> <font color="Maroon" title="c1">D</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#D14" title="RFUNCT_3:def.14">RFUNCT_3:def 14</a></span>;<br/>
<a NAME="E6:51"/>
 <a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font>, <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="rearran1.html#T24" target="_self" title="REARRAN1:th.24">Th24</a></span>;<br/>
<a NAME="E7:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A5</font></span>: 
 <a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">F</font> <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">|</a> <font color="Maroon" title="c1">D</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="classes1.html#T84" title="CLASSES1:th.84">CLASSES1:84</a></span>;<br/>
<a NAME="E8:51"/>
 <a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font>, <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="rearran1.html#T17" target="_self" title="REARRAN1:th.17">Th17</a></span>;<br/>
<a NAME="E9:51"/><span class="kw">then </span>
 <a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">F</font> <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">|</a> <font color="Maroon" title="c1">D</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="classes1.html#T84" title="CLASSES1:th.84">CLASSES1:84</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:51"/>
(  <a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">F</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp;  <a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">F</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E7:51"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">F</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">F</font> )</span><br/>

<span class="kw">hence </span><a NAME="E11:51"/>
(  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K3" title="REARRAN1:func.3">Rland</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">F</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="classes1.html#T83" title="CLASSES1:th.83">CLASSES1:83</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
