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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">F</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">I</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b2">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I</font>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a><br/>  for <font color="Olive" title="b3">J</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> holds <br/> <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Olive" title="b3">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">Sum</a> (<font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">J</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">I</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I</font>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a><br/>  for <font color="Olive" title="b2">J</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> holds <br/> <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Olive" title="b2">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">Sum</a> (<font color="Maroon" title="c2">F</font>,<font color="Olive" title="b2">J</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I</font>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">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">J</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>  st <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> holds <br/> <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Olive" title="b1">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">Sum</a> (<font color="Maroon" title="c2">F</font>,<font color="Olive" title="b1">J</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">J</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</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="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> implies  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">Sum</a> (<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">J</font>) )</span><br/>







<span class="kw">reconsider </span><font color="Maroon" title="c5">J9</font> = <font color="Maroon" title="c4">J</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">J</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D1" onmouseover="rs('zfmisc_1/D1')" onmouseout="rh()">ZFMISC_1:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">f9</font> = <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c5">J9</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c4">J</font> ;<br/>
<a NAME="E3:35"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">J</font></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<span class="kw">assume </span><a NAME="E4:35"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">Sum</a> (<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">J</font>)</span><br/>

<a NAME="E5:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c4">J</font>, <a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">FinS</a> (<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">J</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="#E3:35"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="rfunct_3.html#D13" onmouseover="rs('rfunct_3/D13')" onmouseout="rh()">RFUNCT_3:def 13</a></span>;<br/>
<a NAME="E6:35"/><span class="lab"><font color="Green" title="E35">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">I</font>
 
<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="E7:35"/>
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c6">f9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">J9</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">fs</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E9:35"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c5">J9</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:35"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <font color="Maroon" title="c6">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c7">fs</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T14" onmouseover="rs('uproots/T14')" onmouseout="rh()">UPROOTS:14</a></span>;<br/>
<span class="lab"><font color="Green" title="E38">A6</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c4">J</font></span>)</span> = 
<font color="Maroon" title="c4">J</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>
;<br/>
<a NAME="E12:35"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">fs</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E13:35"/><span class="kw">then </span>
<font color="Maroon" title="c7">fs</font>,<font color="Maroon" title="c6">f9</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="#E9:35"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E11:35"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="classes1.html#T77" onmouseover="rs('classes1/T77')" onmouseout="rh()">CLASSES1:77</a></span>;<br/>
<a NAME="E14:35"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">FinS</a> (<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">J</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="classes1.html#T76" onmouseover="rs('classes1/T76')" onmouseout="rh()">CLASSES1:76</a>, <a class="ref" href="rfinseq.html#T9" onmouseover="rs('rfinseq/T9')" onmouseout="rh()">RFINSEQ:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:35"/>
 <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">Sum</a> (<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">J</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="rfunct_3.html#D14" onmouseover="rs('rfunct_3/D14')" onmouseout="rh()">RFUNCT_3:def 14</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/>


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