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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">X</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b4">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">X</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Olive" title="b2">r</font></span>)</span>,<font color="Olive" title="b3">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Olive" title="b2">r</font></span>)</span>,<font color="Olive" title="b3">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Olive" title="b2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">d</font> iff <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">d</font> )</span><br/><b>let </b><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">D</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="Olive" title="b2">X</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="Olive" title="b2">X</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Olive" title="b1">r</font></span>)</span>,<font color="Olive" title="b2">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Olive" title="b1">r</font></span>)</span>,<font color="Olive" title="b2">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">d</font> iff <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Olive" title="b2">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Olive" title="b2">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">d</font> )</span><br/><b>let </b><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="Olive" title="b1">X</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="Olive" title="b1">X</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Olive" title="b1">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Olive" title="b1">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">d</font> iff <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Olive" title="b1">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Olive" title="b1">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">d</font> )</span><br/><b>let </b><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st  <a href="relset_1.html#K4" title="RELSET_1:func.4">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">X</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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">X</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">d</font> iff <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">d</font> )</span><br/><b>let </b><font color="Maroon" title="c5">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K4" title="RELSET_1:func.4">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">X</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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">X</font></span>)</span> implies ( <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> iff <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> ) )</span><br/>









<b>set </b><font color="Maroon" title="c6">dx</font> =  <a href="relset_1.html#K4" title="RELSET_1:func.4">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">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">drx</font> =  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">frx</font> =  <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font>;<br/>
<b>set </b><font color="Maroon" title="c9">fx</font> =  <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font>;<br/>
<b>assume </b><a NAME="E1:96"/><i><font color="Green" title="E62">A1</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">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">X</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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">X</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> iff <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c10">dx</font> =  <a href="relset_1.html#K4" title="RELSET_1:func.4">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">X</font></span>)</span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E3:96"/><i><font color="Green" title="E63">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></i>;<br/>
<i><font color="Green" title="E64">A3</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></i>
<br/>.= 
<font color="Maroon" title="c10">dx</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/>
<a NAME="E5:96"/><b>then </b><i><font color="Green" title="E65">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <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="c10">dx</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <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="c10">dx</font> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T70" target="_self" title="RFUNCT_3:th.70">Th70</a></i>;<br/>
<a NAME="E6:96"/><i><font color="Green" title="E66">A5</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:96"/>
(  <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  &amp;  <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a>  )
 
<b>by </b><i><a class="txt" href="#E4:96"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="rfunct_3.html#D14" target="_self" title="RFUNCT_3:def.14">Def14</a></i>;<br/>
<a NAME="E8:96"/><b>then </b><i><font color="Green" title="E67">A6</font></i>: 
(  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</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> <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">X</font></span>)</span> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="classes1.html#T83" title="CLASSES1:th.83">CLASSES1:83</a></i>;<br/>
<a NAME="E9:96"/><i><font color="Green" title="E68">A7</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c10">dx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">X</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E10:96"/><b>then </b><i><font color="Green" title="E69">A8</font></i>: 
<font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:96"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="#E4:96"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E11:96"/><b>then </b><i><font color="Green" title="E70">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="#E3:96"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></i>;<br/>
<a NAME="E12:96"/><i><font color="Green" title="E71">A10</font></i>: 
<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">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</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">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:96"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:96"/>
 <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E1:96"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="#E8:96"><i><font color="Green" title="E67">A6</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>
<a NAME="E14:96"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> )
 
;<br/>
<a NAME="E15:96"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E16:96"/><b>then </b><i><font color="Green" title="E72">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E17:96"/>
<font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</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">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:96"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="#E12:96"><i><font color="Green" title="E71">A10</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E19:96"/><i><font color="Green" title="E73">A12</font></i>: 
( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="txt" href="#E8:96"><i><font color="Green" title="E67">A6</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E20:96"/><i><font color="Green" title="E74">A13</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E19:96"><i><font color="Green" title="E73">A12</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>thus </b><a NAME="E21:96"/>
( <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> implies <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> implies <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:96_2"/><i><font color="Green" title="E75">A14</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font>
 <b>and </b><br/><a NAME="E2:96_2"/><i><font color="Green" title="E76">A15</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:96_2"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E16:96"><i><font color="Green" title="E72">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><b> such that </b><br/><a NAME="E5:96_2"/><i><font color="Green" title="E77">A16</font></i>: 
( <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">dx</font> &amp; <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">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E8:96"><i><font color="Green" title="E67">A6</font></i></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E6:96_2"/><i><font color="Green" title="E78">A17</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:96_2"><i><font color="Green" title="E77">A16</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E7:96_2"/><i><font color="Green" title="E79">A18</font></i>: 
<font color="Maroon" title="c12">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:96"><i><font color="Green" title="E64">A3</font></i></a>, <a class="txt" href="#E9:96"><i><font color="Green" title="E68">A7</font></i></a>, <a class="txt" href="#E5:96_2"><i><font color="Green" title="E77">A16</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font>
<b>by </b><i><a class="txt" href="#E4:96"><i><font color="Green" title="E64">A3</font></i></a>, <a class="txt" href="#E5:96_2"><i><font color="Green" title="E77">A16</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font>
<b>by </b><i><a class="txt" href="#E3:96"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="#E7:96_2"><i><font color="Green" title="E79">A18</font></i></a>, <a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></i>
;<br/>
<a NAME="E9:96_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:96"><i><font color="Green" title="E64">A3</font></i></a>, <a class="txt" href="#E5:96_2"><i><font color="Green" title="E77">A16</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:96_2"/><i><font color="Green" title="E80">A19</font></i>: 
( <font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font> )
 <b>by </b><i><a class="txt" href="#E8:96"><i><font color="Green" title="E67">A6</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E12:96_2"/><i><font color="Green" title="E81">A20</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> &amp; <font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E11:96_2"><i><font color="Green" title="E80">A19</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E13:96_2"/>
<font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E19:96"><i><font color="Green" title="E73">A12</font></i></a>, <a class="txt" href="#E20:96"><i><font color="Green" title="E74">A13</font></i></a>, <a class="txt" href="#E2:96_2"><i><font color="Green" title="E76">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E14:96_2"/><b>then </b><i><font color="Green" title="E82">A21</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font>
 
<b>by </b><i><a class="txt" href="#E16:96"><i><font color="Green" title="E72">A11</font></i></a>, <a class="txt" href="#E19:96"><i><font color="Green" title="E73">A12</font></i></a>, <a class="txt" href="#E6:96_2"><i><font color="Green" title="E78">A17</font></i></a>, <a class="ref" href="rfinseq.html#T32" title="RFINSEQ:th.32">RFINSEQ:32</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_2_2_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_2_1_1"><b>case </b></a><a NAME="E1:96_2_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:96_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="#E3:96"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="#E10:96"><i><font color="Green" title="E69">A8</font></i></a>, <a class="txt" href="#E1:96_2"><i><font color="Green" title="E75">A14</font></i></a>, <a class="txt" href="#E11:96_2"><i><font color="Green" title="E80">A19</font></i></a>, <a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></i>;<br/><b>hence </b><a NAME="E3:96_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:96_2"><i><font color="Green" title="E76">A15</font></i></a>, <a class="txt" href="#E5:96_2"><i><font color="Green" title="E77">A16</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_2_2_1_2"><b>case </b></a><a NAME="E1:96_2_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:96_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E12:96_2"><i><font color="Green" title="E81">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:96_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="#E5:96"><i><font color="Green" title="E65">A4</font></i></a>, <a class="txt" href="#E6:96"><i><font color="Green" title="E66">A5</font></i></a>, <a class="txt" href="#E11:96"><i><font color="Green" title="E70">A9</font></i></a>, <a class="txt" href="#E16:96"><i><font color="Green" title="E72">A11</font></i></a>, <a class="txt" href="#E1:96_2"><i><font color="Green" title="E75">A14</font></i></a>, <a class="txt" href="#E11:96_2"><i><font color="Green" title="E80">A19</font></i></a>, <a class="ref" href="rfinseq.html#T32" title="RFINSEQ:th.32">RFINSEQ:32</a></i>;<br/><a NAME="E4:96_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E5:96_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:96_2"><i><font color="Green" title="E76">A15</font></i></a>, <a class="txt" href="#E6:96_2"><i><font color="Green" title="E78">A17</font></i></a>, <a class="txt" href="#E14:96_2"><i><font color="Green" title="E82">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E16:96_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E22:96"/><i><font color="Green" title="E75">A22</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font>
 <b>and </b><br/><a NAME="E23:96"/><i><font color="Green" title="E76">A23</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E24:96"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:96"><i><font color="Green" title="E65">A4</font></i></a>, <a class="txt" href="#E6:96"><i><font color="Green" title="E66">A5</font></i></a>, <a class="txt" href="#E16:96"><i><font color="Green" title="E72">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><b> such that </b><br/><a NAME="E26:96"/><i><font color="Green" title="E77">A24</font></i>: 
( <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E8:96"><i><font color="Green" title="E67">A6</font></i></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E27:96"/><i><font color="Green" title="E78">A25</font></i>: 
<font color="Maroon" title="c12">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E9:96"><i><font color="Green" title="E68">A7</font></i></a>, <a class="txt" href="#E26:96"><i><font color="Green" title="E77">A24</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<i><font color="Green" title="E79">A26</font></i>: <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font>
<b>by </b><i><a class="txt" href="#E26:96"><i><font color="Green" title="E77">A24</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font>
<b>by </b><i><a class="txt" href="#E3:96"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="#E27:96"><i><font color="Green" title="E78">A25</font></i></a>, <a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></i>
;<br/>
<a NAME="E29:96"/>
<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">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font>
 
<b>by </b><i><a class="txt" href="#E4:96"><i><font color="Green" title="E64">A3</font></i></a>, <a class="txt" href="#E26:96"><i><font color="Green" title="E77">A24</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E30:96"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</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">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:96"><i><font color="Green" title="E64">A3</font></i></a>, <a class="txt" href="#E26:96"><i><font color="Green" title="E77">A24</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E32:96"/><i><font color="Green" title="E80">A27</font></i>: 
( <font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font> )
 <b>by </b><i><a class="txt" href="#E8:96"><i><font color="Green" title="E67">A6</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E33:96"/><i><font color="Green" title="E81">A28</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> &amp; <font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E32:96"><i><font color="Green" title="E80">A27</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E34:96"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:96"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="#E4:96"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E35:96"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:96"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="#E4:96"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">n1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E37:96"/><i><font color="Green" title="E82">A29</font></i>: 
( <font color="Maroon" title="c14">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font> )
 <b>by </b><i><a class="txt" href="#E8:96"><i><font color="Green" title="E67">A6</font></i></a>, <a class="txt" href="#E11:96"><i><font color="Green" title="E70">A9</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E38:96"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n1</font> &amp; <font color="Maroon" title="c14">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E37:96"><i><font color="Green" title="E82">A29</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E39:96"/><b>then </b>
<font color="Maroon" title="c14">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_1.html#K15" title="VALUED_1:func.15">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E11:96"><i><font color="Green" title="E70">A9</font></i></a>, <a class="txt" href="#E23:96"><i><font color="Green" title="E76">A23</font></i></a>, <a class="txt" href="#E37:96"><i><font color="Green" title="E82">A29</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E40:96"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="#E5:96"><i><font color="Green" title="E65">A4</font></i></a>, <a class="txt" href="#E6:96"><i><font color="Green" title="E66">A5</font></i></a>, <a class="txt" href="#E16:96"><i><font color="Green" title="E72">A11</font></i></a>, <a class="txt" href="#E28:96"><i><font color="Green" title="E79">A26</font></i></a>, <a class="txt" href="#E37:96"><i><font color="Green" title="E82">A29</font></i></a>, <a class="ref" href="rfinseq.html#T32" title="RFINSEQ:th.32">RFINSEQ:32</a></i>;<br/>
<a NAME="E41:96"/><b>then </b><i><font color="Green" title="E83">A30</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_4_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_4_1_1"><b>case </b></a><a NAME="E1:96_4_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:96_4_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:96"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="#E10:96"><i><font color="Green" title="E69">A8</font></i></a>, <a class="txt" href="#E22:96"><i><font color="Green" title="E75">A22</font></i></a>, <a class="txt" href="#E23:96"><i><font color="Green" title="E76">A23</font></i></a>, <a class="txt" href="#E28:96"><i><font color="Green" title="E79">A26</font></i></a>, <a class="txt" href="#E32:96"><i><font color="Green" title="E80">A27</font></i></a>, <a class="ref" href="valued_1.html#T3" title="VALUED_1:th.3">VALUED_1:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_4_1_2"><b>case </b></a><a NAME="E1:96_4_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:96_4_1_2"/><b>then </b>
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E33:96"><i><font color="Green" title="E81">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:96_4_1_2"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">d1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="#E16:96"><i><font color="Green" title="E72">A11</font></i></a>, <a class="txt" href="#E22:96"><i><font color="Green" title="E75">A22</font></i></a>, <a class="txt" href="#E32:96"><i><font color="Green" title="E80">A27</font></i></a>, <a class="ref" href="rfinseq.html#T32" title="RFINSEQ:th.32">RFINSEQ:32</a></i>;<br/><b>hence </b><a NAME="E4:96_4_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E11:96"><i><font color="Green" title="E70">A9</font></i></a>, <a class="txt" href="#E23:96"><i><font color="Green" title="E76">A23</font></i></a>, <a class="txt" href="#E28:96"><i><font color="Green" title="E79">A26</font></i></a>, <a class="txt" href="#E41:96"><i><font color="Green" title="E83">A30</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E43:96"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
