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<span class="kw">let </span><font color="Maroon" title="c1">Omega</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of  <a href="random_1.html#K1" title="RANDOM_1:func.1">Trivial-SigmaField</a> <font color="Maroon" title="c1">Omega</font><br/>  for <font color="Olive" title="b2">Y</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Omega</font><br/>  for <font color="Olive" title="b3">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">Omega</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ex <font color="Olive" title="b4">G</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>  ex <font color="Olive" title="b5">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Olive" title="b2">Y</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b5">s</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Y</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">Y</font> &amp; (  for <font color="Olive" title="b6">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b6">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> <font color="Olive" title="b4">G</font> holds <br/><font color="Olive" title="b4">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Olive" title="b1">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">Y</font></span>)</span>) <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="Olive" title="b4">G</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of  <a href="random_1.html#K1" title="RANDOM_1:func.1">Trivial-SigmaField</a> <font color="Maroon" title="c1">Omega</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Y</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Omega</font><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">Omega</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ex <font color="Olive" title="b3">G</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>  ex <font color="Olive" title="b4">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Olive" title="b1">Y</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b4">s</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Y</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">Y</font> &amp; (  for <font color="Olive" title="b5">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">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> <font color="Olive" title="b3">G</font> holds <br/><font color="Olive" title="b3">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c2">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">Y</font></span>)</span>) <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="Olive" title="b3">G</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">Y</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Omega</font>; <a class="txt" 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">Omega</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ex <font color="Olive" title="b2">G</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>  ex <font color="Olive" title="b3">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">Y</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Olive" title="b3">s</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Y</font> &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">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> <font color="Olive" title="b2">G</font> holds <br/><font color="Olive" title="b2">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">n</font> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c2">P</font></span>)</span>,<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="Maroon" title="c3">Y</font></span>)</span>) <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="Olive" title="b2">G</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">Omega</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G</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>  ex <font color="Olive" title="b2">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">Y</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Olive" title="b2">s</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Y</font> &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">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> <font color="Olive" title="b1">G</font> holds <br/><font color="Olive" title="b1">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c2">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">Y</font></span>)</span>) <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="Olive" title="b1">G</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">YN</font> = <font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font>;<br/>
<a NAME="E1:24"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font> &amp;  <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="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<span class="lab"><font color="Green" title="E23">A2</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">Omega</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">Omega</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>
;<br/>
<a NAME="E3:24"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c4">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">Omega</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">G</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> , <font color="Maroon" title="c8">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Omega</font><span class="kw"> such that </span><br/><a NAME="E6:24"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">Omega</font> &amp; <font color="Maroon" title="c8">s</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">Omega</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> <font color="Maroon" title="c7">G</font> holds <br/><font color="Maroon" title="c7">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c2">P</font></span>)</span>,<font color="Maroon" title="c6">g</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">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="random_1.html#T13" title="RANDOM_1:th.13">RANDOM_1:13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c6">g</font> as    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of  <a href="random_1.html#K1" title="RANDOM_1:func.1">Trivial-SigmaField</a> <font color="Maroon" title="c1">Omega</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="random_1.html#T29" title="RANDOM_1:th.29">RANDOM_1:29</a></span>;<br/>
<a NAME="E8:24"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E9:24"/>
<font color="Maroon" title="c9">g</font> <a href="random_1.html#R1" title="RANDOM_1:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="random_1.html#T30" title="RANDOM_1:th.30">RANDOM_1:30</a></span>;<br/>
<a NAME="E10:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a>  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="random_1.html#D3" title="RANDOM_1:def.3">RANDOM_1:def 3</a></span>;<br/>
<a NAME="E11:24"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<a NAME="E12:24"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c2">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p2">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c2">P</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">Y</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p2">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c2">P</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="mesfunc6.html#T92" title="MESFUNC6:th.92">MESFUNC6:92</a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span> = 
<font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c1">Omega</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>
<br/>.= 
<font color="Maroon" title="c1">Omega</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>
;<br/>
<a NAME="E14:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c9">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T33" title="FUNCT_2:th.33">FUNCT_2:33</a></span>;<br/>
<a NAME="E15:24"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">Y</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="funct_4.html#T72" title="FUNCT_4:th.72">FUNCT_4:72</a></span>;<br/>
<a NAME="E16:24"/><span class="lab"><font color="Green" title="E31">A10</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="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E17:24"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_3"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:24_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A13</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E3:24_3"/>
<font color="Maroon" title="c9">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_3"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E16:24"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:24_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_3"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span> <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c2">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>) = 
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:24"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="mesfunc6.html#T97" title="MESFUNC6:th.97">MESFUNC6:97</a></span>
<br/>.= 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>

;<br/>
<a NAME="E19:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A14</font></span>: 
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c2">P</font></span>)</span>,<font color="Maroon" title="c9">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c2">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">Y</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:24"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E14:24"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">N1</font> = <font color="Maroon" title="c8">s</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c3">Y</font>;<br/>
<a NAME="E20:24"/>
<font color="Maroon" title="c8">s</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font></span>)</span>,<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T1" title="FUNCT_2:th.1">FUNCT_2:1</a></span>;<br/>
<a NAME="E21:24"/><span class="kw">then </span>
<font color="Maroon" title="c8">s</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c3">Y</font> is   <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 <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T39" title="FUNCT_2:th.39">FUNCT_2:39</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">N1</font> = <font color="Maroon" title="c8">s</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c3">Y</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 <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E23:24"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">N1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E24:24"/><span class="kw">then </span>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">N1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <font color="Maroon" title="c7">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E25:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A15</font></span>: 
 <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">N1</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <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> <font color="Maroon" title="c7">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E26:24"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> <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> <font color="Maroon" title="c7">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E27:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<font color="Maroon" title="c8">s</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">Omega</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="funct_2.html#T1" title="FUNCT_2:th.1">FUNCT_2:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">t1</font> = <font color="Maroon" title="c8">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">N1</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Omega</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:24"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="finseq_2.html#T32" title="FINSEQ_2:th.32">FINSEQ_2:32</a></span>;<br/>
<span class="lab"><font color="Green" title="E36">A17</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">t1</font> = 
<font color="Maroon" title="c8">s</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">N1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T127" title="RELAT_1:th.127">RELAT_1:127</a></span>
<br/>.= 
<font color="Maroon" title="c8">s</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">N1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c3">Y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="funct_1.html#T77" title="FUNCT_1:th.77">FUNCT_1:77</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">N2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c11">N1</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">N2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c11">N1</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 <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E31:24"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c14">N2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E32:24"/><span class="kw">then </span>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c14">N2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <font color="Maroon" title="c7">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E33:24"/><span class="kw">then </span>
 <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c14">N2</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <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> <font color="Maroon" title="c7">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">t2</font> = <font color="Maroon" title="c8">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c14">N2</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Omega</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:24"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="ref" href="finseq_2.html#T32" title="FINSEQ_2:th.32">FINSEQ_2:32</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">t</font> = <font color="Maroon" title="c12">t1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c15">t2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Omega</font> ;<br/>
<a NAME="E36:24"/><span class="lab"><font color="Green" title="E37">A18</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">N1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E37:24"/><span class="lab"><font color="Green" title="E38">A19</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c14">N2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E38:24"/><span class="lab"><font color="Green" title="E39">A20</font></span>: 
( <font color="Maroon" title="c11">N1</font> is   <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 <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font></span>)</span> &amp; <font color="Maroon" title="c14">N2</font> is   <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 <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E39:24"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">N1</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c14">N2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="random_2.html#T1" target="_self" title="RANDOM_2:th.1">Th1</a>, <a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<a NAME="E40:24"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">t1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c14">N2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:24"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="ref" href="random_2.html#T2" target="_self" title="RANDOM_2:th.2">Th2</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<a NAME="E41:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A21</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">t1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c15">t2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:24"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="ref" href="random_2.html#T2" target="_self" title="RANDOM_2:th.2">Th2</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<a NAME="E42:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A22</font></span>: 
<font color="Maroon" title="c16">t</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="finseq_3.html#T91" title="FINSEQ_3:th.91">FINSEQ_3:91</a></span>;<br/>
<a NAME="E43:24"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">N1</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="c11">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T3" title="UPROOTS:th.3">UPROOTS:3</a></span>;<br/>
<a NAME="E44:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A23</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">N1</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E45:24"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c11">N1</font></span>)</span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:24"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E46:24"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">t1</font> <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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:24"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
<a NAME="E47:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A24</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E48:24"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c14">N2</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="c14">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T3" title="UPROOTS:th.3">UPROOTS:3</a></span>;<br/>
<a NAME="E49:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A25</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c14">N2</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">N2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E50:24"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c14">N2</font></span>)</span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:24"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E51:24"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">t2</font> <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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">N2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:24"><span class="lab"><font color="Green" title="E44">A25</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
<a NAME="E52:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A26</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="lab"><font color="Green" title="E46">A27</font></span>: <font color="Maroon" title="c11">N1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c14">N2</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">G</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">N1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>
<br/>.= 
 <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> <font color="Maroon" title="c7">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">t</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">t1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">t2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">N1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c14">N2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:24"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="txt" href="#E52:24"><span class="lab"><font color="Green" title="E45">A26</font></span></a>, <a class="ref" href="card_2.html#T40" title="CARD_2:th.40">CARD_2:40</a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>
;<br/>
<a NAME="E55:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A28</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">N1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c14">N2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E56:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A29</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:24"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="card_1.html#T62" title="CARD_1:th.62">CARD_1:62</a></span>;<br/>
<a NAME="E57:24"/><span class="lab"><font color="Green" title="E49">A30</font></span>: 
 <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> <font color="Maroon" title="c8">s</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> <font color="Maroon" title="c16">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:24"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="txt" href="#E53:24"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="card_1.html#T62" title="CARD_1:th.62">CARD_1:62</a></span>;<br/>
<span class="lab"><font color="Green" title="E50">A31</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> = 
 <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> <font color="Maroon" title="c8">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:24"><span class="lab"><font color="Green" title="E49">A30</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E59:24"/><span class="lab"><font color="Green" title="E51">A32</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">t</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="c1">Omega</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E56:24"><span class="lab"><font color="Green" title="E48">A29</font></span></a>, <a class="ref" href="card_1.html#T62" title="CARD_1:th.62">CARD_1:62</a></span>;<br/>
<a NAME="E60:24"/>
<font color="Maroon" title="c16">t</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">t</font></span>)</span>,<font color="Maroon" title="c1">Omega</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T26" title="FINSEQ_2:th.26">FINSEQ_2:26</a></span>;<br/>
<a NAME="E61:24"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c16">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E42:24"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="txt" href="#E59:24"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="ref" href="finseq_4.html#T63" title="FINSEQ_4:th.63">FINSEQ_4:63</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">PN</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E63:24"/><span class="lab"><font color="Green" title="E52">A33</font></span>: 
( <font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">PN</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">PN</font> <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="c8">s</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c17">PN</font> <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="c8">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E42:24"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="ref" href="bhsp_5.html#T1" title="BHSP_5:th.1">BHSP_5:1</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( ( <font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; (  not <font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) );<br/>
<div><span class="lab"><font color="Green" title="E53">A34</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_9"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">Omega</font></span>)</span> implies  ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Olive" title="b1">x</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:24_9"/>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">Omega</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Olive" title="b1">x</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_9_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_9_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_9_1_1"/>
( <font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> or  not <font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_9_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_9_1_1_1"/>
<font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:24_9_1_1_1"/>
 ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Olive" title="b1">x</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_9_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_9_1_1_2"/>
 not <font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:24_9_1_1_2"/>
 ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Olive" title="b1">x</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:24_9"/>
 ex <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Olive" title="b1">x</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c18">F</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="E66:24"/><span class="lab"><font color="Green" title="E54">A35</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">F</font> <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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">Omega</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E67:24"/><span class="lab"><font color="Green" title="E55">A36</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">Omega</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c18">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="#E64:24"><span class="lab"><font color="Green" title="E53">A34</font></span></a>);<br/></span>
<span class="lab"><font color="Green" title="E56">A37</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> = 
 <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> <font color="Maroon" title="c7">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">G</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E69:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A38</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="c7">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">PN</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">PN</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:24"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
<a NAME="E70:24"/><span class="lab"><font color="Green" title="E58">A39</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">F</font> <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="c8">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E66:24"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_11"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">x</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> <font color="Maroon" title="c18">F</font> implies <font color="Maroon" title="c18">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">PN</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:24_11"/><span class="lab"><font color="Green" title="E59">A40</font></span>: 
<font color="Maroon" title="c19">x</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> <font color="Maroon" title="c18">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">PN</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">nx</font> = <font color="Maroon" title="c19">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E3:24_11"/><span class="lab"><font color="Green" title="E60">A41</font></span>: 
<font color="Maroon" title="c20">nx</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> <font color="Maroon" title="c16">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_11"><span class="lab"><font color="Green" title="E59">A40</font></span></a>, <a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E66:24"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="txt" href="#E58:24"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E4:24_11"/><span class="lab"><font color="Green" title="E61">A42</font></span>: 
<font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">PN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:24"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="txt" href="#E1:24_11"><span class="lab"><font color="Green" title="E59">A40</font></span></a>, <a class="txt" href="#E70:24"><span class="lab"><font color="Green" title="E58">A39</font></span></a>, <a class="txt" href="#E58:24"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E5:24_11"/>
<font color="Maroon" title="c17">PN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</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> <font color="Maroon" title="c7">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:24"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="txt" href="#E3:24_11"><span class="lab"><font color="Green" title="E60">A41</font></span></a>, <a class="txt" href="#E63:24"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><a NAME="E6:24_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A43</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">PN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">PN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">PN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_11_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_11_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_11_1_1"/>
( <font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">nx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> or  not <font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">nx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_11_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_11_1_1_1"/><span class="lab"><font color="Green" title="E63">A44</font></span>: 
<font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">nx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">PN</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c18">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">nx</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">nx</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:24"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E66:24"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="txt" href="#E1:24_11"><span class="lab"><font color="Green" title="E59">A40</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">PN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">PN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_11"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="txt" href="#E1:24_11_1_1_1"><span class="lab"><font color="Green" title="E63">A44</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">PN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">PN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_11_1_1_1"><span class="lab"><font color="Green" title="E63">A44</font></span></a>, <a class="txt" href="#E4:24_11"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="txt" href="#E15:24"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">PN</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:24"><span class="lab"><font color="Green" title="E58">A39</font></span></a>, <a class="txt" href="#E1:24_11"><span class="lab"><font color="Green" title="E59">A40</font></span></a>, <a class="txt" href="#E63:24"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="txt" href="#E6:24_11"><span class="lab"><font color="Green" title="E62">A43</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_11_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_11_1_1_2"/>
 not <font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">nx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">PN</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c18">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">PN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">PN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_11"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="txt" href="#E67:24"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E66:24"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="txt" href="#E1:24_11"><span class="lab"><font color="Green" title="E59">A40</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">PN</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">nx</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:24"><span class="lab"><font color="Green" title="E58">A39</font></span></a>, <a class="txt" href="#E1:24_11"><span class="lab"><font color="Green" title="E59">A40</font></span></a>, <a class="txt" href="#E63:24"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="txt" href="#E6:24_11"><span class="lab"><font color="Green" title="E62">A43</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E8:24_11"/>
<font color="Maroon" title="c18">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">PN</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E72:24"/><span class="kw">then </span>
<font color="Maroon" title="c18">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">PN</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:24"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="txt" href="#E69:24"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E70:24"><span class="lab"><font color="Green" title="E58">A39</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>;<br/>
<a NAME="E73:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A45</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c7">G</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="c18">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:24"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="ref" href="finsop_1.html#T7" title="FINSOP_1:th.7">FINSOP_1:7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">t1</font> = <font color="Maroon" title="c12">t1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">F1</font> = <font color="Maroon" title="c18">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font></span>)</span> as    <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>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">F2</font> = <font color="Maroon" title="c18">F</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font></span>)</span> as    <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>  ;<br/>
<a NAME="E77:24"/><span class="lab"><font color="Green" title="E60">A46</font></span>: 
<font color="Maroon" title="c18">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T8" title="RFINSEQ:th.8">RFINSEQ:8</a></span>;<br/>
<a NAME="E78:24"/><span class="lab"><font color="Green" title="E61">A47</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="finseq_4.html#T62" title="FINSEQ_4:th.62">FINSEQ_4:62</a></span>;<br/>
<a NAME="E79:24"/><span class="lab"><font color="Green" title="E62">A48</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">Omega</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:24"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E80:24"/><span class="lab"><font color="Green" title="E63">A49</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:24"><span class="lab"><font color="Green" title="E48">A29</font></span></a>, <a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E66:24"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E81:24"/><span class="lab"><font color="Green" title="E64">A50</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">t2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E82:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A51</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</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> <font color="Maroon" title="c16">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E66">A52</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">F2</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:24"><span class="lab"><font color="Green" title="E63">A49</font></span></a>, <a class="ref" href="rfinseq.html#D1" title="RFINSEQ:def.1">RFINSEQ:def 1</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">t2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:24"><span class="lab"><font color="Green" title="E64">A50</font></span></a>, <a class="txt" href="#E80:24"><span class="lab"><font color="Green" title="E63">A49</font></span></a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E67">A53</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">F2</font> = 
 <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> <font color="Maroon" title="c15">t2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">t2</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_14"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">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> <font color="Maroon" title="c21">F2</font> implies <font color="Maroon" title="c21">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">t2</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">m</font> )</span><br/><span class="kw">assume </span><a NAME="E1:24_14"/><span class="lab"><font color="Green" title="E68">A54</font></span>: 
<font color="Maroon" title="c22">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> <font color="Maroon" title="c21">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c21">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">t2</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">m</font></span><br/><a NAME="E2:24_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A55</font></span>: 
<font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c15">t2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:24"><span class="lab"><font color="Green" title="E66">A52</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E3:24_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A56</font></span>: 
<font color="Maroon" title="c22">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> <font color="Maroon" title="c15">t2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E4:24_14"/><span class="lab"><font color="Green" title="E71">A57</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">m</font> &amp; <font color="Maroon" title="c22">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> <font color="Maroon" title="c15">t2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_14"><span class="lab"><font color="Green" title="E69">A55</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E5:24_14"/>
<font color="Maroon" title="c22">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E6:24_14"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font></span>)</span> &amp; <font color="Maroon" title="c22">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font></span>)</span> <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> <font color="Maroon" title="c16">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:24"><span class="lab"><font color="Green" title="E64">A50</font></span></a>, <a class="txt" href="#E4:24_14"><span class="lab"><font color="Green" title="E71">A57</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:24_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A58</font></span>: 
<font color="Maroon" title="c22">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">Omega</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E56:24"><span class="lab"><font color="Green" title="E48">A29</font></span></a></span>;<br/><a NAME="E8:24_14"/><span class="lab"><font color="Green" title="E73">A59</font></span>: 
<font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">t2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_14"><span class="lab"><font color="Green" title="E70">A56</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E9:24_14"/><span class="lab"><font color="Green" title="E74">A60</font></span>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">t1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c15">t2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:24"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E10:24_14"/><span class="lab"><font color="Green" title="E75">A61</font></span>: 
<font color="Maroon" title="c15">t2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c15">t2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_14"><span class="lab"><font color="Green" title="E70">A56</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/><a NAME="E11:24_14"/><span class="kw">then </span>
 not <font color="Maroon" title="c15">t2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">t1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_14"><span class="lab"><font color="Green" title="E74">A60</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E12:24_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A62</font></span>: 
<font color="Maroon" title="c15">t2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:24"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E10:24_14"><span class="lab"><font color="Green" title="E75">A61</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A63</font></span>: <font color="Maroon" title="c9">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">t2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">m</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">t2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:24"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E12:24_14"><span class="lab"><font color="Green" title="E76">A62</font></span></a></span>
;<br/><a NAME="E14:24_14"/>
 not <font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E9:24_14"><span class="lab"><font color="Green" title="E74">A60</font></span></a>, <a class="txt" href="#E10:24_14"><span class="lab"><font color="Green" title="E75">A61</font></span></a>, <a class="txt" href="#E8:24_14"><span class="lab"><font color="Green" title="E73">A59</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A64</font></span>: <font color="Maroon" title="c18">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:24"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E7:24_14"><span class="lab"><font color="Green" title="E72">A58</font></span></a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:24_14"><span class="lab"><font color="Green" title="E77">A63</font></span></a>, <a class="txt" href="#E8:24_14"><span class="lab"><font color="Green" title="E73">A59</font></span></a></span>
;<br/><span class="kw">thus </span><font color="Maroon" title="c21">F2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">m</font> = 
<font color="Maroon" title="c18">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:24"><span class="lab"><font color="Green" title="E63">A49</font></span></a>, <a class="txt" href="#E82:24"><span class="lab"><font color="Green" title="E65">A51</font></span></a>, <a class="txt" href="#E1:24_14"><span class="lab"><font color="Green" title="E68">A54</font></span></a>, <a class="ref" href="rfinseq.html#D1" title="RFINSEQ:def.1">RFINSEQ:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">t2</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_14"><span class="lab"><font color="Green" title="E69">A55</font></span></a>, <a class="txt" href="#E15:24_14"><span class="lab"><font color="Green" title="E78">A64</font></span></a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E86:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A65</font></span>: 
<font color="Maroon" title="c21">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">t2</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:24"><span class="lab"><font color="Green" title="E67">A53</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>;<br/>
<a NAME="E87:24"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">Omega</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/>
<a NAME="E88:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A66</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">Omega</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<span class="lab"><font color="Green" title="E70">A67</font></span>:  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c18">F</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c20">F1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c21">F2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:24"><span class="lab"><font color="Green" title="E60">A46</font></span></a>, <a class="ref" href="rvsum_1.html#T75" title="RVSUM_1:th.75">RVSUM_1:75</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c20">F1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E86:24"><span class="lab"><font color="Green" title="E68">A65</font></span></a>, <a class="ref" href="rvsum_1.html#T81" title="RVSUM_1:th.81">RVSUM_1:81</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c20">F1</font>

;<br/>
<a NAME="E90:24"/><span class="lab"><font color="Green" title="E71">A68</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:24"><span class="lab"><font color="Green" title="E62">A48</font></span></a>, <a class="txt" href="#E78:24"><span class="lab"><font color="Green" title="E61">A47</font></span></a>, <a class="txt" href="#E88:24"><span class="lab"><font color="Green" title="E69">A66</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E91:24"/>
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> <font color="Maroon" title="c20">F1</font> holds <br/><font color="Maroon" title="c20">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_16"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">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> <font color="Maroon" title="c20">F1</font> implies <font color="Maroon" title="c20">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_16"/>
<font color="Maroon" title="c22">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> <font color="Maroon" title="c20">F1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/>

<a NAME="E2:24_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A69</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c19">t1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E90:24"><span class="lab"><font color="Green" title="E71">A68</font></span></a>, <a class="txt" href="#E78:24"><span class="lab"><font color="Green" title="E61">A47</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:24_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A70</font></span>: 
<font color="Maroon" title="c22">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> <font color="Maroon" title="c19">t1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:24_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A71</font></span>: 
<font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/>
<a NAME="E5:24_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A72</font></span>: 
<font color="Maroon" title="c16">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E3:24_16"><span class="lab"><font color="Green" title="E73">A70</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<a NAME="E6:24_16"/><span class="lab"><font color="Green" title="E76">A73</font></span>: 
 <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> <font color="Maroon" title="c19">t1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">Omega</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:24"><span class="lab"><font color="Green" title="E61">A47</font></span></a>, <a class="txt" href="#E88:24"><span class="lab"><font color="Green" title="E69">A66</font></span></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c20">F1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">n</font> = 
<font color="Maroon" title="c18">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_16"><span class="lab"><font color="Green" title="E72">A69</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">t1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_16"><span class="lab"><font color="Green" title="E74">A71</font></span></a>, <a class="txt" href="#E6:24_16"><span class="lab"><font color="Green" title="E76">A73</font></span></a>, <a class="txt" href="#E2:24_16"><span class="lab"><font color="Green" title="E72">A69</font></span></a>, <a class="txt" href="#E67:24"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E5:24_16"><span class="lab"><font color="Green" title="E75">A72</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E92:24"/>
 ex <font color="Olive" title="b1">G</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>  ex <font color="Olive" title="b2">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">Y</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Olive" title="b2">s</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Y</font> &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">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> <font color="Olive" title="b1">G</font> holds <br/><font color="Olive" title="b1">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c2">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">Y</font></span>)</span>) <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="Olive" title="b1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E90:24"><span class="lab"><font color="Green" title="E71">A68</font></span></a>, <a class="txt" href="#E78:24"><span class="lab"><font color="Green" title="E61">A47</font></span></a>, <a class="txt" href="#E29:24"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E89:24"><span class="lab"><font color="Green" title="E70">A67</font></span></a>, <a class="txt" href="#E73:24"><span class="lab"><font color="Green" title="E59">A45</font></span></a>, <a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E19:24"><span class="lab"><font color="Green" title="E33">A14</font></span></a></span>; <a class="txt" 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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