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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b4">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b5">F</font> being  <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b6">a</font>, <font color="Olive" title="b7">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <font color="Olive" title="b4">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Olive" title="b1">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b8">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b4">f</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b8">x</font> ) &amp; <font color="Olive" title="b5">F</font>,<font color="Olive" title="b6">a</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Olive" title="b4">f</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b7">x</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="Olive" title="b5">F</font> &amp; ( for <font color="Olive" title="b8">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b8">i</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="b7">x</font> holds <br/><font color="Olive" title="b7">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b8">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b8">i</font></span>)</span> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">n</font> holds <br/> <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">M</font>,<font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Olive" title="b7">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b3">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b4">F</font> being  <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <font color="Olive" title="b3">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b7">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b3">f</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b7">x</font> ) &amp; <font color="Olive" title="b4">F</font>,<font color="Olive" title="b5">a</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Olive" title="b3">f</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b6">x</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="Olive" title="b4">F</font> &amp; ( for <font color="Olive" title="b7">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b7">i</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="b6">x</font> holds <br/><font color="Olive" title="b6">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b7">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b7">i</font></span>)</span> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> holds <br/> <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">M</font>,<font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Olive" title="b6">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b3">F</font> being  <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <font color="Olive" title="b2">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b6">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">f</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b6">x</font> ) &amp; <font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">a</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Olive" title="b2">f</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b5">x</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="Olive" title="b3">F</font> &amp; ( for <font color="Olive" title="b6">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b6">i</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="b5">x</font> holds <br/><font color="Olive" title="b5">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b6">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b6">i</font></span>)</span> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> holds <br/> <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Olive" title="b5">x</font></span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b2">F</font> being  <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <font color="Olive" title="b1">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b5">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b5">x</font> ) &amp; <font color="Olive" title="b2">F</font>,<font color="Olive" title="b3">a</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Olive" title="b1">f</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b4">x</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="Olive" title="b2">F</font> &amp; ( for <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">i</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">x</font> holds <br/><font color="Olive" title="b4">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b5">i</font></span>)</span> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Olive" title="b4">x</font>;<br/>
<a NAME="E1:2"/><span class="lab"><font color="Green" title="E3">A1</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <font color="Maroon" title="c5">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b4">x</font> ) &amp; <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">a</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c5">f</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">x</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="Olive" title="b1">F</font> &amp; ( for <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">i</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">x</font> holds <br/><font color="Olive" title="b3">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b4">i</font></span>)</span> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Olive" title="b3">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">F</font> be   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <font color="Maroon" title="c5">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">x</font> ) &amp; <font color="Maroon" title="c6">F</font>,<font color="Olive" title="b1">a</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c5">f</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">x</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="c6">F</font> &amp; ( for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">i</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">x</font> holds <br/><font color="Olive" title="b2">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">i</font></span>)</span> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Olive" title="b2">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">x</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( 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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> ) &amp; <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">a</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c5">f</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">x</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="c6">F</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</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="c8">x</font> holds <br/><font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:2_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">and </span><br/><a NAME="E3:2_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E4:2_1"/><span class="lab"><font color="Green" title="E7">A5</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c7">a</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E6:2_1"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">x</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="c6">F</font>
 <span class="kw">and </span><br/><a NAME="E7:2_1"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</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="c8">x</font> holds <br/><font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:2_1"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font></span><br/>

<a NAME="E9:2_1"/><span class="lab"><font color="Green" title="E12">A10</font></span>: 
<font color="Maroon" title="c5">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">V58</a>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="mesfunc2.html#D5" title="MESFUNC2:def.5">MESFUNC2:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">n</font> = <font color="Maroon" title="c4">n</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">F1</font> = <font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">F1</font> = <font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span> as   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T36" title="MESFUNC2:th.36">MESFUNC2:36</a></span>;<br/>
<span class="lab"><font color="Green" title="E13">A11</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="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>
;<br/>
<a NAME="E13:2_1"/><span class="lab"><font color="Green" title="E14">A12</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> implies  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_2"/><span class="lab"><font color="Green" title="E15">A13</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font></span><br/>

<a NAME="E2:2_1_2"/><span class="kw">then </span>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:2_1_2"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E1:2_1_2"><span class="lab"><font color="Green" title="E15">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c13">a1</font> = <font color="Maroon" title="c7">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">a1</font> = <font color="Maroon" title="c7">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">x1</font> = <font color="Maroon" title="c8">x</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">x1</font> = <font color="Maroon" title="c8">x</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></span>;<br/>
<span class="lab"><font color="Green" title="E15">A14</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">x1</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">F1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
;<br/>
<a NAME="E17:2_1"/><span class="lab"><font color="Green" title="E16">A15</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a>, <a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></span>;<br/>
<a NAME="E18:2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A16</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="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E19:2_1"/>
 ex <font color="Olive" title="b1">F19</font> being  <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font> st <br/>(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">F19</font></span>)</span> &amp; ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">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">F19</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b4">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c17">F19</font> = <font color="Maroon" title="c12">F1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">F19</font></span>)</span> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  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="c17">F19</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">F19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">F19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">y</font> ) )</span><br/>

<a NAME="E1:2_1_4"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  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="c12">F1</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c18">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="c12">F1</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c19">x</font>, <font color="Maroon" title="c20">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">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="c12">F1</font> &amp; <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">y</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_4_1"/><span class="lab"><font color="Green" title="E18">A17</font></span>: 
<font color="Maroon" title="c18">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="c12">F1</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_4_1"/><span class="lab"><font color="Green" title="E19">A18</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
 <span class="kw">and </span><br/><a NAME="E3:2_1_4_1"/><span class="lab"><font color="Green" title="E20">A19</font></span>: 
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">y</font></span><br/>

<a NAME="E4:2_1_4_1"/>
<font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1"><span class="lab"><font color="Green" title="E17">A16</font></span></a>, <a class="txt" href="#E1:2_1_4_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E5:2_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A20</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c19">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_4_1"><span class="lab"><font color="Green" title="E19">A18</font></span></a>, <a class="txt" href="#E3:2_1_4_1"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<a NAME="E6:2_1_4_1"/><span class="lab"><font color="Green" title="E22">A21</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></span>;<br/>
<a NAME="E7:2_1_4_1"/><span class="lab"><font color="Green" title="E23">A22</font></span>: 
<font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_4_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<a NAME="E8:2_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c18">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E1:2_1_4_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E2:2_1_4_1"><span class="lab"><font color="Green" title="E19">A18</font></span></a>, <a class="txt" href="#E6:2_1_4_1"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:2_1_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E1:2_1_4_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E3:2_1_4_1"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E5:2_1_4_1"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E7:2_1_4_1"><span class="lab"><font color="Green" title="E23">A22</font></span></a>, <a class="txt" href="#E6:2_1_4_1"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:2_1_4"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">F19</font></span>)</span> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  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="c17">F19</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">F19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">F19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="txt" href="#E17:2_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A23</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="ref" href="mesfunc2.html#D5" title="MESFUNC2:def.5">MESFUNC2:def 5</a></span>;<br/>
<span class="lab"><font color="Green" title="E19">A24</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">F1</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">a1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
;<br/>
<a NAME="E22:2_1"/>
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="c12">F1</font> holds <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">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="c12">F1</font> implies 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> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c17">n</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_6"/><span class="lab"><font color="Green" title="E20">A25</font></span>: 
<font color="Maroon" title="c17">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="c12">F1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c17">n</font></span><br/>

<a NAME="E2:2_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A26</font></span>: 
( <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> &amp; <font color="Maroon" title="c14">a1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c17">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:2_1"><span class="lab"><font color="Green" title="E19">A24</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c18">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c17">n</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:2_1_6"/><span class="lab"><font color="Green" title="E22">A27</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c17">n</font></span><br/>

<a NAME="E4:2_1_6"/>
<font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1"><span class="lab"><font color="Green" title="E17">A16</font></span></a>, <a class="txt" href="#E1:2_1_6"><span class="lab"><font color="Green" title="E20">A25</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E5:2_1_6"/><span class="kw">then </span>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c18">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_6"><span class="lab"><font color="Green" title="E22">A27</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:2_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c17">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E1:2_1_6"><span class="lab"><font color="Green" title="E20">A25</font></span></a>, <a class="txt" href="#E2:2_1_6"><span class="lab"><font color="Green" title="E21">A26</font></span></a>, <a class="txt" href="#E3:2_1_6"><span class="lab"><font color="Green" title="E22">A27</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A28</font></span>: 
<font color="Maroon" title="c12">F1</font>,<font color="Maroon" title="c14">a1</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1"><span class="lab"><font color="Green" title="E17">A16</font></span></a>, <a class="txt" href="#E21:2_1"><span class="lab"><font color="Green" title="E19">A24</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_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="E1:2_1_7_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1"/><span class="lab"><font color="Green" title="E21">A29</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="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E3:2_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A30</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:2_1_7_1_1"/><span class="lab"><font color="Green" title="E23">A31</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:2_1_7_1_1_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> or <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_1_1_1"/>
<font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_7_1_1_1_1"/><span class="lab"><font color="Green" title="E24">A32</font></span>: 
<font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_7_1_1_1_1"/><span class="lab"><font color="Green" title="E25">A33</font></span>: 
<font color="Maroon" title="c18">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:2_1_7_1_1_1_1"/><span class="lab"><font color="Green" title="E26">A34</font></span>: 
<font color="Maroon" title="c19">i</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="c6">F</font>
 <span class="kw">and </span><br/><a NAME="E7:2_1_7_1_1_1_1"/><span class="lab"><font color="Green" title="E27">A35</font></span>: 
<font color="Maroon" title="c18">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E25">A33</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">i</font> = <font color="Maroon" title="c19">i</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E26">A34</font></span></a></span>;<br/>
<a NAME="E9:2_1_7_1_1_1_1"/><span class="lab"><font color="Green" title="E28">A36</font></span>: 
<font color="Maroon" title="c20">i</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="c6">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E26">A34</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="E10:2_1_7_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A37</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E11:2_1_7_1_1_1_1"/><span class="lab"><font color="Green" title="E30">A38</font></span>: 
<font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="txt" href="#E9:2_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E28">A36</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_1_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="E1:2_1_7_1_1_1_1_1"/>
( <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c20">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_1_1_1_1"/>
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_7_1_1_1_1_1_1"/>
<font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E24">A32</font></span></a>, <a class="txt" href="#E7:2_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E27">A35</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_1_1_1_2"/>
<font color="Maroon" title="c20">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:2_1_7_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E30">A38</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_7_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_7_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">i</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> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E29">A37</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E5:2_1_7_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E26">A34</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="E6:2_1_7_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">i</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="c12">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E7:2_1_7_1_1_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">A</font> &amp; <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E27">A35</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E8:2_1_7_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E24">A32</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_7_1_1_1_1_1_2"/>
<font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c17">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> or <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:2_1_7_1_1_1"/><span class="lab"><font color="Green" title="E24">A39</font></span>: 
<font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_1_2"><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="E1:2_1_7_1_1_1_2"/>
( <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span> or <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_7_1_1_1"><span class="lab"><font color="Green" title="E24">A39</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_1_2_1"/>
<font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_7_1_1_1_2_1"/><span class="lab"><font color="Green" title="E25">A40</font></span>: 
<font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">A</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_7_1_1_1_2_1"/><span class="lab"><font color="Green" title="E26">A41</font></span>: 
<font color="Maroon" title="c18">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c19">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:2_1_7_1_1_1_2_1"/><span class="lab"><font color="Green" title="E27">A42</font></span>: 
<font color="Maroon" title="c19">i</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="c12">F1</font>
 <span class="kw">and </span><br/><a NAME="E7:2_1_7_1_1_1_2_1"/><span class="lab"><font color="Green" title="E28">A43</font></span>: 
<font color="Maroon" title="c18">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_1_1_2_1"><span class="lab"><font color="Green" title="E26">A41</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E8:2_1_7_1_1_1_2_1"/>
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_7_1_1_1_2_1"><span class="lab"><font color="Green" title="E27">A42</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E9:2_1_7_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A44</font></span>: 
<font color="Maroon" title="c19">i</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="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E10:2_1_7_1_1_1_2_1"/>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_7_1_1_1_2_1"><span class="lab"><font color="Green" title="E27">A42</font></span></a>, <a class="txt" href="#E7:2_1_7_1_1_1_2_1"><span class="lab"><font color="Green" title="E28">A43</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E11:2_1_7_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_1_1_2_1"><span class="lab"><font color="Green" title="E29">A44</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E12:2_1_7_1_1_1_2_1"/>
<font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_1_1_2_1"><span class="lab"><font color="Green" title="E25">A40</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_1_2_2"/><span class="lab"><font color="Green" title="E25">A45</font></span>: 
<font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span></span><br/><div class="add"><a NAME="E2:2_1_7_1_1_1_2_2"/>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_1"><span class="lab"><font color="Green" title="E22">A30</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_1_1_2_2"/>
<font color="Maroon" title="c17">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_1_1_2_2"><span class="lab"><font color="Green" title="E25">A45</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="lab"><font color="Green" title="E24">A46</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">x</font></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</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="E6:2_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A47</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T8" title="FINSEQ_1:th.8">FINSEQ_1:8</a></span>;<br/><a NAME="E7:2_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A48</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">sumx</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw"> such that </span><br/><a NAME="E9:2_1_7_1_1"/><span class="lab"><font color="Green" title="E27">A49</font></span>: 
 <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">sumx</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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">x</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:2_1_7_1_1"/><span class="lab"><font color="Green" title="E28">A50</font></span>: 
<font color="Maroon" title="c17">sumx</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">and </span><br/><a NAME="E11:2_1_7_1_1"/><span class="lab"><font color="Green" title="E29">A51</font></span>: 
for <font color="Olive" title="b1">m</font> being   <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>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">x</font> holds <br/><font color="Maroon" title="c17">sumx</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">sumx</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#D4" title="EXTREAL1:def.4">EXTREAL1:def 4</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">x</font> implies <font color="Maroon" title="c17">sumx</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  );<br/><a NAME="E12:2_1_7_1_1"/><span class="lab"><font color="Green" title="E30">A52</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c12">F1</font> holds <br/><font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">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="c12">F1</font> implies <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_1_3"/>
<font color="Maroon" title="c18">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="c12">F1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E2:2_1_7_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:2_1_7_1_1_3"/>
<font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> <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="#E18:2_1"><span class="lab"><font color="Green" title="E17">A16</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1"><span class="lab"><font color="Green" title="E21">A29</font></span></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E13:2_1_7_1_1"/><span class="lab"><font color="Green" title="E31">A53</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c12">F1</font> holds <br/><font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c18">EMPTY</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T10" title="PROB_1:th.10">PROB_1:10</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c19">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">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="c12">F1</font> implies <font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:2_1_7_1_1_4"/><span class="lab"><font color="Green" title="E32">A54</font></span>: 
<font color="Maroon" title="c19">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="c12">F1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> </span><br/>

<a NAME="E3:2_1_7_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <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="#E12:2_1_7_1_1"><span class="lab"><font color="Green" title="E30">A52</font></span></a></span>;<br/>
<a NAME="E4:2_1_7_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A55</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <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="#E2:2_1_7_1_1_4"><span class="lab"><font color="Green" title="E32">A54</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<a NAME="E5:2_1_7_1_1_4"/>
<font color="Maroon" title="c19">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_7_1_1_4"><span class="lab"><font color="Green" title="E32">A54</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<a NAME="E6:2_1_7_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A56</font></span>: 
<font color="Maroon" title="c19">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="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</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="E7:2_1_7_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A57</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c19">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a></span>;<br/>
<a NAME="E8:2_1_7_1_1_4"/>
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c18">EMPTY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>;<br/>
<a NAME="E9:2_1_7_1_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E6:2_1_7_1_1_4"><span class="lab"><font color="Green" title="E34">A56</font></span></a>, <a class="txt" href="#E4:2_1_7_1_1_4"><span class="lab"><font color="Green" title="E33">A55</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:2_1_7_1_1_4"/>
<font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_4"><span class="lab"><font color="Green" title="E35">A57</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E14:2_1_7_1_1"/><span class="lab"><font color="Green" title="E32">A58</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">m</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_1_5"/><span class="lab"><font color="Green" title="E33">A59</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">m</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:2_1_7_1_1_5"/><span class="lab"><font color="Green" title="E34">A60</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<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">assume </span><a NAME="E3:2_1_7_1_1_5"/><span class="lab"><font color="Green" title="E35">A61</font></span>: 
<font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">sumx</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> </span><br/>

<a NAME="E4:2_1_7_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_7_1_1"><span class="lab"><font color="Green" title="E25">A47</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E5:2_1_7_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A62</font></span>: 
<font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_7_1_1_5"><span class="lab"><font color="Green" title="E34">A60</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E6:2_1_7_1_1_5"/>
<font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c8">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_1_5"><span class="lab"><font color="Green" title="E35">A61</font></span></a>, <a class="txt" href="#E2:2_1_7_1_1_5"><span class="lab"><font color="Green" title="E34">A60</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E7:2_1_7_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c6">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</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="E8:2_1_7_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_7_1_1_5"><span class="lab"><font color="Green" title="E36">A62</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="E9:2_1_7_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A63</font></span>: 
<font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c12">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<a NAME="E10:2_1_7_1_1_5"/><span class="lab"><font color="Green" title="E38">A64</font></span>: 
<font color="Maroon" title="c18">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E11:2_1_7_1_1_5"/>
<font color="Maroon" title="c18">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<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="E12:2_1_7_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c18">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_1_5"><span class="lab"><font color="Green" title="E35">A61</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c17">sumx</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_7_1_1"><span class="lab"><font color="Green" title="E29">A51</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_5"><span class="lab"><font color="Green" title="E33">A59</font></span></a>, <a class="txt" href="#E10:2_1_7_1_1_5"><span class="lab"><font color="Green" title="E38">A64</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E14:2_1_7_1_1_5"/>
<font color="Maroon" title="c17">sumx</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_7_1_1"><span class="lab"><font color="Green" title="E31">A53</font></span></a>, <a class="txt" href="#E9:2_1_7_1_1_5"><span class="lab"><font color="Green" title="E37">A63</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E15:2_1_7_1_1"/><span class="lab"><font color="Green" title="E33">A65</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_7_1_1"><span class="lab"><font color="Green" title="E28">A50</font></span></a></span>;<br/><a NAME="E16:2_1_7_1_1"/><span class="lab"><font color="Green" title="E34">A66</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E15:2_1_7_1_1"><span class="lab"><font color="Green" title="E33">A65</font></span></a>, <a class="txt" href="#E14:2_1_7_1_1"><span class="lab"><font color="Green" title="E32">A58</font></span></a>);<br/></span><span class="lab"><font color="Green" title="E35">A67</font></span>:  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font> = 
<font color="Maroon" title="c17">sumx</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_1"><span class="lab"><font color="Green" title="E27">A49</font></span></a>, <a class="txt" href="#E5:2_1_7_1_1"><span class="lab"><font color="Green" title="E24">A46</font></span></a>, <a class="ref" href="finseq_1.html#T8" title="FINSEQ_1:th.8">FINSEQ_1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">sumx</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_7_1_1"><span class="lab"><font color="Green" title="E29">A51</font></span></a>, <a class="txt" href="#E7:2_1_7_1_1"><span class="lab"><font color="Green" title="E26">A48</font></span></a></span>
<br/>.= 
<a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:2_1_7_1_1"><span class="lab"><font color="Green" title="E34">A66</font></span></a>, <a class="txt" href="#E7:2_1_7_1_1"><span class="lab"><font color="Green" title="E26">A48</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_7"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_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="E1:2_1_7_1_1_7_1"/>
( <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  or <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1"><span class="kw">case </span></a><a NAME="E1:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E36">A68</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font></span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) );<br/><span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) );<br/><a NAME="E2:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E37">A69</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> 2 holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_1"><span class="kw">proof </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="Red">::  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> 2 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#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>3</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="E1:2_1_7_1_1_7_1_1_1"/><span class="lab"><font color="Green" title="E38">A70</font></span>: 
<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> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_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="E1:2_1_7_1_1_7_1_1_1_1"/>
( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_1_7_1_1_1"><span class="lab"><font color="Green" title="E38">A70</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A71</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c19">x</font> =  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">x</font> =  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  ;<br/><span class="kw">take </span>
<font color="Maroon" title="c20">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Maroon" title="c20">x</font>]</span><br/><span class="kw">thus </span><a NAME="E3:2_1_7_1_1_7_1_1_1_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Maroon" title="c20">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_1_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A71</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A72</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c19">x</font> = <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  ;<br/><span class="kw">take </span>
<font color="Maroon" title="c20">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Maroon" title="c20">x</font>]</span><br/><span class="kw">thus </span><a NAME="E3:2_1_7_1_1_7_1_1_1_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Maroon" title="c20">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_1_7_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A72</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c18">b</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw"> such that </span><br/><a NAME="E4:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E38">A73</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> 2 holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</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="#E2:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E37">A69</font></span></a>);<br/></span><a NAME="E5:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E39">A74</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> 2 holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">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> 2 implies  ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_1_7_1_1_2"/><span class="lab"><font color="Green" title="E40">A75</font></span>: 
<font color="Maroon" title="c19">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> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_2_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="E1:2_1_7_1_1_7_1_1_2_1"/>
( <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_1_7_1_1_2"><span class="lab"><font color="Green" title="E40">A75</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_1_2_1_1"/><span class="lab"><font color="Green" title="E41">A76</font></span>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c20">x</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">x</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T34" title="MESFUNC2:th.34">MESFUNC2:34</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c21">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Maroon" title="c21">x</font>]</span><br/><span class="kw">thus </span><a NAME="E3:2_1_7_1_1_7_1_1_2_1_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Maroon" title="c21">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_1_7_1_1_2_1_1"><span class="lab"><font color="Green" title="E41">A76</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_1_2_1_2"/><span class="lab"><font color="Green" title="E41">A77</font></span>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><a NAME="E2:2_1_7_1_1_7_1_1_2_1_2"/>
( <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_1"><span class="lab"><font color="Green" title="E22">A30</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">x</font> = <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c21">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Maroon" title="c21">x</font>]</span><br/><span class="kw">thus </span><a NAME="E4:2_1_7_1_1_7_1_1_2_1_2"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Maroon" title="c21">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_1_7_1_1_2_1_2"><span class="lab"><font color="Green" title="E41">A77</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c19">G</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E7:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E40">A78</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> 2 holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</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="#E5:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E39">A74</font></span></a>);<br/></span><a NAME="E8:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E41">A79</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E9:2_1_7_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A80</font></span>: 
<font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E38">A73</font></span></a></span>;<br/><a NAME="E10:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E43">A81</font></span>: 
<font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E38">A73</font></span></a>, <a class="txt" href="#E8:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E41">A79</font></span></a></span>;<br/><a NAME="E11:2_1_7_1_1_7_1_1"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E12:2_1_7_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A82</font></span>: 
<font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E38">A73</font></span></a></span>;<br/><a NAME="E13:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E45">A83</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E14:2_1_7_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A84</font></span>: 
<font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E40">A78</font></span></a></span>;<br/><a NAME="E15:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E47">A85</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E16:2_1_7_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A86</font></span>: 
<font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E40">A78</font></span></a></span>;<br/><a NAME="E17:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E49">A87</font></span>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">u</font>, <font color="Maroon" title="c21">v</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">v</font> implies <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">v</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_1_7_1_1_3"/><span class="lab"><font color="Green" title="E50">A88</font></span>: 
<font color="Maroon" title="c20">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">v</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_3_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="E1:2_1_7_1_1_7_1_1_3_1"/>
( ( <font color="Maroon" title="c20">u</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">G</font> &amp; <font color="Maroon" title="c21">v</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">G</font> ) or  not <font color="Maroon" title="c20">u</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">G</font> or  not <font color="Maroon" title="c21">v</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">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_1_3_1_1"/><span class="lab"><font color="Green" title="E51">A89</font></span>: 
( <font color="Maroon" title="c20">u</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">G</font> &amp; <font color="Maroon" title="c21">v</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">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">v</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_1_7_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A90</font></span>: 
( <font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E40">A78</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_3_1_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="E1:2_1_7_1_1_7_1_1_3_1_1_1"/>
( ( <font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c21">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c21">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E40">A78</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_1_3"><span class="lab"><font color="Green" title="E50">A88</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_1_3_1_1"><span class="lab"><font color="Green" title="E51">A89</font></span></a>, <a class="txt" href="#E2:2_1_7_1_1_7_1_1_3_1_1"><span class="lab"><font color="Green" title="E52">A90</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E53">A91</font></span>: 
( <font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c21">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">v</font></span><br/><div class="add"><span class="kw">assume </span><a NAME="E2:2_1_7_1_1_7_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">u</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:2_1_7_1_1_7_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E54">A92</font></span>: 
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">u</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1_7_1_1_7_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E55">A93</font></span>: 
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c23">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:2_1_7_1_1_7_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E56">A94</font></span>: 
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">A</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_7_1_1_7_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E57">A95</font></span>: 
<font color="Maroon" title="c23">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E48">A86</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E53">A91</font></span></a>, <a class="txt" href="#E4:2_1_7_1_1_7_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E54">A92</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c24">k9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:2_1_7_1_1_7_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E58">A96</font></span>: 
<font color="Maroon" title="c24">k9</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="c12">F1</font>
 <span class="kw">and </span><br/><a NAME="E11:2_1_7_1_1_7_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A97</font></span>: 
<font color="Maroon" title="c23">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_7_1_1_7_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E57">A95</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">k9</font> = <font color="Maroon" title="c24">k9</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_7_1_1_7_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E58">A96</font></span></a></span>;<br/><a NAME="E13:2_1_7_1_1_7_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A98</font></span>: 
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_7_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E56">A94</font></span></a>, <a class="txt" href="#E10:2_1_7_1_1_7_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E58">A96</font></span></a>, <a class="txt" href="#E11:2_1_7_1_1_7_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E59">A97</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E14:2_1_7_1_1_7_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c25">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_7_1_1_7_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E58">A96</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E15:2_1_7_1_1_7_1_1_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c25">k9</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> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E16:2_1_7_1_1_7_1_1_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c25">k9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E17:2_1_7_1_1_7_1_1_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c25">k9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E18:2_1_7_1_1_7_1_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A99</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k9</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></span>;<br/><a NAME="E19:2_1_7_1_1_7_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E40">A78</font></span></a>, <a class="txt" href="#E13:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E45">A83</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E53">A91</font></span></a>, <a class="txt" href="#E5:2_1_7_1_1_7_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E55">A93</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E20:2_1_7_1_1_7_1_1_3_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_7_1_1_7_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A99</font></span></a>, <a class="txt" href="#E13:2_1_7_1_1_7_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A98</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A100</font></span>: 
( <font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c21">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">v</font></span><br/><div class="add"><span class="kw">assume </span><a NAME="E2:2_1_7_1_1_7_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">u</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:2_1_7_1_1_7_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A101</font></span>: 
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">u</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1_7_1_1_7_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E55">A102</font></span>: 
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c23">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:2_1_7_1_1_7_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E56">A103</font></span>: 
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">A</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_7_1_1_7_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A104</font></span>: 
<font color="Maroon" title="c23">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E48">A86</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E53">A100</font></span></a>, <a class="txt" href="#E5:2_1_7_1_1_7_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E55">A102</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c24">k9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:2_1_7_1_1_7_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E58">A105</font></span>: 
<font color="Maroon" title="c24">k9</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="c12">F1</font>
 <span class="kw">and </span><br/><a NAME="E11:2_1_7_1_1_7_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E59">A106</font></span>: 
<font color="Maroon" title="c23">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_7_1_1_7_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E57">A104</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">k9</font> = <font color="Maroon" title="c24">k9</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_7_1_1_7_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E58">A105</font></span></a></span>;<br/><a NAME="E13:2_1_7_1_1_7_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E60">A107</font></span>: 
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_7_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E56">A103</font></span></a>, <a class="txt" href="#E10:2_1_7_1_1_7_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E58">A105</font></span></a>, <a class="txt" href="#E11:2_1_7_1_1_7_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E59">A106</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E14:2_1_7_1_1_7_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c25">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_7_1_1_7_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E58">A105</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E15:2_1_7_1_1_7_1_1_3_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">k9</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> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E16:2_1_7_1_1_7_1_1_3_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">k9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E17:2_1_7_1_1_7_1_1_3_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">k9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E18:2_1_7_1_1_7_1_1_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A108</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k9</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></span>;<br/><a NAME="E19:2_1_7_1_1_7_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c22">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E40">A78</font></span></a>, <a class="txt" href="#E13:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E45">A83</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E53">A100</font></span></a>, <a class="txt" href="#E4:2_1_7_1_1_7_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E54">A101</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E20:2_1_7_1_1_7_1_1_3_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_7_1_1_7_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E61">A108</font></span></a>, <a class="txt" href="#E13:2_1_7_1_1_7_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E60">A107</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_1_3_1_2"/>
(  not <font color="Maroon" title="c20">u</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">G</font> or  not <font color="Maroon" title="c21">v</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">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">v</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_1_7_1_1_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">v</font> <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="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_1_7_1_1_3_1_2"/>
<font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E18:2_1_7_1_1_7_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E40">A78</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="E19:2_1_7_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A109</font></span>: 
<font color="Maroon" title="c19">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E48">A86</font></span></a>, <a class="txt" href="#E14:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E46">A84</font></span></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1</span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1</span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c20">y</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw"> such that </span><br/><a NAME="E21:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E51">A110</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c20">y</font> holds <br/><font color="Maroon" title="c20">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">m</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span><a NAME="E22:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E52">A111</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E51">A110</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="E23:2_1_7_1_1_7_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E24:2_1_7_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A112</font></span>: 
<font color="Maroon" title="c20">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1</span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E51">A110</font></span></a>, <a class="txt" href="#E22:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E52">A111</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">sumy</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw"> such that </span><br/><a NAME="E26:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E54">A113</font></span>: 
 <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">sumy</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">y</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E27:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E55">A114</font></span>: 
<font color="Maroon" title="c21">sumy</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">and </span><br/><a NAME="E28:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E56">A115</font></span>: 
for <font color="Olive" title="b1">k</font> being   <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>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">y</font> holds <br/><font color="Maroon" title="c21">sumy</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">sumy</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#D4" title="EXTREAL1:def.4">EXTREAL1:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E57">A116</font></span>: <font color="Maroon" title="c21">sumy</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">sumy</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E51">A110</font></span></a>, <a class="txt" href="#E28:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E56">A115</font></span></a></span>
<br/>.= 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E42">A80</font></span></a>, <a class="txt" href="#E24:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E53">A112</font></span></a>, <a class="txt" href="#E27:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E55">A114</font></span></a></span>
;<br/><a NAME="E30:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E58">A117</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 2 = 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E40">A78</font></span></a>, <a class="txt" href="#E14:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E46">A84</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_1"><span class="lab"><font color="Green" title="E22">A30</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
;<br/><a NAME="E32:2_1_7_1_1_7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</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="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E44">A82</font></span></a>, <a class="txt" href="#E21:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E51">A110</font></span></a>, <a class="txt" href="#E22:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E52">A111</font></span></a>, <a class="txt" href="#E30:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E58">A117</font></span></a></span>;<br/><a NAME="E33:2_1_7_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A118</font></span>: 
<font color="Maroon" title="c20">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E3:2_1_7_1_1"><span class="lab"><font color="Green" title="E22">A30</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">G</font> = <font color="Maroon" title="c19">G</font> as   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E49">A87</font></span></a>, <a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></span>;<br/><a NAME="E35:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E60">A119</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">y</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="c22">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E40">A78</font></span></a>, <a class="txt" href="#E21:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E51">A110</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="E36:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E61">A120</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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="c22">G</font> holds <br/>for <font color="Olive" title="b2">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">k</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="c22">G</font> implies for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_1_7_1_1_6"/><span class="lab"><font color="Green" title="E62">A121</font></span>: 
<font color="Maroon" title="c23">k</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="c22">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c24">v</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> implies <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:2_1_7_1_1_7_1_1_6"/><span class="lab"><font color="Green" title="E63">A122</font></span>: 
<font color="Maroon" title="c24">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_6_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="E1:2_1_7_1_1_7_1_1_6_1"/>
( <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E40">A78</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_1_6"><span class="lab"><font color="Green" title="E62">A121</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_1_6_1_1"/>
<font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_7_1_1_7_1_1_6_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1"><span class="lab"><font color="Green" title="E17">A16</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1"><span class="lab"><font color="Green" title="E21">A29</font></span></a>, <a class="txt" href="#E7:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E40">A78</font></span></a>, <a class="txt" href="#E15:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E47">A85</font></span></a>, <a class="txt" href="#E2:2_1_7_1_1_7_1_1_6"><span class="lab"><font color="Green" title="E63">A122</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_1_6_1_2"/>
<font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_7_1_1_7_1_1_6_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E3:2_1_7_1_1"><span class="lab"><font color="Green" title="E22">A30</font></span></a>, <a class="txt" href="#E14:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E46">A84</font></span></a>, <a class="txt" href="#E12:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E44">A82</font></span></a>, <a class="txt" href="#E2:2_1_7_1_1_7_1_1_6"><span class="lab"><font color="Green" title="E63">A122</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E37:2_1_7_1_1_7_1_1"/><span class="lab"><font color="Green" title="E62">A123</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">b</font> holds <br/>(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_7_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">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="Red">::  thesis: </font></span></a><span class="hide"> ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">k</font> &amp; <font color="Maroon" title="c23">k</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">b</font> implies (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> &amp; <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_7_1_1_7_1_1_7"/><span class="lab"><font color="Green" title="E63">A124</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">k</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_7_1_1_7_1_1_7"/><span class="lab"><font color="Green" title="E64">A125</font></span>: 
<font color="Maroon" title="c23">k</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">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> &amp; <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )</span><br/>

<a NAME="E3:2_1_7_1_1_7_1_1_7"/><span class="lab"><font color="Green" title="E65">A126</font></span>: 
( <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E38">A73</font></span></a>, <a class="txt" href="#E2:2_1_7_1_1_7_1_1_7"><span class="lab"><font color="Green" title="E64">A125</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E4:2_1_7_1_1_7_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c22">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E12:2_1"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1"><span class="lab"><font color="Green" title="E21">A29</font></span></a>, <a class="txt" href="#E4:2_1_7_1_1"><span class="lab"><font color="Green" title="E23">A31</font></span></a>, <a class="txt" href="#E14:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E46">A84</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_1_7"><span class="lab"><font color="Green" title="E63">A124</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:2_1_7_1_1_7_1_1_7"/><span class="lab"><font color="Green" title="E66">A127</font></span>: 
<font color="Maroon" title="c24">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E7:2_1_7_1_1_7_1_1_7"/><span class="lab"><font color="Green" title="E67">A128</font></span>: 
<font color="Maroon" title="c24">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E18:2_1"><span class="lab"><font color="Green" title="E17">A16</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1"><span class="lab"><font color="Green" title="E21">A29</font></span></a>, <a class="txt" href="#E4:2_1_7_1_1"><span class="lab"><font color="Green" title="E23">A31</font></span></a>, <a class="txt" href="#E14:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E46">A84</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_1_7"><span class="lab"><font color="Green" title="E63">A124</font></span></a>, <a class="txt" href="#E3:2_1_7_1_1_7_1_1_7"><span class="lab"><font color="Green" title="E65">A126</font></span></a>, <a class="txt" href="#E6:2_1_7_1_1_7_1_1_7"><span class="lab"><font color="Green" title="E66">A127</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/>
<a NAME="E8:2_1_7_1_1_7_1_1_7"/><span class="lab"><font color="Green" title="E68">A129</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E40">A78</font></span></a>, <a class="txt" href="#E4:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E38">A73</font></span></a>, <a class="txt" href="#E36:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E61">A120</font></span></a>, <a class="txt" href="#E2:2_1_7_1_1_7_1_1_7"><span class="lab"><font color="Green" title="E64">A125</font></span></a>, <a class="txt" href="#E6:2_1_7_1_1_7_1_1_7"><span class="lab"><font color="Green" title="E66">A127</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:2_1_7_1_1_7_1_1_7"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E36">A68</font></span></a>, <a class="txt" href="#E12:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E44">A82</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_1_7"><span class="lab"><font color="Green" title="E63">A124</font></span></a>, <a class="txt" href="#E3:2_1_7_1_1_7_1_1_7"><span class="lab"><font color="Green" title="E65">A126</font></span></a>, <a class="txt" href="#E7:2_1_7_1_1_7_1_1_7"><span class="lab"><font color="Green" title="E67">A128</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><br/>

<a NAME="E10:2_1_7_1_1_7_1_1_7"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">v</font> = <font color="Maroon" title="c24">v</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_7_1_1_7"><span class="lab"><font color="Green" title="E67">A128</font></span></a></span>;<br/>
<a NAME="E12:2_1_7_1_1_7_1_1_7"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">v</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="txt" href="#E7:2_1_7_1_1_7_1_1_7"><span class="lab"><font color="Green" title="E67">A128</font></span></a>, <a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:2_1_7_1_1_7_1_1_7"/>
<font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_7_1_1_7_1_1_7"><span class="lab"><font color="Green" title="E68">A129</font></span></a>, <a class="ref" href="extreal2.html#T58" title="EXTREAL2:th.58">EXTREAL2:58</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> = 
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E4:2_1_7_1_1"><span class="lab"><font color="Green" title="E23">A31</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T93" title="ZFMISC_1:th.93">ZFMISC_1:93</a></span>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E50">A109</font></span></a>, <a class="ref" href="finseq_2.html#T147" title="FINSEQ_2:th.147">FINSEQ_2:147</a></span>
;<br/><a NAME="E39:2_1_7_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A130</font></span>: 
<font color="Maroon" title="c22">G</font>,<font color="Maroon" title="c18">b</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E40">A78</font></span></a>, <a class="txt" href="#E4:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E38">A73</font></span></a>, <a class="txt" href="#E36:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E61">A120</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/> <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c20">y</font> = 
<font color="Maroon" title="c21">sumy</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E51">A110</font></span></a>, <a class="txt" href="#E26:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E54">A113</font></span></a></span>
<br/>.= 
<a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E51">A110</font></span></a>, <a class="txt" href="#E33:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E59">A118</font></span></a>, <a class="txt" href="#E28:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E56">A115</font></span></a>, <a class="txt" href="#E29:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E57">A116</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T4" title="XXREAL_3:th.4">XXREAL_3:4</a></span>
;<br/><span class="kw">hence </span><a NAME="E41:2_1_7_1_1_7_1_1"/>
 <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E17:2_1_7_1_1"><span class="lab"><font color="Green" title="E35">A67</font></span></a>, <a class="txt" href="#E21:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E51">A110</font></span></a>, <a class="txt" href="#E37:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E62">A123</font></span></a>, <a class="txt" href="#E39:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E63">A130</font></span></a>, <a class="txt" href="#E10:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E43">A81</font></span></a>, <a class="txt" href="#E35:2_1_7_1_1_7_1_1"><span class="lab"><font color="Green" title="E60">A119</font></span></a>, <a class="ref" href="mesfunc3.html#D2" title="MESFUNC3:def.2">MESFUNC3:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2"><span class="kw">case </span></a><a NAME="E1:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E36">A131</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font></span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  ) );<br/><span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) );<br/><a NAME="E2:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E37">A132</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> 2 holds <br/> ex <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">v</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_1"><span class="kw">proof </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="Red">::  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> 2 implies  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Olive" title="b1">v</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_1_7_1_2_1"/><span class="lab"><font color="Green" title="E38">A133</font></span>: 
<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> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Olive" title="b1">v</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_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="E1:2_1_7_1_1_7_1_2_1_1"/>
( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_1_7_1_2_1"><span class="lab"><font color="Green" title="E38">A133</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E39">A134</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Olive" title="b1">v</font>]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c19">v</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">v</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T34" title="MESFUNC2:th.34">MESFUNC2:34</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c20">v</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Maroon" title="c20">v</font>]</span><br/><span class="kw">thus </span><a NAME="E3:2_1_7_1_1_7_1_2_1_1_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Maroon" title="c20">v</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_1_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E39">A134</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_2_1_1_2"/><span class="lab"><font color="Green" title="E39">A135</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Olive" title="b1">v</font>]</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c19">v</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T10" title="PROB_1:th.10">PROB_1:10</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c19">v</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Maroon" title="c19">v</font>]</span><br/><span class="kw">thus </span><a NAME="E3:2_1_7_1_1_7_1_2_1_1_2"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c18">k</font>,<font color="Maroon" title="c19">v</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_1_7_1_2_1_1_2"><span class="lab"><font color="Green" title="E39">A135</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c18">G</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E4:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E38">A136</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> 2 holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</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="#E2:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E37">A132</font></span></a>);<br/></span><a NAME="E5:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E39">A137</font></span>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">u</font>, <font color="Maroon" title="c20">v</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">v</font> implies <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">v</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_1_7_1_2_2"/><span class="lab"><font color="Green" title="E40">A138</font></span>: 
<font color="Maroon" title="c19">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">v</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_2_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="E1:2_1_7_1_1_7_1_2_2_1"/>
( ( <font color="Maroon" title="c19">u</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">G</font> &amp; <font color="Maroon" title="c20">v</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">G</font> ) or  not <font color="Maroon" title="c19">u</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">G</font> or  not <font color="Maroon" title="c20">v</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">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_2_2_1_1"/><span class="lab"><font color="Green" title="E41">A139</font></span>: 
( <font color="Maroon" title="c19">u</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">G</font> &amp; <font color="Maroon" title="c20">v</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">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">v</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_1_7_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A140</font></span>: 
( <font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E38">A136</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_2_1_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="E1:2_1_7_1_1_7_1_2_2_1_1_1"/>
( ( <font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c20">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c20">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E38">A136</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_2_2"><span class="lab"><font color="Green" title="E40">A138</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E41">A139</font></span></a>, <a class="txt" href="#E2:2_1_7_1_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E42">A140</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_2_2_1_1_1_1"/>
( <font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c20">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">v</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_1_7_1_2_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">v</font> <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="#E4:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E38">A136</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E41">A139</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_1_7_1_2_2_1_1_1_1"/>
<font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_2_2_1_1_1_2"/>
( <font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c20">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">v</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_1_7_1_2_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font> <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="#E4:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E38">A136</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E41">A139</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_1_7_1_2_2_1_1_1_2"/>
<font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_2_2_1_2"/>
(  not <font color="Maroon" title="c19">u</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">G</font> or  not <font color="Maroon" title="c20">v</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">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">v</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_1_7_1_2_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">v</font> <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="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_1_7_1_2_2_1_2"/>
<font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E6:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E40">A141</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> 2 holds <br/> ex <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">v</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">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> 2 implies  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Olive" title="b1">v</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_1_7_1_2_3"/><span class="lab"><font color="Green" title="E41">A142</font></span>: 
<font color="Maroon" title="c19">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> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Olive" title="b1">v</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_3_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="E1:2_1_7_1_1_7_1_2_3_1"/>
( <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_1_7_1_2_3"><span class="lab"><font color="Green" title="E41">A142</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_2_3_1_1"/><span class="lab"><font color="Green" title="E42">A143</font></span>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Olive" title="b1">v</font>]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c20">v</font> =  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">v</font> =  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  ;<br/><span class="kw">take </span>
<font color="Maroon" title="c21">v</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Maroon" title="c21">v</font>]</span><br/><span class="kw">thus </span><a NAME="E3:2_1_7_1_1_7_1_2_3_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Maroon" title="c21">v</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_1_7_1_2_3_1_1"><span class="lab"><font color="Green" title="E42">A143</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_2_3_1_2"/><span class="lab"><font color="Green" title="E42">A144</font></span>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Olive" title="b1">v</font>]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c20">v</font> =  <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">v</font> =  <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  ;<br/><span class="kw">take </span>
<font color="Maroon" title="c21">v</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Maroon" title="c21">v</font>]</span><br/><span class="kw">thus </span><a NAME="E3:2_1_7_1_1_7_1_2_3_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font>,<font color="Maroon" title="c21">v</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_1_7_1_2_3_1_2"><span class="lab"><font color="Green" title="E42">A144</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c19">b</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw"> such that </span><br/><a NAME="E8:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E41">A145</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> 2 holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</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="#E6:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E40">A141</font></span></a>);<br/></span><a NAME="E9:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E42">A146</font></span>: 
2 <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">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E38">A136</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E10:2_1_7_1_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A147</font></span>: 
<font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <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="#E4:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E38">A136</font></span></a></span>;<br/><a NAME="E11:2_1_7_1_1_7_1_2"/>
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E42">A146</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><a NAME="E12:2_1_7_1_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A148</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E43">A147</font></span></a>, <a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>;<br/><a NAME="E13:2_1_7_1_1_7_1_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E14:2_1_7_1_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A149</font></span>: 
<font color="Maroon" title="c18">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E38">A136</font></span></a></span>;<br/><a NAME="E15:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E46">A150</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E16:2_1_7_1_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A151</font></span>: 
<font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E41">A145</font></span></a></span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1</span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1</span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c20">y</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw"> such that </span><br/><a NAME="E18:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E48">A152</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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">y</font> holds <br/><font color="Maroon" title="c20">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span><a NAME="E19:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E49">A153</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E48">A152</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="E20:2_1_7_1_1_7_1_2"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E21:2_1_7_1_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A154</font></span>: 
<font color="Maroon" title="c20">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 2</span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E48">A152</font></span></a>, <a class="txt" href="#E19:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E49">A153</font></span></a></span>;<br/><a NAME="E22:2_1_7_1_1_7_1_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E23:2_1_7_1_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A155</font></span>: 
<font color="Maroon" title="c20">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1</span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E48">A152</font></span></a>, <a class="txt" href="#E19:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E49">A153</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">G</font> = <font color="Maroon" title="c18">G</font> as   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E39">A137</font></span></a>, <a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></span>;<br/><a NAME="E25:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E52">A156</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">y</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="c21">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E38">A136</font></span></a>, <a class="txt" href="#E18:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E48">A152</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="E26:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E53">A157</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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">G</font> holds <br/>for <font color="Olive" title="b2">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">k</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">G</font> implies for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c22">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_1_7_1_2_4"/><span class="lab"><font color="Green" title="E54">A158</font></span>: 
<font color="Maroon" title="c22">k</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">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c22">k</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c23">v</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> implies <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c22">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:2_1_7_1_1_7_1_2_4"/><span class="lab"><font color="Green" title="E55">A159</font></span>: 
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c22">k</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_4_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="E1:2_1_7_1_1_7_1_2_4_1"/>
( <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E38">A136</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_2_4"><span class="lab"><font color="Green" title="E54">A158</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_2_4_1_1"/><span class="lab"><font color="Green" title="E56">A160</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c22">k</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_1_7_1_2_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E18:2_1"><span class="lab"><font color="Green" title="E17">A16</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1"><span class="lab"><font color="Green" title="E21">A29</font></span></a>, <a class="txt" href="#E3:2_1_7_1_1"><span class="lab"><font color="Green" title="E22">A30</font></span></a>, <a class="txt" href="#E4:2_1_7_1_1"><span class="lab"><font color="Green" title="E23">A31</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E36">A131</font></span></a>, <a class="txt" href="#E14:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E45">A149</font></span></a>, <a class="txt" href="#E2:2_1_7_1_1_7_1_2_4"><span class="lab"><font color="Green" title="E55">A159</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_1_7_1_2_4_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c22">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E41">A145</font></span></a>, <a class="txt" href="#E15:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E46">A150</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_2_4_1_1"><span class="lab"><font color="Green" title="E56">A160</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_1_7_1_2_4_1_2"/>
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c22">k</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_7_1_1_7_1_2_4_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c22">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E38">A136</font></span></a>, <a class="txt" href="#E9:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E42">A146</font></span></a>, <a class="txt" href="#E2:2_1_7_1_1_7_1_2_4"><span class="lab"><font color="Green" title="E55">A159</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E27:2_1_7_1_1_7_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E38">A136</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="E28:2_1_7_1_1_7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c21">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E45">A149</font></span></a>, <a class="txt" href="#E10:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E43">A147</font></span></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></span>;<br/><a NAME="E29:2_1_7_1_1_7_1_2"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T147" title="FINSEQ_2:th.147">FINSEQ_2:147</a></span>;<br/><span class="kw">then </span> <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">G</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T93" title="ZFMISC_1:th.93">ZFMISC_1:93</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>
;<br/><a NAME="E31:2_1_7_1_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A161</font></span>: 
<font color="Maroon" title="c21">G</font>,<font color="Maroon" title="c19">b</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E38">A136</font></span></a>, <a class="txt" href="#E8:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E41">A145</font></span></a>, <a class="txt" href="#E26:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E53">A157</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c22">sumy</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw"> such that </span><br/><a NAME="E33:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E55">A162</font></span>: 
 <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">sumy</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">y</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E34:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E56">A163</font></span>: 
<font color="Maroon" title="c22">sumy</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">and </span><br/><a NAME="E35:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E57">A164</font></span>: 
for <font color="Olive" title="b1">k</font> being   <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>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">y</font> holds <br/><font color="Maroon" title="c22">sumy</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">sumy</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#D4" title="EXTREAL1:def.4">EXTREAL1:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E58">A165</font></span>: <font color="Maroon" title="c22">sumy</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">sumy</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E48">A152</font></span></a>, <a class="txt" href="#E35:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E57">A164</font></span></a></span>
<br/>.= 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E47">A151</font></span></a>, <a class="txt" href="#E23:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E51">A155</font></span></a>, <a class="txt" href="#E34:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E56">A163</font></span></a></span>
;<br/><a NAME="E37:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E59">A166</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E38:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E60">A167</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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">b</font> holds <br/>(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_1_7_1_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">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="Red">::  thesis: </font></span></a><span class="hide"> ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">k</font> &amp; <font color="Maroon" title="c23">k</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">b</font> implies (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> &amp; <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_7_1_1_7_1_2_7"/><span class="lab"><font color="Green" title="E61">A168</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">k</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_7_1_1_7_1_2_7"/><span class="lab"><font color="Green" title="E62">A169</font></span>: 
<font color="Maroon" title="c23">k</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">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> &amp; <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )</span><br/>

<a NAME="E3:2_1_7_1_1_7_1_2_7"/>
( <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E41">A145</font></span></a>, <a class="txt" href="#E2:2_1_7_1_1_7_1_2_7"><span class="lab"><font color="Green" title="E62">A169</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:2_1_7_1_1_7_1_2_7"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E41">A145</font></span></a>, <a class="txt" href="#E37:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E59">A166</font></span></a>, <a class="txt" href="#E1:2_1_7_1_1_7_1_2_7"><span class="lab"><font color="Green" title="E61">A168</font></span></a>, <a class="ref" href="mesfunc1.html#D8" title="MESFUNC1:def.8">MESFUNC1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><br/>

<a NAME="E5:2_1_7_1_1_7_1_2_7"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c23">k</font>,<font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E41">A145</font></span></a>, <a class="txt" href="#E15:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E46">A150</font></span></a>, <a class="txt" href="#E37:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E59">A166</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:2_1_7_1_1_7_1_2_7"/>
<font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E41">A145</font></span></a>, <a class="txt" href="#E2:2_1_7_1_1_7_1_2_7"><span class="lab"><font color="Green" title="E62">A169</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="mesfunc1.html#D8" title="MESFUNC1:def.8">MESFUNC1:def 8</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E39:2_1_7_1_1_7_1_2"/><span class="lab"><font color="Green" title="E61">A170</font></span>: 
<font color="Maroon" title="c19">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E41">A145</font></span></a>, <a class="txt" href="#E15:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E46">A150</font></span></a></span>;<br/><a NAME="E40:2_1_7_1_1_7_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_7_1_1"><span class="lab"><font color="Green" title="E24">A46</font></span></a>, <a class="ref" href="finseq_1.html#T8" title="FINSEQ_1:th.8">FINSEQ_1:8</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E41:2_1_7_1_1_7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A171</font></span>: <font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</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="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a></span>
<br/>.= 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E36">A131</font></span></a></span>
;<br/> <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c20">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">sumy</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1</span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E48">A152</font></span></a>, <a class="txt" href="#E33:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E55">A162</font></span></a>, <a class="txt" href="#E35:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E57">A164</font></span></a></span>
<br/>.= 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E44">A148</font></span></a>, <a class="txt" href="#E21:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E50">A154</font></span></a>, <a class="txt" href="#E36:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E58">A165</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E44:2_1_7_1_1_7_1_2"/>
 <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E17:2_1_7_1_1"><span class="lab"><font color="Green" title="E35">A67</font></span></a>, <a class="txt" href="#E42:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E62">A171</font></span></a>, <a class="txt" href="#E18:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E48">A152</font></span></a>, <a class="txt" href="#E38:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E60">A167</font></span></a>, <a class="txt" href="#E31:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E54">A161</font></span></a>, <a class="txt" href="#E39:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E61">A170</font></span></a>, <a class="txt" href="#E25:2_1_7_1_1_7_1_2"><span class="lab"><font color="Green" title="E52">A156</font></span></a>, <a class="ref" href="mesfunc3.html#D2" title="MESFUNC3:def.2">MESFUNC3:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><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="E19:2_1_7_1_1"/>
 <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2"/><span class="lab"><font color="Green" title="E21">A172</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="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2"/>
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E3:2_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A173</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font> <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"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></span>;<br/><a NAME="E4:2_1_7_1_2"/><span class="lab"><font color="Green" title="E23">A174</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">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"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</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="E5:2_1_7_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E6:2_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A175</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_2"><span class="lab"><font color="Green" title="E22">A173</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><a NAME="E7:2_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A176</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</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="kw">consider </span><font color="Maroon" title="c17">G</font> being   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c18">b</font>, <font color="Maroon" title="c19">y</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw"> such that </span><br/><a NAME="E9:2_1_7_1_2"/><span class="lab"><font color="Green" title="E26">A177</font></span>: 
<font color="Maroon" title="c17">G</font>,<font color="Maroon" title="c18">b</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:2_1_7_1_2"/><span class="lab"><font color="Green" title="E27">A178</font></span>: 
<font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">and </span><br/><a NAME="E11:2_1_7_1_2"/><span class="lab"><font color="Green" title="E28">A179</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 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <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="c18">b</font> holds <br/>(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <span class="kw">and </span><br/><a NAME="E12:2_1_7_1_2"/><span class="lab"><font color="Green" title="E29">A180</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">y</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="c17">G</font>
 <span class="kw">and </span><br/><a NAME="E13:2_1_7_1_2"/><span class="lab"><font color="Green" title="E30">A181</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">y</font> holds <br/><font color="Maroon" title="c19">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</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="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:2_1_7_1_2"/><span class="lab"><font color="Green" title="E31">A182</font></span>: 
 <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c19">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1"><span class="lab"><font color="Green" title="E18">A23</font></span></a>, <a class="txt" href="#E13:2_1"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="ref" href="mesfunc3.html#D2" title="MESFUNC3:def.2">MESFUNC3:def 2</a></span>;<br/><a NAME="E15:2_1_7_1_2"/><span class="lab"><font color="Green" title="E32">A183</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</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">x1</font> holds <br/><font color="Maroon" title="c16">x1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">i</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">i</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">x1</font> implies <font color="Maroon" title="c16">x1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_2_1"/><span class="lab"><font color="Green" title="E33">A184</font></span>: 
<font color="Maroon" title="c20">i</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">x1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span></span><br/>

<a NAME="E2:2_1_7_1_2_1"/><span class="lab"><font color="Green" title="E34">A185</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></span>;<br/>
<a NAME="E3:2_1_7_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_1"><span class="lab"><font color="Green" title="E33">A184</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E35">A186</font></span>: <font color="Maroon" title="c16">x1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_1"><span class="lab"><font color="Green" title="E33">A184</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">a1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:2_1"><span class="lab"><font color="Green" title="E19">A24</font></span></a>, <a class="txt" href="#E16:2_1"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_1"><span class="lab"><font color="Green" title="E33">A184</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font> = 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_1"><span class="lab"><font color="Green" title="E33">A184</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_1"><span class="lab"><font color="Green" title="E34">A185</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:2_1"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_1"><span class="lab"><font color="Green" title="E33">A184</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:2_1"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_1"><span class="lab"><font color="Green" title="E33">A184</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:2_1_7_1_2_1"/>
<font color="Maroon" title="c16">x1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">F1</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_1"><span class="lab"><font color="Green" title="E35">A186</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_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="E1:2_1_7_1_2_2_1"/>
( <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  or ( <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1"><span class="kw">case </span></a><a NAME="E1:2_1_7_1_2_2_1_1"/><span class="lab"><font color="Green" title="E33">A187</font></span>: 
( <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font></span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) &amp; ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 ) );<br/><a NAME="E2:2_1_7_1_2_2_1_1"/><span class="lab"><font color="Green" title="E34">A188</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</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 <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">k</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<a NAME="E1:2_1_7_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E35">A189</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</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/>
<span class="kw">assume </span><a NAME="E2:2_1_7_1_2_2_1_1_1"/>
<font color="Maroon" title="c20">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">k</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<a NAME="E3:2_1_7_1_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">k</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="c17">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="E4:2_1_7_1_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A190</font></span>: 
<font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_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="E1:2_1_7_1_2_2_1_1_1_1"/>
( <font color="Maroon" title="c20">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c20">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A191</font></span>: 
<font color="Maroon" title="c20">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">k</font>,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c21">x</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E2:2_1_7_1_2_2_1_1_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E3:2_1_7_1_2_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A192</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_1_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">x</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E36">A190</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E35">A189</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A191</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A192</font></span></a>, <a class="ref" href="finsub_1.html#D1" title="FINSUB_1:def.1">FINSUB_1:def 1</a></span>;<br/><a NAME="E7:2_1_7_1_2_2_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">k</font>,<font color="Maroon" title="c22">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A191</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:2_1_7_1_2_2_1_1_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 <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">k</font>,<font color="Olive" title="b1">x</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E37">A193</font></span>: 
<font color="Maroon" title="c20">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">k</font>,<font color="Olive" title="b1">x</font>]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c21">x</font> = <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">k</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">x</font> = <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E36">A190</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E35">A189</font></span></a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">k</font>,<font color="Maroon" title="c22">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A193</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:2_1_7_1_2_2_1_1_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 <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c20">k</font>,<font color="Olive" title="b1">x</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c20">H</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E4:2_1_7_1_2_2_1_1"/><span class="lab"><font color="Green" title="E35">A194</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">H</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="c17">G</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</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="#E2:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E34">A188</font></span></a>);<br/></span><a NAME="E5:2_1_7_1_2_2_1_1"/><span class="lab"><font color="Green" title="E36">A195</font></span>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">u</font>, <font color="Maroon" title="c22">v</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">v</font> implies <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E37">A196</font></span>: 
<font color="Maroon" title="c21">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_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="E1:2_1_7_1_2_2_1_1_2_1"/>
( ( <font color="Maroon" title="c21">u</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">H</font> &amp; <font color="Maroon" title="c22">v</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">H</font> ) or  not <font color="Maroon" title="c21">u</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">H</font> or  not <font color="Maroon" title="c22">v</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">H</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E38">A197</font></span>: 
( <font color="Maroon" title="c21">u</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">H</font> &amp; <font color="Maroon" title="c22">v</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">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">u1</font> = <font color="Maroon" title="c21">u</font>, <font color="Maroon" title="c24">v1</font> = <font color="Maroon" title="c22">v</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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_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="E1:2_1_7_1_2_2_1_1_2_1_1_1"/>
( <font color="Maroon" title="c21">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c21">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A198</font></span>: 
<font color="Maroon" title="c21">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_2_1_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A197</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E37">A196</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E39">A198</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A199</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A200</font></span>: 
<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A197</font></span></a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A201</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_1_1_1_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="E1:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1"/>
( <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E33">A187</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E43">A202</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font></span><br/><div class="add"><span class="kw">assume </span><a NAME="E2:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A203</font></span>: 
<font color="Maroon" title="c25">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A204</font></span>: 
<font color="Maroon" title="c25">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A205</font></span>: 
<font color="Maroon" title="c24">v1</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="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A197</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="E7:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A206</font></span>: 
<font color="Maroon" title="c24">v1</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">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E8:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E46">A205</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E9:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E45">A204</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E10:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A207</font></span>: <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">v1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E45">A204</font></span></a>, <a class="txt" href="#E6:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E46">A205</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>
;<br/><a NAME="E12:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E13:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E14:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</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="E15:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E43">A202</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E44">A203</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E16:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_7_1_2"><span class="lab"><font color="Green" title="E28">A179</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E40">A199</font></span></a>, <a class="txt" href="#E11:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E48">A207</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_1_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E47">A206</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E7:2_1_7_1_2_2_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E37">A196</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A197</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E39">A198</font></span></a>, <a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:2_1_7_1_2_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_7_1_2_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E41">A200</font></span></a>, <a class="txt" href="#E6:2_1_7_1_2_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E42">A201</font></span></a>, <a class="ref" href="xboole_1.html#T70" title="XBOOLE_1:th.70">XBOOLE_1:70</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A208</font></span>: 
<font color="Maroon" title="c21">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_2_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A197</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_1_2_1_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E39">A208</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A209</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A210</font></span>: 
<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A197</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_1_1_2_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="E1:2_1_7_1_2_2_1_1_2_1_1_1_2_1"/>
( <font color="Maroon" title="c22">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c22">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E42">A211</font></span>: 
<font color="Maroon" title="c22">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E43">A212</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_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="E1:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E33">A187</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A213</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font></span><br/><div class="add"><span class="kw">assume </span><a NAME="E2:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A214</font></span>: 
<font color="Maroon" title="c25">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A215</font></span>: 
<font color="Maroon" title="c25">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E47">A216</font></span>: 
<font color="Maroon" title="c23">u1</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="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A197</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="E7:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A217</font></span>: 
<font color="Maroon" title="c23">u1</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">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E8:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E47">A216</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E9:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E46">A215</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E10:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A218</font></span>: <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">u1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E46">A215</font></span></a>, <a class="txt" href="#E6:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E47">A216</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>
;<br/><a NAME="E12:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E13:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E14:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</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="E15:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E44">A213</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E45">A214</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E16:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_7_1_2"><span class="lab"><font color="Green" title="E28">A179</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E40">A209</font></span></a>, <a class="txt" href="#E11:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E49">A218</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E48">A217</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E3:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1"/>
( <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E37">A196</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A197</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E42">A211</font></span></a>, <a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_7_1_2_2_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E41">A210</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_1_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E43">A212</font></span></a>, <a class="ref" href="xboole_1.html#T70" title="XBOOLE_1:th.70">XBOOLE_1:70</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_2_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E42">A219</font></span>: 
<font color="Maroon" title="c22">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_2_1_1_1_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A197</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_1_2_1_1_1_2_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E42">A219</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_1_2_1_1_1_2_1_2"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_1_2_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A197</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:2_1_7_1_2_2_1_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E37">A196</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E41">A210</font></span></a>, <a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_2_1_2"/>
(  not <font color="Maroon" title="c21">u</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">H</font> or  not <font color="Maroon" title="c22">v</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">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font> <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="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_2_2_1_1_2_1_2"/>
<font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1</span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1</span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">H</font> = <font color="Maroon" title="c20">H</font> as   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E36">A195</font></span></a>, <a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c22">z</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw"> such that </span><br/><a NAME="E8:2_1_7_1_2_2_1_1"/><span class="lab"><font color="Green" title="E37">A220</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">z</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="c19">y</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="c22">z</font> holds <br/><font color="Maroon" title="c22">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span><a NAME="E9:2_1_7_1_2_2_1_1"/>
<font color="Maroon" title="c17">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2"><span class="lab"><font color="Green" title="E21">A172</font></span></a>, <a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></span>;<br/><a NAME="E10:2_1_7_1_2_2_1_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E11:2_1_7_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A221</font></span>: 
1 <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="c17">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E12:2_1_7_1_2_2_1_1"/><span class="lab"><font color="Green" title="E39">A222</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:2_1_7_1_2_2_1_1_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> or <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_2_2_1_1_3_1"/>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font></span>)</span></span><br/>

<a NAME="E2:2_1_7_1_2_2_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:2_1_7_1_2_2_1_1_3_1"/><span class="lab"><font color="Green" title="E40">A223</font></span>: 
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">A</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1_7_1_2_2_1_1_3_1"/><span class="lab"><font color="Green" title="E41">A224</font></span>: 
<font color="Maroon" title="c24">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:2_1_7_1_2_2_1_1_3_1"/><span class="lab"><font color="Green" title="E42">A225</font></span>: 
<font color="Maroon" title="c25">u</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="c6">F</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_7_1_2_2_1_1_3_1"/><span class="lab"><font color="Green" title="E43">A226</font></span>: 
<font color="Maroon" title="c24">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_7_1_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E41">A224</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">u</font> = <font color="Maroon" title="c25">u</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E42">A225</font></span></a></span>;<br/>
<a NAME="E10:2_1_7_1_2_2_1_1_3_1"/><span class="lab"><font color="Green" title="E44">A227</font></span>: 
<font color="Maroon" title="c26">u</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="c6">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E42">A225</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="E11:2_1_7_1_2_2_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A228</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E12:2_1_7_1_2_2_1_1_3_1"/><span class="lab"><font color="Green" title="E46">A229</font></span>: 
<font color="Maroon" title="c26">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="txt" href="#E10:2_1_7_1_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E44">A227</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_3_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="E1:2_1_7_1_2_2_1_1_3_1_1"/>
( <font color="Maroon" title="c26">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c26">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_3_1_1_1"/>
<font color="Maroon" title="c26">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font></span>)</span></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c24">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E11:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E38">A221</font></span></a>, <a class="txt" href="#E8:2_1_7_1_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E43">A226</font></span></a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A230</font></span>: 
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E40">A223</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_1_3_1_1_1"/>
<font color="Maroon" title="c21">H</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>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E11:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E38">A221</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E5:2_1_7_1_2_2_1_1_3_1_1_1"/>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_2_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E47">A230</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_3_1_1_2"/>
<font color="Maroon" title="c26">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font></span>)</span></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c26">u</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E46">A229</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c26">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c26">u</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> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_7_1_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E45">A228</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_1_3_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font> &amp; <font color="Maroon" title="c24">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">u</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_7_1_2"><span class="lab"><font color="Green" title="E24">A175</font></span></a>, <a class="txt" href="#E8:2_1_7_1_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E43">A226</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1_3_1"><span class="lab"><font color="Green" title="E40">A223</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E7:2_1_7_1_2_2_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1"><span class="lab"><font color="Green" title="E17">A16</font></span></a>, <a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:2_1_7_1_2_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E47">A231</font></span>: 
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">B</font>
 <span class="kw">and </span><br/><a NAME="E10:2_1_7_1_2_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E48">A232</font></span>: 
<font color="Maroon" title="c27">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c28">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E12:2_1_7_1_2_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E49">A233</font></span>: 
<font color="Maroon" title="c28">w</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="c17">G</font>
 <span class="kw">and </span><br/><a NAME="E13:2_1_7_1_2_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E50">A234</font></span>: 
<font color="Maroon" title="c27">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_7_1_2_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E48">A232</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c29">w</font> = <font color="Maroon" title="c28">w</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_2_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E49">A233</font></span></a></span>;<br/><a NAME="E15:2_1_7_1_2_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E51">A235</font></span>: 
<font color="Maroon" title="c29">w</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="c17">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_2_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E49">A233</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="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_3_1_1_2_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="E1:2_1_7_1_2_2_1_1_3_1_1_2_1"/>
( <font color="Maroon" title="c29">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c29">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_3_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E52">A236</font></span>: 
<font color="Maroon" title="c29">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font></span>)</span></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E11:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E38">A221</font></span></a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_1_3_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A237</font></span>: 
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E47">A231</font></span></a>, <a class="txt" href="#E13:2_1_7_1_2_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A234</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E52">A236</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c21">H</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>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E11:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E38">A221</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E5:2_1_7_1_2_2_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_2_2_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E53">A237</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_3_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_3_1_1_2_1_2"/><span class="lab"><font color="Green" title="E52">A238</font></span>: 
<font color="Maroon" title="c29">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font></span>)</span></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_3_1_1_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_2_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E49">A233</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_1_3_1_1_2_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_3_1_1_2_1_2"><span class="lab"><font color="Green" title="E52">A238</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_1_3_1_1_2_1_2"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_1_3_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A239</font></span>: 
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E9:2_1_7_1_2_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E47">A231</font></span></a>, <a class="txt" href="#E13:2_1_7_1_2_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A234</font></span></a>, <a class="txt" href="#E15:2_1_7_1_2_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E51">A235</font></span></a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_1_3_1_1_2_1_2"/>
<font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E15:2_1_7_1_2_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E51">A235</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E7:2_1_7_1_2_2_1_1_3_1_1_2_1_2"/>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_7_1_2_2_1_1_3_1_1_2_1_2"><span class="lab"><font color="Green" title="E53">A239</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c23">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font></span>)</span> or <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:2_1_7_1_2_2_1_1_3"/>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:2_1_7_1_2_2_1_1_3"/><span class="lab"><font color="Green" title="E40">A240</font></span>: 
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">A</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1_7_1_2_2_1_1_3"/><span class="lab"><font color="Green" title="E41">A241</font></span>: 
<font color="Maroon" title="c24">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:2_1_7_1_2_2_1_1_3"/><span class="lab"><font color="Green" title="E42">A242</font></span>: 
<font color="Maroon" title="c25">k</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">H</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_7_1_2_2_1_1_3"/><span class="lab"><font color="Green" title="E43">A243</font></span>: 
<font color="Maroon" title="c24">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_7_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E41">A241</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">k</font> = <font color="Maroon" title="c25">k</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E42">A242</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_3_2"><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="E1:2_1_7_1_2_2_1_1_3_2"/>
( <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c26">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_3_2_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_3_2_1"/>
<font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A244</font></span>: 
<font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E42">A242</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_3_2_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="E1:2_1_7_1_2_2_1_1_3_2_1_1"/>
( <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 or <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E40">A240</font></span></a>, <a class="txt" href="#E8:2_1_7_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E43">A243</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_1_3_2_1"><span class="lab"><font color="Green" title="E44">A244</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_3_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_3_2_1_1_1"/><span class="lab"><font color="Green" title="E45">A245</font></span>: 
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_3_2_1_1_1"/>
1 <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="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E38">A221</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="E3:2_1_7_1_2_2_1_1_3_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">G</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>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_1_3_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_3_2_1_1_1"><span class="lab"><font color="Green" title="E45">A245</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_1_3_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_1_3_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><span class="kw">hence </span><a NAME="E7:2_1_7_1_2_2_1_1_3_2_1_1_1"/>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_3_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_3_2_1_1_2"/><span class="lab"><font color="Green" title="E45">A246</font></span>: 
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_3_2_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E3:2_1_7_1_2_2_1_1_3_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_1_3_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_1_3_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_3_2_1_1_2"><span class="lab"><font color="Green" title="E45">A246</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E6:2_1_7_1_2_2_1_1_3_2_1_1_2"/>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_3_2_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_3_2_2"/><span class="lab"><font color="Green" title="E44">A247</font></span>: 
<font color="Maroon" title="c26">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_3_2_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E42">A242</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_1_3_2_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_3_2_2"><span class="lab"><font color="Green" title="E44">A247</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_1_3_2_2"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_1_3_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A248</font></span>: 
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E40">A240</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E42">A242</font></span></a>, <a class="txt" href="#E8:2_1_7_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E43">A243</font></span></a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_1_3_2_2"/>
<font color="Maroon" title="c26">k</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="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_1_3"><span class="lab"><font color="Green" title="E42">A242</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="E7:2_1_7_1_2_2_1_1_3_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E8:2_1_7_1_2_2_1_1_3_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_7_1_2_2_1_1_3_2_2"><span class="lab"><font color="Green" title="E45">A248</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E9:2_1_7_1_2_2_1_1_3_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E10:2_1_7_1_2_2_1_1_3_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><span class="kw">hence </span><a NAME="E11:2_1_7_1_2_2_1_1_3_2_2"/>
<font color="Maroon" title="c23">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E40">A249</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:2_1_7_1_2_2_1_1_4"/>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">x1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_7_1_2_2_1_1_4"/><span class="lab"><font color="Green" title="E41">A250</font></span>: 
<font color="Maroon" title="c23">l</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">x1</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_7_1_2_2_1_1_4"/><span class="lab"><font color="Green" title="E42">A251</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">l</font> = <font color="Maroon" title="c23">l</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_2_2_1_1_4"><span class="lab"><font color="Green" title="E41">A250</font></span></a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_1_4"/>
<font color="Maroon" title="c24">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_2_2_1_1_4"><span class="lab"><font color="Green" title="E41">A250</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E7:2_1_7_1_2_2_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A252</font></span>: 
<font color="Maroon" title="c24">l</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="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E8:2_1_7_1_2_2_1_1_4"/>
<font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_2_2_1_1_4"><span class="lab"><font color="Green" title="E41">A250</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_1_4"><span class="lab"><font color="Green" title="E42">A251</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E9:2_1_7_1_2_2_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A253</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">l</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_1_4"><span class="lab"><font color="Green" title="E43">A252</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_4_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="E1:2_1_7_1_2_2_1_1_4_1"/>
( <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E45">A254</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c25">EMPTY</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T21" title="MEASURE1:th.21">MEASURE1:21</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c26">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:2_1_7_1_2_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E46">A255</font></span>: 
<font color="Maroon" title="c26">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E45">A254</font></span></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E47">A256</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_1_4"><span class="lab"><font color="Green" title="E43">A252</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c26">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E46">A255</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E7:2_1_7_1_2_2_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A257</font></span>: 
<font color="Maroon" title="c26">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E8:2_1_7_1_2_2_1_1_4_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">Fl</font> = <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_7_1_2_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E47">A256</font></span></a></span>;<br/><a NAME="E10:2_1_7_1_2_2_1_1_4_1_1"/>
<font color="Maroon" title="c25">EMPTY</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/><a NAME="E11:2_1_7_1_2_2_1_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c27">Fl</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></span>;<br/><a NAME="E12:2_1_7_1_2_2_1_1_4_1_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c27">Fl</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>;<br/><a NAME="E13:2_1_7_1_2_2_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A258</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_1_4"><span class="lab"><font color="Green" title="E43">A252</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><a NAME="E14:2_1_7_1_2_2_1_1_4_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_1_4"><span class="lab"><font color="Green" title="E43">A252</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E46">A255</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E15:2_1_7_1_2_2_1_1_4_1_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E48">A257</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:2_1_7_1_2_2_1_1_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2_2_1_1_4"><span class="lab"><font color="Green" title="E44">A253</font></span></a>, <a class="txt" href="#E13:2_1_7_1_2_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E49">A258</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_4_1_2"/><span class="lab"><font color="Green" title="E45">A259</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">l</font> = 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_1_4"><span class="lab"><font color="Green" title="E43">A252</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_4_1_2"><span class="lab"><font color="Green" title="E45">A259</font></span></a>, <a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_2_2_1_1_4_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2_2_1_1_4"><span class="lab"><font color="Green" title="E44">A253</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E14:2_1_7_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E15:2_1_7_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A260</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E16:2_1_7_1_2_2_1_1"/><span class="lab"><font color="Green" title="E42">A261</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_5"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_5_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="E1:2_1_7_1_2_2_1_1_5_1"/>
( <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E33">A187</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_5_1_1"/><span class="lab"><font color="Green" title="E43">A262</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> </span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E41">A260</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A263</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_5_1_1"><span class="lab"><font color="Green" title="E43">A262</font></span></a>, <a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_1_5_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</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="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E15:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E41">A260</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:2_1_7_1_2_2_1_1_5_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_2_2_1_1_5_1_1"><span class="lab"><font color="Green" title="E44">A263</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_5_1_2"/><span class="lab"><font color="Green" title="E43">A264</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> </span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_5_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</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="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E15:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E41">A260</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_2_2_1_1_5_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_5_1_2"><span class="lab"><font color="Green" title="E43">A264</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E43">A265</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:2_1_7_1_2_2_1_1_6"/>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:2_1_7_1_2_2_1_1_6"/><span class="kw">then </span>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_2_2_1_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E42">A261</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E18:2_1_7_1_2_2_1_1"/><span class="lab"><font color="Green" title="E44">A266</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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">H</font> holds <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">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">H</font> implies 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> <font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">m</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_2_2_1_1_7"/><span class="lab"><font color="Green" title="E45">A267</font></span>: 
<font color="Maroon" title="c23">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">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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> <font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">m</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c24">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">m</font> implies <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:2_1_7_1_2_2_1_1_7"/><span class="lab"><font color="Green" title="E46">A268</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_7_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="E1:2_1_7_1_2_2_1_1_7_1"/>
( <font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c23">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_7_1_1"/><span class="lab"><font color="Green" title="E47">A269</font></span>: 
<font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A270</font></span>: 
<font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_7"><span class="lab"><font color="Green" title="E45">A267</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_7_1_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="E1:2_1_7_1_2_2_1_1_7_1_1_1"/>
( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 or <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_7_1_2_2_1_1_7"><span class="lab"><font color="Green" title="E46">A268</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_1_7_1_1"><span class="lab"><font color="Green" title="E48">A270</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_7_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_7_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A271</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_7_1_1_1_1"/><span class="lab"><font color="Green" title="E50">A272</font></span>: 
<font color="Maroon" title="c23">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="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_7"><span class="lab"><font color="Green" title="E45">A267</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="E3:2_1_7_1_2_2_1_1_7_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">G</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>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_7_1_1"><span class="lab"><font color="Green" title="E47">A269</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_1_7_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E49">A271</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_1_7_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A273</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_1_7_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_7_1_1"><span class="lab"><font color="Green" title="E47">A269</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E49">A271</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E50">A272</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:2_1_7_1_2_2_1_1_7_1_1_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_7_1_2_2_1_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E51">A273</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_7_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_7_1_1_1_2"/><span class="lab"><font color="Green" title="E49">A274</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_7_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E3:2_1_7_1_2_2_1_1_7_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">hence </span><a NAME="E4:2_1_7_1_2_2_1_1_7_1_1_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E10:2_1_7_1_2"><span class="lab"><font color="Green" title="E27">A178</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E33">A187</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_7_1_1"><span class="lab"><font color="Green" title="E47">A269</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_7_1_1_1_2"><span class="lab"><font color="Green" title="E49">A274</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_7_1_2"/><span class="lab"><font color="Green" title="E47">A275</font></span>: 
<font color="Maroon" title="c23">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_7_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_7"><span class="lab"><font color="Green" title="E45">A267</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_1_7_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_7_1_2"><span class="lab"><font color="Green" title="E47">A275</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_1_7_1_2"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A276</font></span>: 
<font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_7"><span class="lab"><font color="Green" title="E45">A267</font></span></a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_1_7_1_2"/><span class="lab"><font color="Green" title="E49">A277</font></span>: 
<font color="Maroon" title="c23">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="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_7"><span class="lab"><font color="Green" title="E45">A267</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="E7:2_1_7_1_2_2_1_1_7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E8:2_1_7_1_2_2_1_1_7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_7_1_2_2_1_1_7"><span class="lab"><font color="Green" title="E46">A268</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_1_7_1_2"><span class="lab"><font color="Green" title="E48">A276</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E9:2_1_7_1_2_2_1_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A278</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E10:2_1_7_1_2_2_1_1_7_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_1_7"><span class="lab"><font color="Green" title="E46">A268</font></span></a>, <a class="txt" href="#E6:2_1_7_1_2_2_1_1_7_1_2"><span class="lab"><font color="Green" title="E49">A277</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_1_7_1_2"><span class="lab"><font color="Green" title="E48">A276</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E11:2_1_7_1_2_2_1_1_7_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2_2_1_1_7_1_2"><span class="lab"><font color="Green" title="E50">A278</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E19:2_1_7_1_2_2_1_1"/><span class="lab"><font color="Green" title="E45">A279</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">z</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="c19">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E37">A220</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/><a NAME="E20:2_1_7_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A280</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">z</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="c21">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_2"><span class="lab"><font color="Green" title="E29">A180</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</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="E21:2_1_7_1_2_2_1_1"/><span class="lab"><font color="Green" title="E47">A281</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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="c22">z</font> holds <br/><font color="Maroon" title="c22">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">k</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="c22">z</font> implies <font color="Maroon" title="c22">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_2_2_1_1_8"/><span class="lab"><font color="Green" title="E48">A282</font></span>: 
<font color="Maroon" title="c23">k</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="c22">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font></span><br/>

<a NAME="E2:2_1_7_1_2_2_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A283</font></span>: 
<font color="Maroon" title="c22">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E37">A220</font></span></a></span>;<br/>
<a NAME="E3:2_1_7_1_2_2_1_1_8"/><span class="lab"><font color="Green" title="E50">A284</font></span>: 
<font color="Maroon" title="c19">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_7_1_2"><span class="lab"><font color="Green" title="E30">A181</font></span></a>, <a class="txt" href="#E19:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E45">A279</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_8"><span class="lab"><font color="Green" title="E48">A282</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_8_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="E1:2_1_7_1_2_2_1_1_8_1"/>
( <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c23">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_8_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_8_1_1"/><span class="lab"><font color="Green" title="E51">A285</font></span>: 
<font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_7_1_2"><span class="lab"><font color="Green" title="E27">A178</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_1_8"><span class="lab"><font color="Green" title="E49">A283</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_2_2_1_1_8_1_1"/>
<font color="Maroon" title="c22">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_7_1_2"><span class="lab"><font color="Green" title="E27">A178</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_1_8"><span class="lab"><font color="Green" title="E50">A284</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_8_1_1"><span class="lab"><font color="Green" title="E51">A285</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_1_8_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_1_8_1_2"/><span class="lab"><font color="Green" title="E51">A286</font></span>: 
<font color="Maroon" title="c23">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_1_8_1_2"/><span class="lab"><font color="Green" title="E52">A287</font></span>: 
<font color="Maroon" title="c23">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_2"><span class="lab"><font color="Green" title="E29">A180</font></span></a>, <a class="txt" href="#E19:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E45">A279</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_8"><span class="lab"><font color="Green" title="E48">A282</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="E3:2_1_7_1_2_2_1_1_8_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_1_8_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_1_8_1_2"><span class="lab"><font color="Green" title="E51">A286</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_1_8_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A288</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> = 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E46">A280</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_8"><span class="lab"><font color="Green" title="E48">A282</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_1_8_1_2"><span class="lab"><font color="Green" title="E52">A287</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_1_8_1_2"><span class="lab"><font color="Green" title="E53">A288</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_2"><span class="lab"><font color="Green" title="E29">A180</font></span></a>, <a class="txt" href="#E19:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E45">A279</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_8"><span class="lab"><font color="Green" title="E48">A282</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:2_1_7_1_2_2_1_1_8_1_2"/>
<font color="Maroon" title="c22">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_7_1_2"><span class="lab"><font color="Green" title="E30">A181</font></span></a>, <a class="txt" href="#E19:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E45">A279</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_1_8"><span class="lab"><font color="Green" title="E48">A282</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_1_8"><span class="lab"><font color="Green" title="E49">A283</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E22:2_1_7_1_2_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c8">x</font> = 
<font color="Maroon" title="c8">x</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<font color="Maroon" title="c16">x1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E15:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E41">A260</font></span></a>, <a class="ref" href="finseq_5.html#T11" title="FINSEQ_5:th.11">FINSEQ_5:11</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A289</font></span>:  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font> = 
<span class="p1">(<span class="default"><a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c16">x1</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E40">A249</font></span></a>, <a class="txt" href="#E17:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E43">A265</font></span></a>, <a class="ref" href="extreal1.html#T43" title="EXTREAL1:th.43">EXTREAL1:43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c16">x1</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E42">A261</font></span></a>, <a class="ref" href="extreal1.html#T41" title="EXTREAL1:th.41">EXTREAL1:41</a></span>
;<br/> <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">H</font> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E35">A194</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="c18">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>
;<br/><a NAME="E26:2_1_7_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">H</font>,<font color="Maroon" title="c18">b</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E39">A222</font></span></a>, <a class="txt" href="#E18:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E44">A266</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><span class="kw">then </span> <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> = 
 <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c22">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E10:2_1_7_1_2"><span class="lab"><font color="Green" title="E27">A178</font></span></a>, <a class="txt" href="#E11:2_1_7_1_2"><span class="lab"><font color="Green" title="E28">A179</font></span></a>, <a class="txt" href="#E8:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E37">A220</font></span></a>, <a class="txt" href="#E20:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E46">A280</font></span></a>, <a class="ref" href="mesfunc3.html#D2" title="MESFUNC3:def.2">MESFUNC3:def 2</a></span>
<br/>.= 
 <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_7_1_2"><span class="lab"><font color="Green" title="E31">A182</font></span></a>, <a class="txt" href="#E19:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E45">A279</font></span></a>, <a class="txt" href="#E21:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E47">A281</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>
<br/>.= 
 <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c16">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E20:2_1"><span class="lab"><font color="Green" title="E18">A23</font></span></a>, <a class="txt" href="#E13:2_1"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="txt" href="#E23:2_1"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="txt" href="#E16:2_1"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2"><span class="lab"><font color="Green" title="E21">A172</font></span></a>, <a class="txt" href="#E15:2_1_7_1_2"><span class="lab"><font color="Green" title="E32">A183</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2"><span class="lab"><font color="Green" title="E25">A176</font></span></a></span>
<br/>.= 
 <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:2_1_7_1_2_2_1_1"><span class="lab"><font color="Green" title="E48">A289</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">hence </span><a NAME="E28:2_1_7_1_2_2_1_1"/>
 <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2"><span class="kw">case </span></a><a NAME="E1:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E33">A290</font></span>: 
( <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font></span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( ( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">b</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1 ) &amp; ( $1 <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="c18">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) );<br/><a NAME="E2:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E34">A291</font></span>: 
<font color="Maroon" title="c5">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">V58</a>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="mesfunc2.html#D5" title="MESFUNC2:def.5">MESFUNC2:def 5</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c20">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E35">A292</font></span>: 
<font color="Maroon" title="c20">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E33">A290</font></span></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E36">A293</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/> ex <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">v</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">k</font>,<font color="Olive" title="b1">v</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_2_2_1_2_1"/>
<font color="Maroon" title="c21">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">k</font>,<font color="Olive" title="b1">v</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_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="E1:2_1_7_1_2_2_1_2_1_1"/>
( <font color="Maroon" title="c21">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c21">k</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="c18">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E37">A294</font></span>: 
<font color="Maroon" title="c21">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">k</font>,<font color="Olive" title="b1">v</font>]</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c22">v</font> = <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c21">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  ;<br/><span class="kw">take </span>
<font color="Maroon" title="c22">v</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">k</font>,<font color="Maroon" title="c22">v</font>]</span><br/><span class="kw">thus </span><a NAME="E3:2_1_7_1_2_2_1_2_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">k</font>,<font color="Maroon" title="c22">v</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E37">A294</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E37">A295</font></span>: 
<font color="Maroon" title="c21">k</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="c18">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">k</font>,<font color="Olive" title="b1">v</font>]</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c22">v</font> = <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  ;<br/><span class="kw">take </span>
<font color="Maroon" title="c22">v</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">k</font>,<font color="Maroon" title="c22">v</font>]</span><br/><span class="kw">thus </span><a NAME="E3:2_1_7_1_2_2_1_2_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">k</font>,<font color="Maroon" title="c22">v</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E37">A295</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c21">c</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw"> such that </span><br/><a NAME="E7:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E37">A296</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">c</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</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="#E5:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E36">A293</font></span></a>);<br/></span><a NAME="E8:2_1_7_1_2_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E9:2_1_7_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A297</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E10:2_1_7_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E11:2_1_7_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E35">A292</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E12:2_1_7_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A298</font></span>: 
<font color="Maroon" title="c20">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E13:2_1_7_1_2_2_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">v</font> = <font color="Maroon" title="c20">v</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E39">A298</font></span></a></span>;<br/><a NAME="E15:2_1_7_1_2_2_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c22">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E9:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E38">A297</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E35">A292</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E16:2_1_7_1_2_2_1_2"/><span class="kw">then </span>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E34">A291</font></span></a>, <a class="txt" href="#E12:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E39">A298</font></span></a>, <a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 1</a></span>;<br/><a NAME="E17:2_1_7_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A299</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extreal2.html#T58" title="EXTREAL2:th.58">EXTREAL2:58</a></span>;<br/><a NAME="E18:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E41">A300</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">c</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="c18">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E37">A296</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="E19:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E42">A301</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c22">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E12:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E39">A298</font></span></a></span>;<br/><a NAME="E20:2_1_7_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A302</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E33">A290</font></span></a>, <a class="txt" href="#E9:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E38">A297</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E35">A292</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E21:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E44">A303</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">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">c</font> holds <br/>(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">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="Red">::  thesis: </font></span></a><span class="hide"> ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">m</font> &amp; <font color="Maroon" title="c23">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">c</font> implies (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font> &amp; <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_7_1_2_2_1_2_2"/><span class="lab"><font color="Green" title="E45">A304</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">m</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_7_1_2_2_1_2_2"/><span class="lab"><font color="Green" title="E46">A305</font></span>: 
<font color="Maroon" title="c23">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">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font> &amp; <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )</span><br/>

<a NAME="E3:2_1_7_1_2_2_1_2_2"/><span class="lab"><font color="Green" title="E47">A306</font></span>: 
<font color="Maroon" title="c23">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="c21">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_7_1_2_2_1_2_2"><span class="lab"><font color="Green" title="E46">A305</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_2_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="E1:2_1_7_1_2_2_1_2_2_1"/>
( <font color="Maroon" title="c23">m</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="c21">c</font> or <font color="Maroon" title="c23">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">c</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_2_1_1"/>
<font color="Maroon" title="c23">m</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="c21">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font> &amp; <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_7_1_2_2_1_2_2_1_1"/>
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font> &amp; <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E43">A302</font></span></a>, <a class="txt" href="#E17:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E40">A299</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E37">A296</font></span></a>, <a class="txt" href="#E18:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E41">A300</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_2_2"><span class="lab"><font color="Green" title="E46">A305</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_2_1_2"/>
<font color="Maroon" title="c23">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font> &amp; <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )</span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c23">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E41">A300</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_2_2"><span class="lab"><font color="Green" title="E47">A306</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A307</font></span>: 
<font color="Maroon" title="c23">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="c18">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_2_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_2"><span class="lab"><font color="Green" title="E45">A304</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A308</font></span>: 
<font color="Maroon" title="c23">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="c18">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_2_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E48">A307</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_2_2_1_2"/>
<font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E37">A296</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_2_2"><span class="lab"><font color="Green" title="E46">A305</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E48">A307</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:2_1_7_1_2_2_1_2_2_1_2"/>
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font> &amp; <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_7_1_2"><span class="lab"><font color="Green" title="E28">A179</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_2"><span class="lab"><font color="Green" title="E45">A304</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E49">A308</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E22:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E45">A309</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E9:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E38">A297</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E35">A292</font></span></a>, <a class="txt" href="#E19:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E42">A301</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><div><span class="lab"><font color="Green" title="E46">A310</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:2_1_7_1_2_2_1_2_3"/><span class="lab"><font color="Green" title="E47">A311</font></span>: 
(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">y</font> or  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_3_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="E1:2_1_7_1_2_2_1_2_3_1"/>
(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">y</font> or  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_3"><span class="lab"><font color="Green" title="E47">A311</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_3_1_1"/>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_7_1_2_2_1_2_3_1_1"/><span class="lab"><font color="Green" title="E48">A312</font></span>: 
<font color="Maroon" title="c23">k</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">y</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_7_1_2_2_1_2_3_1_1"/><span class="lab"><font color="Green" title="E49">A313</font></span>: 
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">k</font> = <font color="Maroon" title="c23">k</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_2_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E48">A312</font></span></a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_2_3_1_1"/><span class="lab"><font color="Green" title="E50">A314</font></span>: 
<font color="Maroon" title="c19">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">k</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_7_1_2"><span class="lab"><font color="Green" title="E30">A181</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E48">A312</font></span></a></span>;<br/><a NAME="E7:2_1_7_1_2_2_1_2_3_1_1"/>
<font color="Maroon" title="c24">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_2_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E48">A312</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="E8:2_1_7_1_2_2_1_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A315</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_3_1_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="E1:2_1_7_1_2_2_1_2_3_1_1_1"/>
( <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c24">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_3_1_1_1_1"/>
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_7_1_2_2_1_2_3_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_7_1_2"><span class="lab"><font color="Green" title="E27">A178</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E49">A313</font></span></a>, <a class="txt" href="#E6:2_1_7_1_2_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E50">A314</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_3_1_1_1_2"/>
<font color="Maroon" title="c24">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_2_3_1_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_7_1_2_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E51">A315</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_2_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A316</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_2_3_1_1_1_2"/>
<font color="Maroon" title="c24">k</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">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="txt" href="#E12:2_1_7_1_2"><span class="lab"><font color="Green" title="E29">A180</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E48">A312</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_2_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A317</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_7_1_2"><span class="lab"><font color="Green" title="E28">A179</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_2_3_1_1_1_2"><span class="lab"><font color="Green" title="E52">A316</font></span></a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_2_3_1_1_1_2"/>
( <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_2"><span class="lab"><font color="Green" title="E29">A180</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E48">A312</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">Gk</font> = <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">EMPTY</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T10" title="PROB_1:th.10">PROB_1:10</a></span>;<br/><a NAME="E9:2_1_7_1_2_2_1_2_3_1_1_1_2"/>
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">EMPTY</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">Gk</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/><a NAME="E10:2_1_7_1_2_2_1_2_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A318</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">Gk</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>;<br/><a NAME="E11:2_1_7_1_2_2_1_2_3_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_2"><span class="lab"><font color="Green" title="E29">A180</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E48">A312</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><span class="kw">hence </span><a NAME="E12:2_1_7_1_2_2_1_2_3_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E49">A313</font></span></a>, <a class="txt" href="#E6:2_1_7_1_2_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E50">A314</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_2_3_1_1_1_2"><span class="lab"><font color="Green" title="E53">A317</font></span></a>, <a class="txt" href="#E10:2_1_7_1_2_2_1_2_3_1_1_1_2"><span class="lab"><font color="Green" title="E54">A318</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_3_1_2"/><span class="lab"><font color="Green" title="E48">A319</font></span>: 
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c23">EMPTY</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T10" title="PROB_1:th.10">PROB_1:10</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_2_3_1_2"/><span class="lab"><font color="Green" title="E49">A320</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</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="E4:2_1_7_1_2_2_1_2_3_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E5:2_1_7_1_2_2_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A321</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_2_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">Fn1</font> = <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_2_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E49">A320</font></span></a></span>;<br/><a NAME="E8:2_1_7_1_2_2_1_2_3_1_2"/><span class="lab"><font color="Green" title="E51">A322</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">Fn1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_7_1_2_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E50">A321</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><a NAME="E9:2_1_7_1_2_2_1_2_3_1_2"/>
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">EMPTY</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">Fn1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/><a NAME="E10:2_1_7_1_2_2_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A323</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">Fn1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>;<br/><a NAME="E11:2_1_7_1_2_2_1_2_3_1_2"/>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E48">A319</font></span></a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></span>;<br/><span class="kw">hence </span><a NAME="E12:2_1_7_1_2_2_1_2_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E45">A309</font></span></a>, <a class="txt" href="#E10:2_1_7_1_2_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E52">A323</font></span></a>, <a class="txt" href="#E8:2_1_7_1_2_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E51">A322</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E24:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E47">A324</font></span>: 
not  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">x1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c23">EMPTY</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T10" title="PROB_1:th.10">PROB_1:10</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:2_1_7_1_2_2_1_2_4"/>
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">x1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:2_1_7_1_2_2_1_2_4"/><span class="lab"><font color="Green" title="E48">A325</font></span>: 
<font color="Maroon" title="c24">k</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">x1</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1_7_1_2_2_1_2_4"/><span class="lab"><font color="Green" title="E49">A326</font></span>: 
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c25">k</font> = <font color="Maroon" title="c24">k</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_2_4"><span class="lab"><font color="Green" title="E48">A325</font></span></a></span>;<br/>
<a NAME="E7:2_1_7_1_2_2_1_2_4"/>
<font color="Maroon" title="c25">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_2_4"><span class="lab"><font color="Green" title="E48">A325</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<a NAME="E8:2_1_7_1_2_2_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A327</font></span>: 
<font color="Maroon" title="c25">k</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="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E9:2_1_7_1_2_2_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A328</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">k</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a></span>;<br/>
<a NAME="E10:2_1_7_1_2_2_1_2_4"/><span class="lab"><font color="Green" title="E52">A329</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E8:2_1_7_1_2_2_1_2_4"><span class="lab"><font color="Green" title="E50">A327</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E11:2_1_7_1_2_2_1_2_4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">Fk</font> = <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_7_1_2_2_1_2_4"><span class="lab"><font color="Green" title="E52">A329</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_4_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="E1:2_1_7_1_2_2_1_2_4_1"/>
( <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_4_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_7_1_2_2_1_2_4_1_1"/><span class="lab"><font color="Green" title="E53">A330</font></span>: 
<font color="Maroon" title="c27">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_2_4_1_1"/>
<font color="Maroon" title="c27">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_7_1_2_2_1_2_4"><span class="lab"><font color="Green" title="E52">A329</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E53">A330</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A331</font></span>: 
<font color="Maroon" title="c27">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_2_4_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c27">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E8:2_1_7_1_2_2_1_2_4"><span class="lab"><font color="Green" title="E50">A327</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E53">A330</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E7:2_1_7_1_2_2_1_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A332</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E54">A331</font></span></a></span>;<br/><a NAME="E8:2_1_7_1_2_2_1_2_4_1_1"/>
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c23">EMPTY</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">Fk</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/><a NAME="E9:2_1_7_1_2_2_1_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A333</font></span>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">Fk</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>;<br/><a NAME="E10:2_1_7_1_2_2_1_2_4_1_1"/>
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E8:2_1_7_1_2_2_1_2_4"><span class="lab"><font color="Green" title="E50">A327</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><span class="kw">hence </span><a NAME="E11:2_1_7_1_2_2_1_2_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_2_4"><span class="lab"><font color="Green" title="E48">A325</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_2_4"><span class="lab"><font color="Green" title="E49">A326</font></span></a>, <a class="txt" href="#E9:2_1_7_1_2_2_1_2_4"><span class="lab"><font color="Green" title="E51">A328</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E55">A332</font></span></a>, <a class="txt" href="#E9:2_1_7_1_2_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E56">A333</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_4_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  = 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D19" title="VALUED_0:def.19">VALUED_0:def 19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E8:2_1_7_1_2_2_1_2_4"><span class="lab"><font color="Green" title="E50">A327</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_2_2_1_2_4_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_2_4"><span class="lab"><font color="Green" title="E48">A325</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_2_4"><span class="lab"><font color="Green" title="E49">A326</font></span></a>, <a class="txt" href="#E9:2_1_7_1_2_2_1_2_4"><span class="lab"><font color="Green" title="E51">A328</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( ( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 ) &amp; ( $1 <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="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) );<br/><a NAME="E25:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E48">A334</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">G</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="c18">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E49">A335</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="c17">G</font></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">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 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="c18">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E48">A334</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="E27:2_1_7_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A336</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</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="c18">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T8" title="FINSEQ_1:th.8">FINSEQ_1:8</a></span>;<br/><a NAME="E28:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E51">A337</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/> ex <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">v</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c23">k</font>,<font color="Olive" title="b1">v</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_2_2_1_2_6"/><span class="lab"><font color="Green" title="E52">A338</font></span>: 
<font color="Maroon" title="c23">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c23">k</font>,<font color="Olive" title="b1">v</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_6_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="E1:2_1_7_1_2_2_1_2_6_1"/>
( <font color="Maroon" title="c23">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c23">k</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="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_6_1_1"/><span class="lab"><font color="Green" title="E53">A339</font></span>: 
<font color="Maroon" title="c23">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c23">k</font>,<font color="Olive" title="b1">v</font>]</span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_2_6_1_1"/>
<font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_6"><span class="lab"><font color="Green" title="E52">A338</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_2_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_6_1_1"><span class="lab"><font color="Green" title="E53">A339</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_2_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A340</font></span>: 
<font color="Maroon" title="c23">k</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="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_2_6_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_6"><span class="lab"><font color="Green" title="E52">A338</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E6:2_1_7_1_2_2_1_2_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">k</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="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_2_6_1_1"><span class="lab"><font color="Green" title="E54">A340</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E7:2_1_7_1_2_2_1_2_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A341</font></span>: 
<font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E8:2_1_7_1_2_2_1_2_6_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">v</font> = <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_2_2_1_2_6_1_1"><span class="lab"><font color="Green" title="E55">A341</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c24">v</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c23">k</font>,<font color="Maroon" title="c24">v</font>]</span><br/><span class="kw">thus </span><a NAME="E10:2_1_7_1_2_2_1_2_6_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c23">k</font>,<font color="Maroon" title="c24">v</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_6_1_1"><span class="lab"><font color="Green" title="E53">A339</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_6_1_2"/><span class="lab"><font color="Green" title="E53">A342</font></span>: 
<font color="Maroon" title="c23">k</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="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c23">k</font>,<font color="Olive" title="b1">v</font>]</span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_2_6_1_2"/>
( <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E38">A297</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">v</font> = <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c24">v</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c23">k</font>,<font color="Maroon" title="c24">v</font>]</span><br/><span class="kw">thus </span><a NAME="E4:2_1_7_1_2_2_1_2_6_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c23">k</font>,<font color="Maroon" title="c24">v</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_6_1_2"><span class="lab"><font color="Green" title="E53">A342</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c23">H</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E30:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E52">A343</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">H</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</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="#E28:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E51">A337</font></span></a>);<br/></span><a NAME="E31:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E53">A344</font></span>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">i</font>, <font color="Maroon" title="c25">j</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">j</font> implies <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_2_2_1_2_7"/><span class="lab"><font color="Green" title="E54">A345</font></span>: 
<font color="Maroon" title="c24">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_7_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="E1:2_1_7_1_2_2_1_2_7_1"/>
( ( <font color="Maroon" title="c24">i</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="c23">H</font> &amp; <font color="Maroon" title="c25">j</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="c23">H</font> ) or  not <font color="Maroon" title="c24">i</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="c23">H</font> or  not <font color="Maroon" title="c25">j</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="c23">H</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_7_1_1"/><span class="lab"><font color="Green" title="E55">A346</font></span>: 
( <font color="Maroon" title="c24">i</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="c23">H</font> &amp; <font color="Maroon" title="c25">j</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="c23">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">i</font> = <font color="Maroon" title="c24">i</font>, <font color="Maroon" title="c27">j</font> = <font color="Maroon" title="c25">j</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="E3:2_1_7_1_2_2_1_2_7_1_1"/><span class="lab"><font color="Green" title="E56">A347</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E55">A346</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_2_7_1_1"/><span class="lab"><font color="Green" title="E57">A348</font></span>: 
<font color="Maroon" title="c27">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E55">A346</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_2_7_1_1"/><span class="lab"><font color="Green" title="E58">A349</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c17">G</font> holds <br/><font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_7_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:2_1_7_1_2_2_1_2_7_1_1_1"/><span class="lab"><font color="Green" title="E59">A350</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_7_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_1_7_1_2_2_1_2_7_1_1_1_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c28">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_7_1_2_2_1_2_7_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A351</font></span>: 
<font color="Maroon" title="c28">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:2_1_7_1_2_2_1_2_7_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A352</font></span>: 
<font color="Maroon" title="c28">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c29">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:2_1_7_1_2_2_1_2_7_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A353</font></span>: 
<font color="Maroon" title="c28">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">A</font>
 <span class="kw">and </span><br/><a NAME="E7:2_1_7_1_2_2_1_2_7_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A354</font></span>: 
<font color="Maroon" title="c29">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_2_2_1_2_7_1_1_1_1"><span class="lab"><font color="Green" title="E60">A351</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c30">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:2_1_7_1_2_2_1_2_7_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A355</font></span>: 
<font color="Maroon" title="c30">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="c12">F1</font>
 <span class="kw">and </span><br/><a NAME="E10:2_1_7_1_2_2_1_2_7_1_1_1_1"/><span class="lab"><font color="Green" title="E65">A356</font></span>: 
<font color="Maroon" title="c29">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_2_2_1_2_7_1_1_1_1"><span class="lab"><font color="Green" title="E63">A354</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c31">m</font> = <font color="Maroon" title="c30">m</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2_2_1_2_7_1_1_1_1"><span class="lab"><font color="Green" title="E64">A355</font></span></a></span>;<br/>
<a NAME="E12:2_1_7_1_2_2_1_2_7_1_1_1_1"/>
<font color="Maroon" title="c31">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="c12">F1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2_2_1_2_7_1_1_1_1"><span class="lab"><font color="Green" title="E64">A355</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="E13:2_1_7_1_2_2_1_2_7_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c31">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_2"><span class="lab"><font color="Green" title="E25">A176</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E14:2_1_7_1_2_2_1_2_7_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A357</font></span>: 
<font color="Maroon" title="c31">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E15:2_1_7_1_2_2_1_2_7_1_1_1_1"/>
<font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2_2_1_2_7_1_1_1_1"><span class="lab"><font color="Green" title="E64">A355</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<a NAME="E16:2_1_7_1_2_2_1_2_7_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c29">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_7_1_2_2_1_2_7_1_1_1_1"><span class="lab"><font color="Green" title="E65">A356</font></span></a>, <a class="txt" href="#E14:2_1_7_1_2_2_1_2_7_1_1_1_1"><span class="lab"><font color="Green" title="E66">A357</font></span></a>, <a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></span>;<br/>
<a NAME="E17:2_1_7_1_2_2_1_2_7_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c29">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:2_1_7_1_2_2_1_2_7_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_2_7_1_1_1_1"><span class="lab"><font color="Green" title="E61">A352</font></span></a>, <a class="txt" href="#E6:2_1_7_1_2_2_1_2_7_1_1_1_1"><span class="lab"><font color="Green" title="E62">A353</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">let </span><font color="Maroon" title="c28">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="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">k</font> &amp; <font color="Maroon" title="c28">k</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="c17">G</font> implies <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">k</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:2_1_7_1_2_2_1_2_7_1_1_1"/><span class="lab"><font color="Green" title="E60">A358</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">k</font>
 <span class="kw">and </span><br/><a NAME="E3:2_1_7_1_2_2_1_2_7_1_1_1"/><span class="lab"><font color="Green" title="E61">A359</font></span>: 
<font color="Maroon" title="c28">k</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="c17">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">k</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E4:2_1_7_1_2_2_1_2_7_1_1_1"/>
<font color="Maroon" title="c28">k</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="c17">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_7_1_2_2_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E60">A358</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E61">A359</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E5:2_1_7_1_2_2_1_2_7_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E6:2_1_7_1_2_2_1_2_7_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/>
<a NAME="E7:2_1_7_1_2_2_1_2_7_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A360</font></span>: 
<font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<a NAME="E8:2_1_7_1_2_2_1_2_7_1_1_1"/>
<font color="Maroon" title="c28">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_2_2_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E61">A359</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E9:2_1_7_1_2_2_1_2_7_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_7_1_2_2_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E60">A358</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E10:2_1_7_1_2_2_1_2_7_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E61">A359</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:2_1_7_1_2_2_1_2_7_1_1_1"/>
<font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">k</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_2_2_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E62">A360</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_7_1_1_1"><span class="lab"><font color="Green" title="E59">A350</font></span></a>, <a class="ref" href="xboole_1.html#T18" title="XBOOLE_1:th.18">XBOOLE_1:18</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:2_1_7_1_2_2_1_2_7_1_1"/><span class="lab"><font color="Green" title="E59">A361</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E55">A346</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E7:2_1_7_1_2_2_1_2_7_1_1"/><span class="lab"><font color="Green" title="E60">A362</font></span>: 
<font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E55">A346</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E8:2_1_7_1_2_2_1_2_7_1_1"/><span class="lab"><font color="Green" title="E61">A363</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c26">i</font>,<font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E55">A346</font></span></a></span>;<br/><a NAME="E9:2_1_7_1_2_2_1_2_7_1_1"/><span class="lab"><font color="Green" title="E62">A364</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c27">j</font>,<font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">j</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E55">A346</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_7_1_1_2"><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="E1:2_1_7_1_2_2_1_2_7_1_1_2"/>
( <font color="Maroon" title="c26">i</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="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c26">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_7_1_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_7_1_1_2_1"/><span class="lab"><font color="Green" title="E63">A365</font></span>: 
<font color="Maroon" title="c26">i</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="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_2_7_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_7"><span class="lab"><font color="Green" title="E54">A345</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E57">A348</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_2_7_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">j</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="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:2_1_7_1_2_2_1_2_7_1_1_2_1"/>
<font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E59">A361</font></span></a>, <a class="txt" href="#E8:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E61">A363</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E58">A349</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_7_1_1_2_1"><span class="lab"><font color="Green" title="E63">A365</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_7_1_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_7_1_1_2_2"/>
<font color="Maroon" title="c26">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_2_7_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A366</font></span>: 
<font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E60">A362</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_2_7_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A367</font></span>: 
<font color="Maroon" title="c26">i</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="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_7_1_1_2_2_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="E1:2_1_7_1_2_2_1_2_7_1_1_2_2_1"/>
( <font color="Maroon" title="c27">j</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="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c27">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_7_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_7_1_1_2_2_1_1"/>
<font color="Maroon" title="c27">j</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="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_7_1_2_2_1_2_7_1_1_2_2_1_1"/>
<font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E56">A347</font></span></a>, <a class="txt" href="#E9:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E62">A364</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E58">A349</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_2_7_1_1_2_2"><span class="lab"><font color="Green" title="E64">A367</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_7_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_7_1_1_2_2_1_2"/>
<font color="Maroon" title="c27">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_2_7_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c27">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E57">A348</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_2_2_1_2_7_1_1_2_2_1_2"/>
<font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_7"><span class="lab"><font color="Green" title="E54">A345</font></span></a>, <a class="txt" href="#E8:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E61">A363</font></span></a>, <a class="txt" href="#E9:2_1_7_1_2_2_1_2_7_1_1"><span class="lab"><font color="Green" title="E62">A364</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_2_7_1_1_2_2"><span class="lab"><font color="Green" title="E63">A366</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_7_1_2"/><span class="lab"><font color="Green" title="E55">A368</font></span>: 
(  not <font color="Maroon" title="c24">i</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="c23">H</font> or  not <font color="Maroon" title="c25">j</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="c23">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_7_1_2_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="E1:2_1_7_1_2_2_1_2_7_1_2_1"/>
(  not <font color="Maroon" title="c24">i</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="c23">H</font> or  not <font color="Maroon" title="c25">j</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="c23">H</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_7_1_2"><span class="lab"><font color="Green" title="E55">A368</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_7_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_7_1_2_1_1"/>
not <font color="Maroon" title="c24">i</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="c23">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_2_7_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <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="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_2_2_1_2_7_1_2_1_1"/>
<font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_7_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_7_1_2_1_2"/>
not <font color="Maroon" title="c25">j</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="c23">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_2_7_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <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="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_2_2_1_2_7_1_2_1_2"/>
<font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E32:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E54">A369</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">H</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="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</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="E33:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E55">A370</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="c23">H</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( ( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">y</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1</span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> $1</span>)</span> ) &amp; ( $1 <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">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</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="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) );<br/><a NAME="E34:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E56">A371</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/> ex <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">v</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c24">k</font>,<font color="Olive" title="b1">v</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_2_2_1_2_8"/>
<font color="Maroon" title="c24">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c24">k</font>,<font color="Olive" title="b1">v</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_8_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="E1:2_1_7_1_2_2_1_2_8_1"/>
( <font color="Maroon" title="c24">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c24">k</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">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_8_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_8_1_1"/><span class="lab"><font color="Green" title="E57">A372</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c24">k</font>,<font color="Olive" title="b1">v</font>]</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c25">v</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">k</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c24">k</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  ;<br/><span class="kw">take </span>
<font color="Maroon" title="c25">v</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c24">k</font>,<font color="Maroon" title="c25">v</font>]</span><br/><span class="kw">thus </span><a NAME="E3:2_1_7_1_2_2_1_2_8_1_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c24">k</font>,<font color="Maroon" title="c25">v</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_8_1_1"><span class="lab"><font color="Green" title="E57">A372</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_8_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_8_1_2"/><span class="lab"><font color="Green" title="E57">A373</font></span>: 
<font color="Maroon" title="c24">k</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">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c24">k</font>,<font color="Olive" title="b1">v</font>]</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c25">v</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</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="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  ;<br/><span class="kw">take </span>
<font color="Maroon" title="c25">v</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c24">k</font>,<font color="Maroon" title="c25">v</font>]</span><br/><span class="kw">thus </span><a NAME="E3:2_1_7_1_2_2_1_2_8_1_2"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c24">k</font>,<font color="Maroon" title="c25">v</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_8_1_2"><span class="lab"><font color="Green" title="E57">A373</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c24">z</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <span class="kw"> such that </span><br/><a NAME="E36:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E57">A374</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">z</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</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="#E34:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E56">A371</font></span></a>);<br/></span><a NAME="E37:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E58">A375</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">y</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="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_2"><span class="lab"><font color="Green" title="E29">A180</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/><a NAME="E38:2_1_7_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A376</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">z</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="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E57">A374</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="E39:2_1_7_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A377</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">z</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="c23">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></span>;<br/><a NAME="E40:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E61">A378</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">z</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E59">A376</font></span></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></span>;<br/><span class="lab"><font color="Green" title="E62">A379</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2"><span class="lab"><font color="Green" title="E23">A174</font></span></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="txt" href="#E38:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E59">A376</font></span></a>, <a class="txt" href="#E40:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E61">A378</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E60">A377</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
;<br/><a NAME="E42:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E63">A380</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c24">z</font> holds <br/><font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">m</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">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="c24">z</font> implies <font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_2_2_1_2_10"/><span class="lab"><font color="Green" title="E64">A381</font></span>: 
<font color="Maroon" title="c25">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="c24">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font></span>)</span></span><br/>

<a NAME="E2:2_1_7_1_2_2_1_2_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A382</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c25">m</font>,<font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E37">A296</font></span></a>, <a class="txt" href="#E27:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E50">A336</font></span></a>, <a class="txt" href="#E36:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E57">A374</font></span></a>, <a class="txt" href="#E37:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E58">A375</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_10_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="E1:2_1_7_1_2_2_1_2_10_1"/>
( <font color="Maroon" title="c25">m</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">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c25">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_10_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_10_1_1"/>
<font color="Maroon" title="c25">m</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">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_7_1_2_2_1_2_10_1_1"/>
<font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E49">A335</font></span></a>, <a class="txt" href="#E36:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E57">A374</font></span></a>, <a class="txt" href="#E37:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E58">A375</font></span></a>, <a class="txt" href="#E38:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E59">A376</font></span></a>, <a class="txt" href="#E41:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E62">A379</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_10"><span class="lab"><font color="Green" title="E64">A381</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_2_10"><span class="lab"><font color="Green" title="E65">A382</font></span></a>, <a class="ref" href="finseq_1.html#T8" title="FINSEQ_1:th.8">FINSEQ_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_10_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_10_1_2"/><span class="lab"><font color="Green" title="E66">A383</font></span>: 
<font color="Maroon" title="c25">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font></span>)</span></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_2_10_1_2"/>
<font color="Maroon" title="c25">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E57">A374</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_10"><span class="lab"><font color="Green" title="E64">A381</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_2_10_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_10_1_2"><span class="lab"><font color="Green" title="E66">A383</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_2_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A384</font></span>: 
<font color="Maroon" title="c25">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="c18">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E50">A336</font></span></a>, <a class="txt" href="#E37:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E58">A375</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_2_10_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E37">A296</font></span></a>, <a class="txt" href="#E27:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E50">A336</font></span></a>, <a class="txt" href="#E36:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E57">A374</font></span></a>, <a class="txt" href="#E37:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E58">A375</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_10"><span class="lab"><font color="Green" title="E64">A381</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:2_1_7_1_2_2_1_2_10_1_2"/>
<font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c25">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E50">A336</font></span></a>, <a class="txt" href="#E36:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E57">A374</font></span></a>, <a class="txt" href="#E37:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E58">A375</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_10"><span class="lab"><font color="Green" title="E64">A381</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_2_10_1_2"><span class="lab"><font color="Green" title="E67">A384</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">reconsider </span><font color="Maroon" title="c25">H</font> = <font color="Maroon" title="c23">H</font> as   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E53">A344</font></span></a>, <a class="ref" href="prob_2.html#D3" title="PROB_2:def.3">PROB_2:def 3</a></span>;<br/><a NAME="E44:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E64">A385</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</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="c19">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_2"><span class="lab"><font color="Green" title="E29">A180</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/><a NAME="E45:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E65">A386</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">z</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">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E57">A374</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="E46:2_1_7_1_2_2_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">z</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></span>;<br/><a NAME="E47:2_1_7_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A387</font></span>: 
<font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</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="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E57">A374</font></span></a>, <a class="txt" href="#E45:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E65">A386</font></span></a></span>;<br/><a NAME="E48:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E67">A388</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c24">z</font> holds <br/><font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">y</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">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="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">k</font> &amp; <font color="Maroon" title="c26">k</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="c24">z</font> implies <font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">y</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_7_1_2_2_1_2_11"/><span class="lab"><font color="Green" title="E68">A389</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">k</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_7_1_2_2_1_2_11"/><span class="lab"><font color="Green" title="E69">A390</font></span>: 
<font color="Maroon" title="c26">k</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="c24">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">y</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_11_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="E1:2_1_7_1_2_2_1_2_11_1"/>
( <font color="Maroon" title="c26">k</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="c24">z</font> or <font color="Maroon" title="c26">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">z</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_11_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_11_1_1"/>
<font color="Maroon" title="c26">k</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="c24">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">y</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_7_1_2_2_1_2_11_1_1"/>
<font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">y</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E65">A386</font></span></a>, <a class="txt" href="#E47:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E66">A387</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_11_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_11_1_2"/>
<font color="Maroon" title="c26">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">y</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_2_11_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_7_1_2_2_1_2_11"><span class="lab"><font color="Green" title="E69">A390</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_2_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A391</font></span>: 
<font color="Maroon" title="c26">k</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="c19">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E65">A386</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_2_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A392</font></span>: 
<font color="Maroon" title="c26">k</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">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_11"><span class="lab"><font color="Green" title="E68">A389</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A393</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c19">y</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font> = 
<font color="Maroon" title="c19">y</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</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/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_7_1_2"><span class="lab"><font color="Green" title="E30">A181</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_2_11_1_2"><span class="lab"><font color="Green" title="E71">A392</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">b</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_2"><span class="lab"><font color="Green" title="E29">A180</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_2_11_1_2"><span class="lab"><font color="Green" title="E71">A392</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
;<br/><a NAME="E6:2_1_7_1_2_2_1_2_11_1_2"/><span class="lab"><font color="Green" title="E73">A394</font></span>: 
<font color="Maroon" title="c26">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_11"><span class="lab"><font color="Green" title="E68">A389</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_2_11"><span class="lab"><font color="Green" title="E69">A390</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E7:2_1_7_1_2_2_1_2_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A395</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c25">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="txt" href="#E44:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E64">A385</font></span></a>, <a class="txt" href="#E45:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E65">A386</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><a NAME="E8:2_1_7_1_2_2_1_2_11_1_2"/>
<font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="txt" href="#E44:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E64">A385</font></span></a>, <a class="txt" href="#E45:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E65">A386</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_2_11_1_2"><span class="lab"><font color="Green" title="E70">A391</font></span></a>, <a class="txt" href="#E6:2_1_7_1_2_2_1_2_11_1_2"><span class="lab"><font color="Green" title="E73">A394</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_7_1_2_2_1_2_11_1_2"/>
<font color="Maroon" title="c24">z</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">y</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E57">A374</font></span></a>, <a class="txt" href="#E45:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E65">A386</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_2_11_1_2"><span class="lab"><font color="Green" title="E70">A391</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_2_11_1_2"><span class="lab"><font color="Green" title="E72">A393</font></span></a>, <a class="txt" href="#E6:2_1_7_1_2_2_1_2_11_1_2"><span class="lab"><font color="Green" title="E73">A394</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_2_11_1_2"><span class="lab"><font color="Green" title="E74">A395</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">y</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">y</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E57">A374</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="E50:2_1_7_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A396</font></span>: 
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E67">A388</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/><a NAME="E51:2_1_7_1_2_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A397</font></span>: <font color="Maroon" title="c8">x</font> = 
<font color="Maroon" title="c8">x</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<font color="Maroon" title="c16">x1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E9:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E38">A297</font></span></a>, <a class="ref" href="finseq_5.html#T11" title="FINSEQ_5:th.11">FINSEQ_5:11</a></span>
<br/>.= 
<font color="Maroon" title="c16">x1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E9:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E38">A297</font></span></a></span>
;<br/><a NAME="E53:2_1_7_1_2_2_1_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_1_7_1_2_2_1_2_14"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:2_1_7_1_2_2_1_2_14"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2"><span class="lab"><font color="Green" title="E21">A172</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E54:2_1_7_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></span>;<br/><a NAME="E55:2_1_7_1_2_2_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">b</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="c18">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/><a NAME="E56:2_1_7_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A398</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E57:2_1_7_1_2_2_1_2"/><span class="lab"><font color="Green" title="E71">A399</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</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="c25">H</font> holds <br/><font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">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="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">k</font> &amp; <font color="Maroon" title="c26">k</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="c25">H</font> implies <font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_7_1_2_2_1_2_15"/><span class="lab"><font color="Green" title="E72">A400</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">k</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_7_1_2_2_1_2_15"/><span class="lab"><font color="Green" title="E73">A401</font></span>: 
<font color="Maroon" title="c26">k</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="c25">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font></span><br/>

<a NAME="E3:2_1_7_1_2_2_1_2_15"/>
<font color="Maroon" title="c26">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E55">A370</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_15"><span class="lab"><font color="Green" title="E72">A400</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_2_15"><span class="lab"><font color="Green" title="E73">A401</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E4:2_1_7_1_2_2_1_2_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A402</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c26">k</font>,<font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_15_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="E1:2_1_7_1_2_2_1_2_15_1"/>
( <font color="Maroon" title="c26">k</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="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c26">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_15_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_15_1_1"/>
<font color="Maroon" title="c26">k</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="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_7_1_2_2_1_2_15_1_1"/>
<font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_2_15"><span class="lab"><font color="Green" title="E74">A402</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_15_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_15_1_2"/>
<font color="Maroon" title="c26">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_2_15_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A403</font></span>: 
<font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E54">A369</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_2_15"><span class="lab"><font color="Green" title="E73">A401</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_2_15_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c26">k</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="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_2_15_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c26">k</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="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_15"><span class="lab"><font color="Green" title="E72">A400</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">hence </span><a NAME="E5:2_1_7_1_2_2_1_2_15_1_2"/>
<font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2_2_1_2_15"><span class="lab"><font color="Green" title="E74">A402</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_2_15_1_2"><span class="lab"><font color="Green" title="E75">A403</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">H</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
;<br/><a NAME="E59:2_1_7_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E71">A399</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A404</font></span>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">H</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></span>
;<br/><a NAME="E61:2_1_7_1_2_2_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2"><span class="lab"><font color="Green" title="E23">A174</font></span></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a>, <a class="ref" href="finseq_5.html#T11" title="FINSEQ_5:th.11">FINSEQ_5:11</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c12">F1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> = 
<font color="Maroon" title="c6">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<font color="Maroon" title="c6">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></span>
;<br/><span class="kw">then </span> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A405</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</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="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E18:2_1"><span class="lab"><font color="Green" title="E17">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</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="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">H</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E72">A404</font></span></a>, <a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>
;<br/><a NAME="E65:2_1_7_1_2_2_1_2"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c25">H</font> holds <br/>for <font color="Olive" title="b2">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_21"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">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="c25">H</font> implies for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">m</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_7_1_2_2_1_2_21"/><span class="lab"><font color="Green" title="E74">A406</font></span>: 
<font color="Maroon" title="c26">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="c25">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">m</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">m</font></span><br/>

<a NAME="E2:2_1_7_1_2_2_1_2_21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A407</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">m</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a></span>;<br/>
<a NAME="E3:2_1_7_1_2_2_1_2_21"/><span class="lab"><font color="Green" title="E76">A408</font></span>: 
<font color="Maroon" title="c26">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E74">A406</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c27">v</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">m</font> implies <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c27">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:2_1_7_1_2_2_1_2_21"/><span class="lab"><font color="Green" title="E77">A409</font></span>: 
<font color="Maroon" title="c27">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c27">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">m</font></span><br/>

<a NAME="E5:2_1_7_1_2_2_1_2_21"/><span class="lab"><font color="Green" title="E78">A410</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c26">m</font>,<font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">m</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E37">A296</font></span></a>, <a class="txt" href="#E27:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E50">A336</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E74">A406</font></span></a></span>;<br/>
<a NAME="E6:2_1_7_1_2_2_1_2_21"/><span class="lab"><font color="Green" title="E79">A411</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E74">A406</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_21_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="E1:2_1_7_1_2_2_1_2_21_1"/>
( <font color="Maroon" title="c26">m</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="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c26">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_21_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_21_1_1"/><span class="lab"><font color="Green" title="E80">A412</font></span>: 
<font color="Maroon" title="c26">m</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="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c27">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">m</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_2_21_1_1"/>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_7_1_2"><span class="lab"><font color="Green" title="E23">A174</font></span></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_7_1_2_2_1_2_21_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c27">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E26:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E49">A335</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E75">A407</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E78">A410</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E77">A409</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2_2_1_2_21_1_1"><span class="lab"><font color="Green" title="E80">A412</font></span></a>, <a class="ref" href="finseq_1.html#T8" title="FINSEQ_1:th.8">FINSEQ_1:8</a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_7_1_2_2_1_2_21_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_7_1_2_2_1_2_21_1_2"/>
<font color="Maroon" title="c26">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c27">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">m</font></span><br/><div class="add"><a NAME="E2:2_1_7_1_2_2_1_2_21_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A413</font></span>: 
<font color="Maroon" title="c26">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E76">A408</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_7_1_2_2_1_2_21_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A414</font></span>: 
<font color="Maroon" title="c26">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="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_7_1_2_2_1_2_21_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c26">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="c17">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E79">A411</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E5:2_1_7_1_2_2_1_2_21_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c26">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="c17">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="E6:2_1_7_1_2_2_1_2_21_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A415</font></span>: 
<font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E7:2_1_7_1_2_2_1_2_21_1_2"/>
<font color="Maroon" title="c25">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E74">A406</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E8:2_1_7_1_2_2_1_2_21_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A416</font></span>: 
<font color="Maroon" title="c27">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E73">A405</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E77">A409</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E9:2_1_7_1_2_2_1_2_21_1_2"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1"><span class="lab"><font color="Green" title="E17">A16</font></span></a>, <a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E10:2_1_7_1_2_2_1_2_21_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c27">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E75">A407</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E77">A409</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_2_21_1_2"><span class="lab"><font color="Green" title="E80">A413</font></span></a>, <a class="txt" href="#E6:2_1_7_1_2_2_1_2_21_1_2"><span class="lab"><font color="Green" title="E82">A415</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E11:2_1_7_1_2_2_1_2_21_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c27">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">F1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_7_1_2_2_1_2_21_1_2"><span class="lab"><font color="Green" title="E83">A416</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="E12:2_1_7_1_2_2_1_2_21_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A417</font></span>: 
<font color="Maroon" title="c27">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E13:2_1_7_1_2_2_1_2_21_1_2"/>
<font color="Maroon" title="c26">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="c17">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E79">A411</font></span></a>, <a class="txt" href="#E3:2_1_7_1_2_2_1_2_21_1_2"><span class="lab"><font color="Green" title="E81">A414</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E14:2_1_7_1_2_2_1_2_21_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c26">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="c17">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="E15:2_1_7_1_2_2_1_2_21_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c27">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_7_1_2"><span class="lab"><font color="Green" title="E26">A177</font></span></a>, <a class="txt" href="#E27:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E50">A336</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E75">A407</font></span></a>, <a class="txt" href="#E5:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E78">A410</font></span></a>, <a class="txt" href="#E4:2_1_7_1_2_2_1_2_21"><span class="lab"><font color="Green" title="E77">A409</font></span></a>, <a class="txt" href="#E2:2_1_7_1_2_2_1_2_21_1_2"><span class="lab"><font color="Green" title="E80">A413</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E16:2_1_7_1_2_2_1_2_21_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c27">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c26">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_7_1_2_2_1_2_21_1_2"><span class="lab"><font color="Green" title="E84">A417</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E66:2_1_7_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A418</font></span>: 
<font color="Maroon" title="c25">H</font>,<font color="Maroon" title="c21">c</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E37">A296</font></span></a>, <a class="txt" href="#E27:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E50">A336</font></span></a>, <a class="txt" href="#E64:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E73">A405</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>;<br/><a NAME="E67:2_1_7_1_2_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E68:2_1_7_1_2_2_1_2"/><span class="kw">then </span>
1 <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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E69:2_1_7_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c21">c</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_7_1_2"><span class="lab"><font color="Green" title="E27">A178</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E37">A296</font></span></a>, <a class="txt" href="#E56:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E70">A398</font></span></a></span>;<br/><span class="kw">then </span> <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> = 
 <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c24">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E30:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A343</font></span></a>, <a class="txt" href="#E21:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E44">A303</font></span></a>, <a class="txt" href="#E36:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E57">A374</font></span></a>, <a class="txt" href="#E37:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E58">A375</font></span></a>, <a class="txt" href="#E42:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E63">A380</font></span></a>, <a class="txt" href="#E66:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E74">A418</font></span></a>, <a class="ref" href="mesfunc3.html#D2" title="MESFUNC3:def.2">MESFUNC3:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c19">y</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c25">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">z</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E62">A379</font></span></a>, <a class="txt" href="#E23:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E46">A310</font></span></a>, <a class="txt" href="#E50:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="ref" href="extreal1.html#T43" title="EXTREAL1:th.43">EXTREAL1:43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c25">H</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_7_1_2"><span class="lab"><font color="Green" title="E31">A182</font></span></a>, <a class="ref" href="extreal1.html#T41" title="EXTREAL1:th.41">EXTREAL1:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c16">x1</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E20:2_1"><span class="lab"><font color="Green" title="E18">A23</font></span></a>, <a class="txt" href="#E13:2_1"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="txt" href="#E23:2_1"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="txt" href="#E16:2_1"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E1:2_1_7_1_2"><span class="lab"><font color="Green" title="E21">A172</font></span></a>, <a class="txt" href="#E15:2_1_7_1_2"><span class="lab"><font color="Green" title="E32">A183</font></span></a>, <a class="txt" href="#E7:2_1_7_1_2"><span class="lab"><font color="Green" title="E25">A176</font></span></a>, <a class="txt" href="#E41:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E62">A379</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c16">x1</font></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#T41" title="EXTREAL1:th.41">EXTREAL1:41</a></span>
<br/>.= 
 <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E46">A310</font></span></a>, <a class="txt" href="#E24:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E47">A324</font></span></a>, <a class="txt" href="#E52:2_1_7_1_2_2_1_2"><span class="lab"><font color="Green" title="E69">A397</font></span></a>, <a class="ref" href="extreal1.html#T43" title="EXTREAL1:th.43">EXTREAL1:43</a></span>
;<br/><span class="kw">hence </span><a NAME="E71:2_1_7_1_2_2_1_2"/>
 <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><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="E17:2_1_7_1_2"/>
 <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><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="E25:2_1"/>
 <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c8">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:2"/><span class="lab"><font color="Green" title="E4">A419</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <font color="Maroon" title="c4">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b4">x</font> ) &amp; <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">a</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">x</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="Olive" title="b1">F</font> &amp; ( for <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">i</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">x</font> holds <br/><font color="Olive" title="b3">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b4">i</font></span>)</span> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Olive" title="b3">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">F</font> be   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <font color="Maroon" title="c4">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">x</font> ) &amp; <font color="Maroon" title="c5">F</font>,<font color="Olive" title="b1">a</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">x</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="c5">F</font> &amp; ( for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">i</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">x</font> holds <br/><font color="Olive" title="b2">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b3">i</font></span>)</span> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Olive" title="b2">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">x</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( 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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> ) &amp; <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">a</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">x</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="c5">F</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</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">x</font> holds <br/><font color="Maroon" title="c7">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c7">x</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_2"/>
<font color="Maroon" title="c4">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">and </span><br/><a NAME="E2:2_2"/><span class="lab"><font color="Green" title="E5">A420</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E3:2_2"/>
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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">and </span><br/><a NAME="E4:2_2"/><span class="lab"><font color="Green" title="E6">A421</font></span>: 
<font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">a</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E5:2_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">x</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="c5">F</font>
 <span class="kw">and </span><br/><a NAME="E6:2_2"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</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">x</font> holds <br/><font color="Maroon" title="c7">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:2_2"/><span class="lab"><font color="Green" title="E7">A422</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c7">x</font></span><br/>

<a NAME="E8:2_2"/>
 <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="c5">F</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="#E7:2_2"><span class="lab"><font color="Green" title="E7">A422</font></span></a></span>;<br/>
<a NAME="E9:2_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font> <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="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E10:2_2"/><span class="kw">then </span>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</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="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:2_2"/>
 <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_2"><span class="lab"><font color="Green" title="E5">A420</font></span></a>, <a class="txt" href="#E4:2_2"><span class="lab"><font color="Green" title="E6">A421</font></span></a>, <a class="ref" href="mesfunc3.html#D1" title="MESFUNC3:def.1">MESFUNC3:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:2"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E4">A419</font></span></a>, <a class="txt" href="#E1:2"><span class="lab"><font color="Green" title="E3">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:2"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b3">F</font> being  <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <font color="Olive" title="b2">f</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b6">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">f</font> holds <br/> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b6">x</font> ) &amp; <font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">a</font> <a href="mesfunc3.html#R1" title="MESFUNC3:pred.1">are_Re-presentation_of</a> <font color="Olive" title="b2">f</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b5">x</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="Olive" title="b3">F</font> &amp; ( for <font color="Olive" title="b6">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b6">i</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="b5">x</font> holds <br/><font color="Olive" title="b5">x</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b6">i</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">F</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b6">i</font></span>)</span> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> holds <br/> <a href="mesfunc3.html#K1" title="MESFUNC3:func.1">integral</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="extreal1.html#K4" title="EXTREAL1:func.4">Sum</a> <font color="Olive" title="b5">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
