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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b4">F</font> being  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <font color="Olive" title="b3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Olive" title="b4">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Olive" title="b4">F</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; ( for <font color="Olive" title="b5">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b4">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">E</font> ) &amp; ( for <font color="Olive" title="b5">n</font>, <font color="Olive" title="b6">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">m</font> holds <br/>for <font color="Olive" title="b7">x</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="b7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">E</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b6">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b7">x</font> ) &amp; ( for <font color="Olive" title="b5">x</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="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">E</font> holds <br/><font color="Olive" title="b4">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b5">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> ) holds <br/> ex <font color="Olive" title="b5">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b6">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b5">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Olive" title="b2">M</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b6">n</font></span>)</span> ) &amp; <font color="Olive" title="b5">I</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Olive" title="b2">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Olive" title="b4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b5">I</font> )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b3">F</font> being  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <font color="Olive" title="b2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Olive" title="b3">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Olive" title="b3">F</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; ( for <font color="Olive" title="b4">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b3">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b2">E</font> ) &amp; ( for <font color="Olive" title="b4">n</font>, <font color="Olive" title="b5">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">m</font> holds <br/>for <font color="Olive" title="b6">x</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="b6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">E</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b6">x</font> ) &amp; ( for <font color="Olive" title="b4">x</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="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">E</font> holds <br/><font color="Olive" title="b3">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b4">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> ) holds <br/> ex <font color="Olive" title="b4">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b5">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b4">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Olive" title="b1">M</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font></span>)</span> ) &amp; <font color="Olive" title="b4">I</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Olive" title="b1">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Olive" title="b3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b4">I</font> )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">F</font> being  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Olive" title="b2">F</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> ) &amp; ( for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">m</font> holds <br/>for <font color="Olive" title="b5">x</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="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b5">x</font> ) &amp; ( for <font color="Olive" title="b3">x</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="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> holds <br/><font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> ) holds <br/> ex <font color="Olive" title="b3">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b4">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b3">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font></span>)</span> ) &amp; <font color="Olive" title="b3">I</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b3">I</font> )</span><br/><b>let </b><font color="Maroon" title="c4">E</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st <font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Olive" title="b1">F</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> ) &amp; ( for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/>for <font color="Olive" title="b4">x</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="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b4">x</font> ) &amp; ( for <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="c1">X</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> ) holds <br/> ex <font color="Olive" title="b2">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b2">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font></span>)</span> ) &amp; <font color="Olive" title="b2">I</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b2">I</font> )</span><br/><b>let </b><font color="Maroon" title="c5">F</font> be   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c5">F</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; ( 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" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> ) &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/>for <font color="Olive" title="b3">x</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="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b3">x</font> <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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b3">x</font> ) &amp; ( for <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="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> ) implies  ex <font color="Olive" title="b1">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) &amp; <font color="Olive" title="b1">I</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b1">I</font> ) )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green" title="E52">A0</font></i>: 
<font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>and </b><br/><a NAME="E3:68"/><i><font color="Green" title="E54">S1</font></i>: 
<font color="Maroon" title="c5">F</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>
 <b>and </b><br/><a NAME="E4:68"/><i><font color="Green" title="E55">A2</font></i>: 
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" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font>
 <b>and </b><br/><a NAME="E5:68"/><i><font color="Green" title="E56">A3</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/>for <font color="Olive" title="b3">x</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="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b3">x</font> <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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b3">x</font>
 <b>and </b><br/><a NAME="E6:68"/><i><font color="Green" title="E57">A4</font></i>: 
for <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="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) &amp; <font color="Olive" title="b1">I</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b1">I</font> )</span><br/>

<a NAME="E7:68"/><i><font color="Green" title="E58">E0</font></i>: 
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" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a></span><br/>

<a NAME="E1:68_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font></span>)</span> holds <br/> <a href="numbers.html#K6" title="NUMBERS:func.6">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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font></span>)</span> implies  <a href="numbers.html#K6" title="NUMBERS:func.6">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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/>

<b>assume </b><a NAME="E1:68_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/>

<a NAME="E2:68_1_1"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font>
 
<b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="txt" href="#E3:68"><i><font color="Green" title="E54">S1</font></i></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/>
<a NAME="E3:68_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">x</font> <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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">x</font>
 
<b>by </b><i><a class="txt" href="#E5:68"><i><font color="Green" title="E56">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:68_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="#E2:68"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:68_1"/>
<font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T71" title="SUPINF_2:th.71">SUPINF_2:71</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:68"/><i><font color="Green" title="E59">E13</font></i>: 
 <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font>
 
<b>by </b><i><a class="txt" href="#E6:68"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="txt" href="#E3:68"><i><font color="Green" title="E54">S1</font></i></a>, <a class="txt" href="#E4:68"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="mesfunc8.html#T25" title="MESFUNC8:th.25">MESFUNC8:25</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_2"/>
(  ex <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" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or 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" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1"><b>suppose </b></a><a NAME="E1:68_2_1"/>
 ex <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" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) &amp; <font color="Olive" title="b1">I</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b1">I</font> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">N</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:68_2_1"/><i><font color="Green" title="E60">B1</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E4:68_2_1"/><i><font color="Green" title="E61">B0</font></i>: 
<font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="txt" href="#E3:68"><i><font color="Green" title="E54">S1</font></i></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">EE</font> = <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="#E4:68"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="mesfunc1.html#T37" title="MESFUNC1:th.37">MESFUNC1:37</a></i>;<br/><a NAME="E6:68_2_1"/><i><font color="Green" title="E62">B2</font></i>: 
<font color="Maroon" title="c7">EE</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:68_2_1"/><b>then </b>
<font color="Maroon" title="c7">EE</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="txt" href="#E3:68"><i><font color="Green" title="E54">S1</font></i></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/><a NAME="E8:68_2_1"/><b>then </b><i><font color="Green" title="E63">B3</font></i>: 
<font color="Maroon" title="c7">EE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">EE</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E9:68_2_1"/><i><font color="Green" title="E64">B4</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c7">EE</font>
 <b>by </b><i><a class="txt" href="#E4:68"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="#E6:68_2_1"><i><font color="Green" title="E62">B2</font></i></a>, <a class="ref" href="mesfunc1.html#T34" title="MESFUNC1:th.34">MESFUNC1:34</a></i>;<br/><a NAME="E10:68_2_1"/>
<font color="Maroon" title="c7">EE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">EE</font>
 <b>by </b><i><a class="txt" href="#E8:68_2_1"><i><font color="Green" title="E63">B3</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><a NAME="E11:68_2_1"/><b>then </b><i><font color="Green" title="E65">B5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">EE</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c7">EE</font>
 <b>by </b><i><a class="txt" href="#E9:68_2_1"><i><font color="Green" title="E64">B4</font></i></a>, <a class="ref" href="mesfunc5.html#T48" title="MESFUNC5:th.48">MESFUNC5:48</a></i>;<br/><a NAME="E12:68_2_1"/><i><font color="Green" title="E66">B9</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="#E7:68"><i><font color="Green" title="E58">E0</font></i></a></i>;<br/><a NAME="E13:68_2_1"/>
<font color="Maroon" title="c7">EE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">EE</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">EE</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">EE</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )</span><br/><b>assume </b><a NAME="E1:68_2_1_1_1"/><i><font color="Green" title="E67">C1</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">EE</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">EE</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><br/><a NAME="E2:68_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:68_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 <b>by </b><i><a class="ref" href="mesfunc1.html#D16" title="MESFUNC1:def.16">MESFUNC1:def 16</a></i>;<br/><a NAME="E4:68_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">EE</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="#E8:68_2_1"><i><font color="Green" title="E63">B3</font></i></a>, <a class="txt" href="#E1:68_2_1_1_1"><i><font color="Green" title="E67">C1</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E5:68_2_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">EE</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="#E1:68_2_1_1_1"><i><font color="Green" title="E67">C1</font></i></a>, <a class="txt" href="#E8:68_2_1"><i><font color="Green" title="E63">B3</font></i></a>, <a class="ref" href="mesfunc1.html#D16" title="MESFUNC1:def.16">MESFUNC1:def 16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E2:68_2_1_1"/><b>then </b><i><font color="Green" title="E67">C2</font></i>: 
<font color="Maroon" title="c7">EE</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">EE</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:68_2_1_1"/>
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="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">EE</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">EE</font>
 
<b>by </b><i><a class="txt" href="#E8:68_2_1"><i><font color="Green" title="E63">B3</font></i></a>, <a class="ref" href="mesfunc1.html#D16" title="MESFUNC1:def.16">MESFUNC1:def 16</a></i>;<br/>
<a NAME="E4:68_2_1_1"/><b>then </b>
 <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">EE</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">EE</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:68_2_1_1"/>
<font color="Maroon" title="c7">EE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">EE</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="#E2:68_2_1_1"><i><font color="Green" title="E67">C2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E14:68_2_1"/><b>then </b>
<font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">EE</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">EE</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E3:68_2_1"><i><font color="Green" title="E60">B1</font></i></a></i>;<br/><a NAME="E15:68_2_1"/><b>then </b><i><font color="Green" title="E67">C0</font></i>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">EE</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="#E8:68_2_1"><i><font color="Green" title="E63">B3</font></i></a>, <a class="txt" href="#E11:68_2_1"><i><font color="Green" title="E65">B5</font></i></a>, <a class="txt" href="#E12:68_2_1"><i><font color="Green" title="E66">B9</font></i></a>, <a class="ref" href="mesfunc9.html#T13" target="_self" title="MESFUNC9:th.13">Lem13</a>, <a class="ref" href="mesfunc5.html#T21" title="MESFUNC5:th.21">MESFUNC5:21</a></i>;<br/><a NAME="E16:68_2_1"/>
<font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font>
 <b>by </b><i><a class="txt" href="#E4:68"><i><font color="Green" title="E55">A2</font></i></a></i>;<br/><a NAME="E17:68_2_1"/><b>then </b>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">EE</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:68_2_1"><i><font color="Green" title="E61">B0</font></i></a>, <a class="txt" href="#E7:68"><i><font color="Green" title="E58">E0</font></i></a>, <a class="txt" href="#E6:68_2_1"><i><font color="Green" title="E62">B2</font></i></a>, <a class="ref" href="mesfunc5.html#T99" title="MESFUNC5:th.99">MESFUNC5:99</a></i>;<br/><a NAME="E18:68_2_1"/><b>then </b>
 <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E15:68_2_1"><i><font color="Green" title="E67">C0</font></i></a>, <a class="txt" href="#E4:68_2_1"><i><font color="Green" title="E61">B0</font></i></a>, <a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></i>;<br/><a NAME="E19:68_2_1"/><b>then </b><i><font color="Green" title="E68">D1</font></i>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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> ) <b>-&gt; </b>   <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>  =  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c8">I</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> <b> such that </b><br/><a NAME="E21:68_2_1"/><i><font color="Green" title="E69">D2</font></i>: 
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon" title="c8">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</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>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i><b>reconsider </b><font color="Maroon" title="c9">I</font> = <font color="Maroon" title="c8">I</font> as   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> ;<br/><b>take </b>
<font color="Maroon" title="c9">I</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) &amp; <font color="Maroon" title="c9">I</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c9">I</font> )</span><br/><a NAME="E23:68_2_1"/><i><font color="Green" title="E70">D3</font></i>: 
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" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">n1</font> = <font color="Maroon" title="c10">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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:68_2_1_2"/>
<font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">n1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E21:68_2_1"><i><font color="Green" title="E69">D2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:68_2_1_2"/>
<font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E24:68_2_1"/><i><font color="Green" title="E71">E5</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">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="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">n</font>, <font color="Maroon" title="c11">m</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> implies <font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">n</font> )</span><br/>

<b>assume </b><a NAME="E1:68_2_1_3"/><i><font color="Green" title="E72">E2</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">n</font></span><br/>

<a NAME="E2:68_2_1_3"/><i><font color="Green" title="E73">E51</font></i>: 
( <font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">m</font></span>)</span> &amp; <font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">m</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="#E7:68"><i><font color="Green" title="E58">E0</font></i></a>, <a class="txt" href="#E3:68"><i><font color="Green" title="E54">S1</font></i></a>, <a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/>
<a NAME="E3:68_2_1_3"/><i><font color="Green" title="E74">E52</font></i>: 
( <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> &amp; <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">m</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> )
 
<b>by </b><i><a class="txt" href="#E4:68"><i><font color="Green" title="E55">A2</font></i></a></i>;<br/>
<a NAME="E4:68_2_1_3"/>
for <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="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font> <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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="txt" href="#E1:68_2_1_3"><i><font color="Green" title="E72">E2</font></i></a>, <a class="txt" href="#E5:68"><i><font color="Green" title="E56">A3</font></i></a></i>;<br/>
<a NAME="E5:68_2_1_3"/><b>then </b>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:68_2_1_3"><i><font color="Green" title="E73">E51</font></i></a>, <a class="txt" href="#E3:68_2_1_3"><i><font color="Green" title="E74">E52</font></i></a>, <a class="ref" href="mesfunc9.html#T15" target="_self" title="MESFUNC9:th.15">Prop3</a></i>;<br/>
<a NAME="E6:68_2_1_3"/><b>then </b>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:68_2_1_3"><i><font color="Green" title="E73">E51</font></i></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/>
<a NAME="E7:68_2_1_3"/><b>then </b>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:68_2_1_3"><i><font color="Green" title="E73">E51</font></i></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/>
<a NAME="E8:68_2_1_3"/><b>then </b>
<font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E21:68_2_1"><i><font color="Green" title="E69">D2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:68_2_1_3"/>
<font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">n</font>
 <b>by </b><i><a class="txt" href="#E21:68_2_1"><i><font color="Green" title="E69">D2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E25:68_2_1"/><b>then </b><i><font color="Green" title="E72">E4</font></i>: 
<font color="Maroon" title="c9">I</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>
 <b>by </b><i><a class="ref" href="rinfsup2.html#T7" title="RINFSUP2:th.7">RINFSUP2:7</a></i>;<br/><a NAME="E26:68_2_1"/><b>then </b><i><font color="Green" title="E73">E6</font></i>: 
( <font color="Maroon" title="c9">I</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K1" title="RINFSUP2:func.1">sup</a> <font color="Maroon" title="c9">I</font> )
 <b>by </b><i><a class="ref" href="rinfsup2.html#T37" title="RINFSUP2:th.37">RINFSUP2:37</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">N1</font> = <font color="Maroon" title="c6">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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E28:68_2_1"/><i><font color="Green" title="E74">E7</font></i>: 
( <font color="Maroon" title="c9">I</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c10">N1</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> &amp;  <a href="rinfsup2.html#K1" title="RINFSUP2:func.1">sup</a> <font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K1" title="RINFSUP2:func.1">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c10">N1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E25:68_2_1"><i><font color="Green" title="E72">E4</font></i></a>, <a class="ref" href="rinfsup2.html#T26" title="RINFSUP2:th.26">RINFSUP2:26</a></i>;<br/><a NAME="E29:68_2_1"/>
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>  holds   <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c10">N1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">k</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c10">N1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">k</font></span><br/>

<a NAME="E1:68_2_1_4"/>
<font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">N1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">N1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E24:68_2_1"><i><font color="Green" title="E71">E5</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E2:68_2_1_4"/><b>then </b>
<font color="Maroon" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">N1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c10">N1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:68_2_1_4"/>
 <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c10">N1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">k</font>
 <b>by </b><i><a class="txt" href="#E19:68_2_1"><i><font color="Green" title="E68">D1</font></i></a>, <a class="txt" href="#E23:68_2_1"><i><font color="Green" title="E70">D3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E30:68_2_1"/><b>then </b>
<font color="Maroon" title="c9">I</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c10">N1</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>
 <b>by </b><i><a class="ref" href="rinfsup2.html#T32" title="RINFSUP2:th.32">RINFSUP2:32</a></i>;<br/><a NAME="E31:68_2_1"/><b>then </b>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c10">N1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="ref" href="mesfunc9.html#T7" target="_self" title="MESFUNC9:th.7">LIM</a></i>;<br/><a NAME="E32:68_2_1"/><b>then </b><i><font color="Green" title="E75">E9</font></i>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="#E26:68_2_1"><i><font color="Green" title="E73">E6</font></i></a>, <a class="txt" href="#E28:68_2_1"><i><font color="Green" title="E74">E7</font></i></a>, <a class="ref" href="rinfsup2.html#T37" title="RINFSUP2:th.37">RINFSUP2:37</a></i>;<br/><a NAME="E33:68_2_1"/><i><font color="Green" title="E76">E12</font></i>: 
<font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="ref" href="mesfunc8.html#D10" title="MESFUNC8:def.10">MESFUNC8:def 10</a></i>;<br/><a NAME="E34:68_2_1"/>
(  <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p3">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:68_2_1_5"/><i><font color="Green" title="E77">E10</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc8.html#D10" title="MESFUNC8:def.10">MESFUNC8:def 10</a></i>;<br/>
<a NAME="E2:68_2_1_5"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> implies <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>assume </b><a NAME="E1:68_2_1_5_1"/><i><font color="Green" title="E78">E30</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<b>then reconsider </b><font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:68_2_1_5_1"/><i><font color="Green" title="E79">E32</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 
<b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="txt" href="#E1:68_2_1_5"><i><font color="Green" title="E77">E10</font></i></a>, <a class="txt" href="#E1:68_2_1_5_1"><i><font color="Green" title="E78">E30</font></i></a>, <a class="txt" href="#E6:68"><i><font color="Green" title="E57">A4</font></i></a></i>;<br/>
<a NAME="E4:68_2_1_5_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E1:68_2_1_5_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">x1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="txt" href="#E5:68"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="#E1:68_2_1_5"><i><font color="Green" title="E77">E10</font></i></a>, <a class="txt" href="#E1:68_2_1_5_1"><i><font color="Green" title="E78">E30</font></i></a>, <a class="txt" href="#E2:68"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/>
<b>hence </b><a NAME="E2:68_2_1_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:68_2_1_5_1"/><b>then </b>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E3:68_2_1_5_1"><i><font color="Green" title="E79">E32</font></i></a>, <a class="ref" href="mesfunc9.html#T10" target="_self" title="MESFUNC9:th.10">LIM3a</a></i>;<br/>
<b>hence </b><a NAME="E6:68_2_1_5_1"/>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:68_2_1_5_1"><i><font color="Green" title="E78">E30</font></i></a>, <a class="ref" href="mesfunc8.html#D10" title="MESFUNC8:def.10">MESFUNC8:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:68_2_1_5"/>
 <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T71" title="SUPINF_2:th.71">SUPINF_2:71</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p3">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E4:68_2_1_5"/><i><font color="Green" title="E78">E14</font></i>: 
<font color="Maroon" title="c7">EE</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">EE</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:68_2_1_5_2"/><i><font color="Green" title="E79">E40</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">EE</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span></span><br/>

<a NAME="E2:68_2_1_5_2"/><b>then </b><i><font color="Green" title="E80">E41</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="#E1:68_2_1_5_2"><i><font color="Green" title="E79">E40</font></i></a></i>;<br/>
<a NAME="E4:68_2_1_5_2"/>
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 
<b>by </b><i><a class="txt" href="#E2:68_2_1_5_2"><i><font color="Green" title="E80">E41</font></i></a>, <a class="ref" href="mesfunc1.html#D16" title="MESFUNC1:def.16">MESFUNC1:def 16</a></i>;<br/>
<a NAME="E5:68_2_1_5_2"/><b>then </b><i><font color="Green" title="E81">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c6">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E6:68_2_1_5_2"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">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="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">m</font> <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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_5_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">n</font>, <font color="Maroon" title="c14">m</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">m</font> <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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font> )</span><br/>

<b>assume </b><a NAME="E1:68_2_1_5_2_1"/>
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">m</font> <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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font></span><br/>

<a NAME="E2:68_2_1_5_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">x1</font> <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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">x1</font>
 
<b>by </b><i><a class="txt" href="#E2:68_2_1_5_2"><i><font color="Green" title="E80">E41</font></i></a>, <a class="txt" href="#E5:68"><i><font color="Green" title="E56">A3</font></i></a></i>;<br/>
<a NAME="E3:68_2_1_5_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">m</font> <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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">x1</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<b>hence </b><a NAME="E4:68_2_1_5_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">m</font> <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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:68_2_1_5_2"/><b>then </b><i><font color="Green" title="E82">E46</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>
 
<b>by </b><i><a class="ref" href="rinfsup2.html#T7" title="RINFSUP2:th.7">RINFSUP2:7</a></i>;<br/>
<a NAME="E8:68_2_1_5_2"/><b>then </b><i><font color="Green" title="E83">E47</font></i>: 
( <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K1" title="RINFSUP2:func.1">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rinfsup2.html#T37" title="RINFSUP2:th.37">RINFSUP2:37</a></i>;<br/>
<a NAME="E9:68_2_1_5_2"/><i><font color="Green" title="E84">E48</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c10">N1</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> &amp;  <a href="rinfsup2.html#K1" title="RINFSUP2:func.1">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K1" title="RINFSUP2:func.1">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c10">N1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E7:68_2_1_5_2"><i><font color="Green" title="E82">E46</font></i></a>, <a class="ref" href="rinfsup2.html#T26" title="RINFSUP2:th.26">RINFSUP2:26</a></i>;<br/>
<a NAME="E10:68_2_1_5_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c10">N1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<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> <font color="Maroon" title="c6">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></i>;<br/>
<a NAME="E11:68_2_1_5_2"/><b>then </b>
for <font color="Olive" title="b1">n</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>  holds   <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c10">N1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<b>by </b><i><a class="txt" href="#E5:68_2_1_5_2"><i><font color="Green" title="E81">E44</font></i></a>, <a class="txt" href="#E9:68_2_1_5_2"><i><font color="Green" title="E84">E48</font></i></a>, <a class="ref" href="rinfsup2.html#T7" title="RINFSUP2:th.7">RINFSUP2:7</a></i>;<br/>
<a NAME="E12:68_2_1_5_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c10">N1</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>
 
<b>by </b><i><a class="ref" href="rinfsup2.html#T32" title="RINFSUP2:th.32">RINFSUP2:32</a></i>;<br/>
<a NAME="E13:68_2_1_5_2"/><b>then </b>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c10">N1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="ref" href="mesfunc9.html#T7" target="_self" title="MESFUNC9:th.7">LIM</a></i>;<br/>
<a NAME="E14:68_2_1_5_2"/><b>then </b>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c12">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="#E8:68_2_1_5_2"><i><font color="Green" title="E83">E47</font></i></a>, <a class="txt" href="#E9:68_2_1_5_2"><i><font color="Green" title="E84">E48</font></i></a>, <a class="ref" href="rinfsup2.html#T37" title="RINFSUP2:th.37">RINFSUP2:37</a></i>;<br/>
<a NAME="E15:68_2_1_5_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="#E1:68_2_1_5"><i><font color="Green" title="E77">E10</font></i></a>, <a class="txt" href="#E2:68_2_1_5_2"><i><font color="Green" title="E80">E41</font></i></a>, <a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="ref" href="mesfunc8.html#D10" title="MESFUNC8:def.10">MESFUNC8:def 10</a></i>;<br/>
<a NAME="E16:68_2_1_5_2"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="#E33:68_2_1"><i><font color="Green" title="E76">E12</font></i></a>, <a class="txt" href="#E2:68_2_1_5_2"><i><font color="Green" title="E80">E41</font></i></a>, <a class="ref" href="mesfunc1.html#D16" title="MESFUNC1:def.16">MESFUNC1:def 16</a></i>;<br/>
<b>hence </b><a NAME="E17:68_2_1_5_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E2:68_2_1_5_2"><i><font color="Green" title="E80">E41</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:68_2_1_5"/><i><font color="Green" title="E79">E16</font></i>: 
<font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="#E33:68_2_1"><i><font color="Green" title="E76">E12</font></i></a>, <a class="txt" href="#E8:68"><i><font color="Green" title="E59">E13</font></i></a>, <a class="ref" href="mesfunc1.html#T37" title="MESFUNC1:th.37">MESFUNC1:37</a></i>;<br/>
<a NAME="E6:68_2_1_5"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">N</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:68_2_1"><i><font color="Green" title="E60">B1</font></i></a>, <a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/>
<b>hence </b><a NAME="E7:68_2_1_5"/>
<font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p3">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E5:68_2_1_5"><i><font color="Green" title="E79">E16</font></i></a>, <a class="txt" href="#E4:68_2_1_5"><i><font color="Green" title="E78">E14</font></i></a>, <a class="ref" href="measure1.html#T62" title="MEASURE1:th.62">MEASURE1:62</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E35:68_2_1"/>
( ( 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" title="c9">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) &amp; <font color="Maroon" title="c9">I</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c9">I</font> )
 <b>by </b><i><a class="txt" href="#E32:68_2_1"><i><font color="Green" title="E75">E9</font></i></a>, <a class="txt" href="#E33:68_2_1"><i><font color="Green" title="E76">E12</font></i></a>, <a class="txt" href="#E8:68"><i><font color="Green" title="E59">E13</font></i></a>, <a class="txt" href="#E25:68_2_1"><i><font color="Green" title="E72">E4</font></i></a>, <a class="txt" href="#E23:68_2_1"><i><font color="Green" title="E70">D3</font></i></a>, <a class="ref" href="mesfunc9.html#T13" target="_self" title="MESFUNC9:th.13">Lem13</a>, <a class="ref" href="rinfsup2.html#T37" title="RINFSUP2:th.37">RINFSUP2:37</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_2"><b>suppose </b></a><a NAME="E1:68_2_2"/><i><font color="Green" title="E60">F0</font></i>: 
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" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span></span>)</span> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">I</font> being  <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) &amp; <font color="Olive" title="b1">I</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b1">I</font> )</span><br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1</span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span>;<br/><a NAME="E2:68_2_2"/><i><font color="Green" title="E61">F1</font></i>: 
for <font color="Olive" title="b1">n</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>   ex <font color="Olive" title="b2">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">A</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>,<font color="Olive" title="b1">A</font>]</span><br/>

<a NAME="E1:68_2_2_1"/>
<font color="Maroon" title="c4">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="txt" href="#E3:68"><i><font color="Green" title="E54">S1</font></i></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="#E4:68"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="mesfunc1.html#T37" title="MESFUNC1:th.37">MESFUNC1:37</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c7">A</font>]</span><br/>

<b>thus </b><a NAME="E3:68_2_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c7">A</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c6">L</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> ,<font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E4:68_2_2"/><i><font color="Green" title="E62">F2</font></i>: 
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c6">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:68_2_2"><i><font color="Green" title="E61">F1</font></i></a>);<br/></i><a NAME="E5:68_2_2"/><i><font color="Green" title="E63">F3</font></i>: 
 <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c6">L</font> is   <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="measure1.html#T52" title="MEASURE1:th.52">MEASURE1:52</a></i>;<br/><a NAME="E6:68_2_2"/>
 <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c6">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">E0</font> =  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c6">L</font> as    <a href="measure2.html#M1" title="MEASURE2:mode.1">N_Measure_fam</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="#E5:68_2_2"><i><font color="Green" title="E63">F3</font></i></a>, <a class="ref" href="measure2.html#D1" title="MEASURE2:def.1">MEASURE2:def 1</a></i>;<br/><a NAME="E8:68_2_2"/><i><font color="Green" title="E64">F5</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">L</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="ref" href="measure2.html#T1" title="MEASURE2:th.1">MEASURE2:1</a></i>;<br/><a NAME="E9:68_2_2"/><i><font color="Green" title="E65">F6</font></i>: 
for <font color="Olive" title="b1">n</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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <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">L</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font> implies <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">L</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>assume </b><a NAME="E1:68_2_2_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">L</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:68_2_2_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:68_2_2_2"/><b>then </b>
<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">L</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E4:68_2_2_2"/><b>then </b>
<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">L</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:68_2_2"><i><font color="Green" title="E62">F2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:68_2_2_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="c6">L</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:68_2_2"><i><font color="Green" title="E60">F0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E10:68_2_2"/><b>then </b>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</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">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <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">L</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E8:68_2_2"><i><font color="Green" title="E64">F5</font></i></a>, <a class="ref" href="supinf_2.html#T67" title="SUPINF_2:th.67">SUPINF_2:67</a></i>;<br/><a NAME="E11:68_2_2"/><b>then </b>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</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">L</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="c6">L</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="supinf_2.html#T63" title="SUPINF_2:th.63">SUPINF_2:63</a></i>;<br/><a NAME="E12:68_2_2"/><b>then </b>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</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">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E9:68_2_2"><i><font color="Green" title="E65">F6</font></i></a></i>;<br/><a NAME="E13:68_2_2"/><b>then </b><i><font color="Green" title="E66">F7</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="measure2.html#T13" title="MEASURE2:th.13">MEASURE2:13</a></i>;<br/><a NAME="E14:68_2_2"/><i><font color="Green" title="E67">F8</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E13:68_2_2"><i><font color="Green" title="E66">F7</font></i></a>, <a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/><a NAME="E15:68_2_2"/><i><font color="Green" title="E68">F9</font></i>: 
<font color="Maroon" title="c4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="ref" href="mesfunc8.html#D10" title="MESFUNC8:def.10">MESFUNC8:def 10</a></i>;<br/><b>set </b><font color="Maroon" title="c8">E1</font> = <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>;<br/><a NAME="E16:68_2_2"/><i><font color="Green" title="E69">LL1</font></i>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:68"><i><font color="Green" title="E59">E13</font></i></a>, <a class="txt" href="#E14:68_2_2"><i><font color="Green" title="E67">F8</font></i></a>, <a class="txt" href="#E15:68_2_2"><i><font color="Green" title="E68">F9</font></i></a>, <a class="ref" href="mesfunc5.html#T101" title="MESFUNC5:th.101">MESFUNC5:101</a></i>;<br/><a NAME="E17:68_2_2"/><i><font color="Green" title="E70">L4</font></i>: 
<font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E18:68_2_2"/><b>then </b><i><font color="Green" title="E71">AA2</font></i>: 
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" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:68"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="mesfunc1.html#T34" title="MESFUNC1:th.34">MESFUNC1:34</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c9">H</font> being   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E20:68_2_2"/><i><font color="Green" title="E72">G1</font></i>: 
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" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seqfunc.html#S1" title="SEQFUNC:sch.1">SEQFUNC:sch 1</a>();<br/></i><a NAME="E21:68_2_2"/><i><font color="Green" title="E73">G2</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> (  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> &amp; <font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> &amp; <font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">without+infty</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> &amp; <font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> &amp; <font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font> is <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">without+infty</a> )</span><br/>

<a NAME="E1:68_2_2_3"/><i><font color="Green" title="E74">G20</font></i>: 
<font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E17:68_2_2"><i><font color="Green" title="E70">L4</font></i></a>, <a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="txt" href="#E3:68"><i><font color="Green" title="E54">S1</font></i></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/>
<a NAME="E2:68_2_2_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E20:68_2_2"><i><font color="Green" title="E72">G1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:68_2_2_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:68_2_2_3"><i><font color="Green" title="E74">G20</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> &amp; <font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font> is <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">without+infty</a> )</span><br/>

<a NAME="E4:68_2_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="#E7:68"><i><font color="Green" title="E58">E0</font></i></a>, <a class="ref" href="mesfunc5.html#T21" title="MESFUNC5:th.21">MESFUNC5:21</a></i>;<br/>
<a NAME="E5:68_2_2_3"/><b>then </b>
<font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="#E20:68_2_2"><i><font color="Green" title="E72">G1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:68_2_2_3"/>
<font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T18" title="MESFUNC5:th.18">MESFUNC5:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font> is <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">without+infty</a></span><br/>

<a NAME="E7:68_2_2_3"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</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="68_2_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">x</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/>

<b>assume </b><a NAME="E1:68_2_2_3_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">x</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="E2:68_2_2_3_1"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E20:68_2_2"><i><font color="Green" title="E72">G1</font></i></a></i>;<br/>
<a NAME="E3:68_2_2_3_1"/><b>then </b><i><font color="Green" title="E75">G5</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/>
<a NAME="E4:68_2_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="txt" href="#E20:68_2_2"><i><font color="Green" title="E72">G1</font></i></a></i>;<br/>
<a NAME="E5:68_2_2_3_1"/><b>then </b><i><font color="Green" title="E76">G4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="txt" href="#E3:68_2_2_3_1"><i><font color="Green" title="E75">G5</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>assume </b><a NAME="E6:68_2_2_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:68_2_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="#E5:68_2_2_3_1"><i><font color="Green" title="E76">G4</font></i></a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>;<br/>
<a NAME="E8:68_2_2_3_1"/><b>then </b><i><font color="Green" title="E77">G7</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<b>by </b><i><a class="txt" href="#E3:68_2_2_3_1"><i><font color="Green" title="E75">G5</font></i></a>, <a class="ref" href="mesfunc1.html#D16" title="MESFUNC1:def.16">MESFUNC1:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">n1</font> = <font color="Maroon" title="c10">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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E10:68_2_2_3_1"/>
( <font color="Maroon" title="c6">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">eq_dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">n1</font></span>)</span>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="txt" href="#E4:68_2_2"><i><font color="Green" title="E62">F2</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:68_2_2_3_1"/><b>then </b>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c6">L</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c6">L</font> )
 
<b>by </b><i><a class="txt" href="#E8:68_2_2_3_1"><i><font color="Green" title="E77">G7</font></i></a>, <a class="txt" href="#E3:68_2_2_3_1"><i><font color="Green" title="E75">G5</font></i></a>, <a class="txt" href="#E17:68_2_2"><i><font color="Green" title="E70">L4</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E12:68_2_2_3_1"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E13:68_2_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:68_2_2_3_1"><i><font color="Green" title="E75">G5</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:68_2_2_3"/>
<font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font> is <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">without+infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T17" title="MESFUNC5:th.17">MESFUNC5:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E22:68_2_2"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds   <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">n</font>, <font color="Maroon" title="c11">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span><br/>

<a NAME="E1:68_2_2_4"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E21:68_2_2"><i><font color="Green" title="E73">G2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:68_2_2_4"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E21:68_2_2"><i><font color="Green" title="E73">G2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c10">H</font> = <font color="Maroon" title="c9">H</font> as  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  <b>by </b><i><a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</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> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1</span>)</span>;<br/><a NAME="E24:68_2_2"/><i><font color="Green" title="E74">K1</font></i>: 
for <font color="Olive" title="b1">n</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> <br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c11">G</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E26:68_2_2"/><i><font color="Green" title="E75">K3</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c11">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c11">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c11">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S1" title="RECDEF_1:sch.1">RECDEF_1:sch 1</a>(<a class="txt" href="#E24:68_2_2"><i><font color="Green" title="E74">K1</font></i></a>);<br/></i><a NAME="E27:68_2_2"/><i><font color="Green" title="E76">J3</font></i>: 
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" title="c11">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c13">n1</font> = <font color="Maroon" title="c12">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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:68_2_2_5"/>
<font color="Maroon" title="c11">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E26:68_2_2"><i><font color="Green" title="E75">K3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:68_2_2_5"/>
<font color="Maroon" title="c11">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">f</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="c11">G</font> implies <font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  )</span><br/><b>assume </b><a NAME="E1:68_2_2_6"/>
<font color="Maroon" title="c12">f</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="c11">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> </span><br/><b>then consider </b><font color="Maroon" title="c13">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:68_2_2_6"/><i><font color="Green" title="E77">K41</font></i>: 
( <font color="Maroon" title="c13">m</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="c11">G</font> &amp; <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">m</font> = <font color="Maroon" title="c13">m</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="#E3:68_2_2_6"><i><font color="Green" title="E77">K41</font></i></a>, <a class="txt" href="#E26:68_2_2"><i><font color="Green" title="E75">K3</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon" title="c11">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 is   <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/><a NAME="E5:68_2_2_6"/><i><font color="Green" title="E78">K42</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="txt" href="#E26:68_2_2"><i><font color="Green" title="E75">K3</font></i></a></i>;<br/><a NAME="E6:68_2_2_6"/><i><font color="Green" title="E79">K43</font></i>: 
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>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>3</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="68_2_2_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c15">n</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:68_2_2_6_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c15">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:68_2_2_6_1"/>
<font color="Maroon" title="c11">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E27:68_2_2"><i><font color="Green" title="E76">J3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:68_2_2_6_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E7:68_2_2_6"/>
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>3</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E5:68_2_2_6"><i><font color="Green" title="E78">K42</font></i></a>, <a class="txt" href="#E6:68_2_2_6"><i><font color="Green" title="E79">K43</font></i></a>);<br/></i><a NAME="E8:68_2_2_6"/><b>then </b>
<font color="Maroon" title="c11">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font> is   <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/><b>hence </b><a NAME="E9:68_2_2_6"/>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> 
 <b>by </b><i><a class="txt" href="#E3:68_2_2_6"><i><font color="Green" title="E77">K41</font></i></a>, <a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E29:68_2_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">G</font> = <font color="Maroon" title="c11">G</font> as   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  <b>by </b><i><a class="txt" href="#E26:68_2_2"><i><font color="Green" title="E75">K3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/><a NAME="E31:68_2_2"/><i><font color="Green" title="E77">Y1</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds   <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_2_2_7_1"/>
<font color="Maroon" title="c13">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span><br/><b>then consider </b><font color="Maroon" title="c14">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:68_2_2_7_1"/><i><font color="Green" title="E78">Y11</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><a NAME="E4:68_2_2_7_1"/><i><font color="Green" title="E79">M0</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E27:68_2_2"><i><font color="Green" title="E76">J3</font></i></a></i>;<br/><a NAME="E5:68_2_2_7_1"/>
( <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> &amp; <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">without+infty</a> &amp; <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">k</font> is <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">without+infty</a> &amp; <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">k</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> )
 <b>by </b><i><a class="txt" href="#E21:68_2_2"><i><font color="Green" title="E73">G2</font></i></a></i>;<br/><a NAME="E6:68_2_2_7_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:68_2_2_7_1"><i><font color="Green" title="E79">M0</font></i></a>, <a class="ref" href="mesfunc5.html#T23" title="MESFUNC5:th.23">MESFUNC5:23</a></i>;<br/><a NAME="E7:68_2_2_7_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E21:68_2_2"><i><font color="Green" title="E73">G2</font></i></a></i>;<br/><a NAME="E8:68_2_2_7_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E21:68_2_2"><i><font color="Green" title="E73">G2</font></i></a></i>;<br/><b>hence </b><a NAME="E9:68_2_2_7_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:68_2_2_7_1"><i><font color="Green" title="E78">Y11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:68_2_2_7"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E26:68_2_2"><i><font color="Green" title="E75">K3</font></i></a>, <a class="txt" href="#E21:68_2_2"><i><font color="Green" title="E73">G2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E32:68_2_2"/><i><font color="Green" title="E78">Z1</font></i>: 
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" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a></span><br/>

<a NAME="E1:68_2_2_8"/><i><font color="Green" title="E79">P27</font></i>: 
( <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_2_2_8_1"/><i><font color="Green" title="E80">P28</font></i>: 
<font color="Maroon" title="c13">n</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a></span><br/>

<a NAME="E2:68_2_2_8_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="#E7:68"><i><font color="Green" title="E58">E0</font></i></a>, <a class="ref" href="mesfunc5.html#T21" title="MESFUNC5:th.21">MESFUNC5:21</a></i>;<br/>
<b>hence </b><a NAME="E3:68_2_2_8_1"/>
<font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="#E1:68_2_2_8_1"><i><font color="Green" title="E80">P28</font></i></a>, <a class="txt" href="#E26:68_2_2"><i><font color="Green" title="E75">K3</font></i></a>, <a class="txt" href="#E20:68_2_2"><i><font color="Green" title="E72">G1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:68_2_2_8"/>
( <font color="Maroon" title="c13">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_8_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_2_2_8_2"/>
<font color="Maroon" title="c13">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a></span><br/>

<b>then consider </b><font color="Maroon" title="c14">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:68_2_2_8_2"/><i><font color="Green" title="E80">P28</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<a NAME="E4:68_2_2_8_2"/><i><font color="Green" title="E81">M0</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E27:68_2_2"><i><font color="Green" title="E76">J3</font></i></a></i>;<br/>
<a NAME="E5:68_2_2_8_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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> holds <br/> <a href="numbers.html#K6" title="NUMBERS:func.6">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="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_8_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> implies  <a href="numbers.html#K6" title="NUMBERS:func.6">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="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">x</font> )</span><br/>

<b>assume </b><a NAME="E1:68_2_2_8_2_1"/><i><font color="Green" title="E82">M21</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">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="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">x</font></span><br/>

<a NAME="E2:68_2_2_8_2_1"/><i><font color="Green" title="E83">M23</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E31:68_2_2"><i><font color="Green" title="E77">Y1</font></i></a></i>;<br/>
<a NAME="E3:68_2_2_8_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">x</font> )
 
<b>by </b><i><a class="txt" href="#E20:68_2_2"><i><font color="Green" title="E72">G1</font></i></a></i>;<br/>
<a NAME="E4:68_2_2_8_2_1"/><b>then </b><i><font color="Green" title="E84">M25</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">x</font> )
 
<b>by </b><i><a class="txt" href="#E1:68_2_2_8_2_1"><i><font color="Green" title="E82">M21</font></i></a>, <a class="txt" href="#E2:68_2_2_8_2_1"><i><font color="Green" title="E83">M23</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E5:68_2_2_8_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">x</font> <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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">x</font>
 
<b>by </b><i><a class="txt" href="#E5:68"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="#E17:68_2_2"><i><font color="Green" title="E70">L4</font></i></a>, <a class="txt" href="#E1:68_2_2_8_2_1"><i><font color="Green" title="E82">M21</font></i></a>, <a class="txt" href="#E2:68_2_2_8_2_1"><i><font color="Green" title="E83">M23</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E6:68_2_2_8_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E4:68_2_2_8_2_1"><i><font color="Green" title="E84">M25</font></i></a>, <a class="ref" href="supinf_2.html#T72" title="SUPINF_2:th.72">SUPINF_2:72</a></i>;<br/>
<b>hence </b><a NAME="E7:68_2_2_8_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">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="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="#E4:68_2_2_8_2"><i><font color="Green" title="E81">M0</font></i></a>, <a class="txt" href="#E1:68_2_2_8_2_1"><i><font color="Green" title="E82">M21</font></i></a>, <a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:68_2_2_8_2"/>
<font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="#E3:68_2_2_8_2"><i><font color="Green" title="E80">P28</font></i></a>, <a class="ref" href="supinf_2.html#T71" title="SUPINF_2:th.71">SUPINF_2:71</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:68_2_2_8"/>
<font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="#E1:68_2_2_8"><i><font color="Green" title="E79">P27</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E33:68_2_2"/><i><font color="Green" title="E79">X0</font></i>: 
for <font color="Olive" title="b1">n1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c10">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">n1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <font color="Maroon" title="c10">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n1</font> )</span><br/>

<b>assume </b><a NAME="E1:68_2_2_9"/>
<font color="Maroon" title="c13">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n1</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c14">n</font> = <font color="Maroon" title="c13">n1</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1 <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1;<br/>
<a NAME="E3:68_2_2_9"/><i><font color="Green" title="E80">P21</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="#E26:68_2_2"><i><font color="Green" title="E75">K3</font></i></a>, <a class="ref" href="mesfunc9.html#D4" target="_self" title="MESFUNC9:def.4">Def0</a></i>;<br/>
<a NAME="E4:68_2_2_9"/><i><font color="Green" title="E81">P22</font></i>: 
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="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c15">k</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:68_2_2_9_1"/><i><font color="Green" title="E82">P23</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c15">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:68_2_2_9_1"/><i><font color="Green" title="E83">M0</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E27:68_2_2"><i><font color="Green" title="E76">J3</font></i></a></i>;<br/>
<a NAME="E3:68_2_2_9_1"/>
<font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_9_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:68_2_2_9_1_1"/><i><font color="Green" title="E84">M3</font></i>: 
( <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> &amp; <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">without+infty</a> &amp; <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font> is <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">without+infty</a> &amp; <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> )
 
<b>by </b><i><a class="txt" href="#E21:68_2_2"><i><font color="Green" title="E73">G2</font></i></a></i>;<br/>
<a NAME="E2:68_2_2_9_1_1"/><i><font color="Green" title="E85">M2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E31:68_2_2"><i><font color="Green" title="E77">Y1</font></i></a></i>;<br/>
<a NAME="E3:68_2_2_9_1_1"/><i><font color="Green" title="E86">M6</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a>
 
<b>by </b><i><a class="txt" href="#E32:68_2_2"><i><font color="Green" title="E78">Z1</font></i></a>, <a class="ref" href="mesfunc5.html#T18" title="MESFUNC5:th.18">MESFUNC5:18</a></i>;<br/>
<a NAME="E4:68_2_2_9_1_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:68_2_2_9_1_1"><i><font color="Green" title="E84">M3</font></i></a>, <a class="ref" href="mesfunc5.html#T22" title="MESFUNC5:th.22">MESFUNC5:22</a></i>;<br/>
<a NAME="E5:68_2_2_9_1_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:68_2_2_9_1_1"><i><font color="Green" title="E85">M2</font></i></a>, <a class="txt" href="#E21:68_2_2"><i><font color="Green" title="E73">G2</font></i></a></i>;<br/>
<a NAME="E6:68_2_2_9_1_1"/><b>then </b><i><font color="Green" title="E87">M1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E21:68_2_2"><i><font color="Green" title="E73">G2</font></i></a></i>;<br/>
<a NAME="E7:68_2_2_9_1_1"/>
for <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="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_9_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">x</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font> )</span><br/>

<b>assume </b><a NAME="E1:68_2_2_9_1_1_1"/><i><font color="Green" title="E88">M4</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></span><br/>

<a NAME="E2:68_2_2_9_1_1_1"/><b>then </b><i><font color="Green" title="E89">M5</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E21:68_2_2"><i><font color="Green" title="E73">G2</font></i></a></i>;<br/>
<a NAME="E3:68_2_2_9_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 
<b>by </b><i><a class="txt" href="#E1:68_2_2_9_1_1"><i><font color="Green" title="E84">M3</font></i></a>, <a class="txt" href="#E3:68_2_2_9_1_1"><i><font color="Green" title="E86">M6</font></i></a>, <a class="ref" href="mesfunc5.html#D5" title="MESFUNC5:def.5">MESFUNC5:def 5</a>, <a class="ref" href="mesfunc5.html#D6" title="MESFUNC5:def.6">MESFUNC5:def 6</a></i>;<br/>
<a NAME="E4:68_2_2_9_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></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="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T41" title="EXTREAL2:th.41">EXTREAL2:41</a></i>;<br/>
<a NAME="E5:68_2_2_9_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></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="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="ref" href="extreal2.html#T13" title="EXTREAL2:th.13">EXTREAL2:13</a></i>;<br/>
<a NAME="E6:68_2_2_9_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></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="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font>
 
<b>by </b><i><a class="ref" href="extreal1.html#T24" title="EXTREAL1:th.24">EXTREAL1:24</a>, <a class="ref" href="supinf_2.html#T18" title="SUPINF_2:th.18">SUPINF_2:18</a></i>;<br/>
<a NAME="E7:68_2_2_9_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</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="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:68_2_2_9_1"><i><font color="Green" title="E83">M0</font></i></a>, <a class="txt" href="#E2:68_2_2_9_1_1_1"><i><font color="Green" title="E89">M5</font></i></a>, <a class="txt" href="#E2:68_2_2_9_1_1"><i><font color="Green" title="E85">M2</font></i></a>, <a class="ref" href="mesfunc1.html#D4" title="MESFUNC1:def.4">MESFUNC1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:68_2_2_9_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font>
 <b>by </b><i><a class="txt" href="#E1:68_2_2_9_1_1_1"><i><font color="Green" title="E88">M4</font></i></a>, <a class="txt" href="#E6:68_2_2_9_1_1"><i><font color="Green" title="E87">M1</font></i></a>, <a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:68_2_2_9_1_1"/>
<font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:68_2_2_9_1_1"><i><font color="Green" title="E87">M1</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:68_2_2_9_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="#E1:68_2_2_9_1"><i><font color="Green" title="E82">P23</font></i></a>, <a class="ref" href="mesfunc9.html#D4" target="_self" title="MESFUNC9:def.4">Def0</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:68_2_2_9"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E3:68_2_2_9"><i><font color="Green" title="E80">P21</font></i></a>, <a class="txt" href="#E4:68_2_2_9"><i><font color="Green" title="E81">P22</font></i></a>);<br/></i>
<a NAME="E6:68_2_2_9"/><b>then </b>
<font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font>
 
;<br/>
<b>hence </b><a NAME="E7:68_2_2_9"/>
<font color="Maroon" title="c10">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E34:68_2_2"/><i><font color="Green" title="E80">P2</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> &amp;  <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c10">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c12">G</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E33:68_2_2"><i><font color="Green" title="E79">X0</font></i></a>, <a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/><a NAME="E35:68_2_2"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds   <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">n</font>, <font color="Maroon" title="c14">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font></span>)</span></span><br/>

<a NAME="E1:68_2_2_10"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E31:68_2_2"><i><font color="Green" title="E77">Y1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:68_2_2_10"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E31:68_2_2"><i><font color="Green" title="E77">Y1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c13">G</font> = <font color="Maroon" title="c12">G</font> as  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  <b>by </b><i><a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">G</font> = <font color="Maroon" title="c13">G</font> as  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  <b>by </b><i><a class="txt" href="#E32:68_2_2"><i><font color="Green" title="E78">Z1</font></i></a>, <a class="ref" href="mesfunc9.html#T30" target="_self" title="MESFUNC9:th.30">LIM4</a></i>;<br/><a NAME="E38:68_2_2"/><i><font color="Green" title="E81">V1</font></i>: 
<font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E21:68_2_2"><i><font color="Green" title="E73">G2</font></i></a>, <a class="txt" href="#E26:68_2_2"><i><font color="Green" title="E75">K3</font></i></a></i>;<br/><a NAME="E39:68_2_2"/><i><font color="Green" title="E82">T20</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">k</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a></span><br/>

<a NAME="E1:68_2_2_11"/>
for <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="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> holds <br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">x</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> implies <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</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><br/>

<b>assume </b><a NAME="E1:68_2_2_11_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</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><br/>

<a NAME="E2:68_2_2_11_1"/>
( <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> &amp; <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font> is <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">without+infty</a> )
 
<b>by </b><i><a class="txt" href="#E21:68_2_2"><i><font color="Green" title="E73">G2</font></i></a></i>;<br/>
<a NAME="E3:68_2_2_11_1"/><b>then </b>
(  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</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>  )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D5" title="MESFUNC5:def.5">MESFUNC5:def 5</a>, <a class="ref" href="mesfunc5.html#D6" title="MESFUNC5:def.6">MESFUNC5:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:68_2_2_11_1"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</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> 
 <b>by </b><i><a class="ref" href="extreal2.html#T101" title="EXTREAL2:th.101">EXTREAL2:101</a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:68_2_2_11"/>
<font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>
 <b>by </b><i><a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E40:68_2_2"/><i><font color="Green" title="E83">V3</font></i>: 
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" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span><br/>

<a NAME="E1:68_2_2_12"/>
( <font color="Maroon" title="c15">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_2_2_12_1"/>
<font color="Maroon" title="c15">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c16">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:68_2_2_12_1"/><i><font color="Green" title="E84">T2</font></i>: 
<font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<a NAME="E4:68_2_2_12_1"/><i><font color="Green" title="E85">T21</font></i>: 
( <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> &amp; <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">k</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E17:68_2_2"><i><font color="Green" title="E70">L4</font></i></a>, <a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="txt" href="#E3:68"><i><font color="Green" title="E54">S1</font></i></a>, <a class="txt" href="#E20:68_2_2"><i><font color="Green" title="E72">G1</font></i></a>, <a class="txt" href="#E18:68_2_2"><i><font color="Green" title="E71">AA2</font></i></a>, <a class="ref" href="mesfunc9.html#T20" target="_self" title="MESFUNC9:th.20">Prop1</a></i>;<br/>
<a NAME="E5:68_2_2_12_1"/><i><font color="Green" title="E86">T22</font></i>: 
<font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E21:68_2_2"><i><font color="Green" title="E73">G2</font></i></a></i>;<br/>
<a NAME="E6:68_2_2_12_1"/><i><font color="Green" title="E87">T23</font></i>: 
( <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> &amp; <font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">k</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> )
 
<b>by </b><i><a class="txt" href="#E39:68_2_2"><i><font color="Green" title="E82">T20</font></i></a></i>;<br/>
<a NAME="E7:68_2_2_12_1"/>
<font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E27:68_2_2"><i><font color="Green" title="E76">J3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:68_2_2_12_1"/>
<font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:68_2_2_12_1"><i><font color="Green" title="E84">T2</font></i></a>, <a class="txt" href="#E6:68_2_2_12_1"><i><font color="Green" title="E87">T23</font></i></a>, <a class="txt" href="#E4:68_2_2_12_1"><i><font color="Green" title="E85">T21</font></i></a>, <a class="txt" href="#E5:68_2_2_12_1"><i><font color="Green" title="E86">T22</font></i></a>, <a class="ref" href="mesfunc2.html#T13" title="MESFUNC2:th.13">MESFUNC2:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:68_2_2_12"/>
<font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E17:68_2_2"><i><font color="Green" title="E70">L4</font></i></a>, <a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="txt" href="#E3:68"><i><font color="Green" title="E54">S1</font></i></a>, <a class="txt" href="#E20:68_2_2"><i><font color="Green" title="E72">G1</font></i></a>, <a class="txt" href="#E18:68_2_2"><i><font color="Green" title="E71">AA2</font></i></a>, <a class="ref" href="mesfunc9.html#T20" target="_self" title="MESFUNC9:th.20">Prop1</a>, <a class="txt" href="#E26:68_2_2"><i><font color="Green" title="E75">K3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E41:68_2_2"/>
for <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="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> holds <br/><font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> implies <font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c15">x</font> is <a href="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</a> )</span><br/>

<b>assume </b><a NAME="E1:68_2_2_13"/><i><font color="Green" title="E84">Q0</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c15">x</font> is <a href="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</a></span><br/>

<a NAME="E2:68_2_2_13"/>
<font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">E</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E3:68_2_2_13"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c15">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 
<b>by </b><i><a class="txt" href="#E6:68"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="#E1:68_2_2_13"><i><font color="Green" title="E84">Q0</font></i></a></i>;<br/>
<a NAME="E4:68_2_2_13"/><b>then </b>
<font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c15">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 
<b>by </b><i><a class="txt" href="#E1:68_2_2_13"><i><font color="Green" title="E84">Q0</font></i></a>, <a class="txt" href="#E20:68_2_2"><i><font color="Green" title="E72">G1</font></i></a>, <a class="ref" href="mesfunc9.html#T12" target="_self" title="MESFUNC9:th.12">Lm16</a></i>;<br/>
<a NAME="E5:68_2_2_13"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c14">G</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c15">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 
<b>by </b><i><a class="txt" href="#E33:68_2_2"><i><font color="Green" title="E79">X0</font></i></a>, <a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>
<a NAME="E6:68_2_2_13"/><b>then </b>
 <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c15">x</font></span>)</span> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 
<b>by </b><i><a class="txt" href="#E1:68_2_2_13"><i><font color="Green" title="E84">Q0</font></i></a>, <a class="txt" href="#E38:68_2_2"><i><font color="Green" title="E81">V1</font></i></a>, <a class="ref" href="mesfunc9.html#T33" target="_self" title="MESFUNC9:th.33">Cor01</a></i>;<br/>
<b>hence </b><a NAME="E7:68_2_2_13"/>
<font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c15">x</font> is <a href="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</a>
 <b>by </b><i><a class="ref" href="mesfunc9.html#D2" target="_self" title="MESFUNC9:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c15">J</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E43:68_2_2"/><i><font color="Green" title="E84">Q2</font></i>: 
( ( 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" title="c15">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span></span>)</span> ) &amp; <font color="Maroon" title="c15">J</font> is <a href="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</a> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c14">G</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc9.html#K3" title="MESFUNC9:func.3">Sum</a> <font color="Maroon" title="c15">J</font> )
 <b>by </b><i><a class="txt" href="#E38:68_2_2"><i><font color="Green" title="E81">V1</font></i></a>, <a class="txt" href="#E40:68_2_2"><i><font color="Green" title="E83">V3</font></i></a>, <a class="txt" href="#E32:68_2_2"><i><font color="Green" title="E78">Z1</font></i></a>, <a class="ref" href="mesfunc9.html#T51" target="_self" title="MESFUNC9:th.51">Th131</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <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> ) <b>-&gt; </b>   <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>  =  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c16">I</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> <b> such that </b><br/><a NAME="E45:68_2_2"/><i><font color="Green" title="E85">P3</font></i>: 
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon" title="c16">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i><b>reconsider </b><font color="Maroon" title="c17">I</font> = <font color="Maroon" title="c16">I</font> as   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> ;<br/><b>take </b>
<font color="Maroon" title="c17">I</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) &amp; <font color="Maroon" title="c17">I</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c17">I</font> )</span><br/><a NAME="E47:68_2_2"/><i><font color="Green" title="E86">P4</font></i>: 
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" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_14"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c19">n1</font> = <font color="Maroon" title="c18">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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:68_2_2_14"/><i><font color="Green" title="E87">P41</font></i>: 
<font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E45:68_2_2"><i><font color="Green" title="E85">P3</font></i></a></i>;<br/>
<a NAME="E3:68_2_2_14"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font>
 
<b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="txt" href="#E3:68"><i><font color="Green" title="E54">S1</font></i></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:68_2_2_14"/>
<font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:68"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="#E2:68_2_2_14"><i><font color="Green" title="E87">P41</font></i></a>, <a class="txt" href="#E14:68_2_2"><i><font color="Green" title="E67">F8</font></i></a>, <a class="ref" href="mesfunc5.html#T101" title="MESFUNC5:th.101">MESFUNC5:101</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E48:68_2_2"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">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="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">n</font>, <font color="Maroon" title="c19">m</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">n</font> implies <font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c18">n</font> )</span><br/>

<b>assume </b><a NAME="E1:68_2_2_15"/><i><font color="Green" title="E87">P51</font></i>: 
<font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c18">n</font></span><br/>

<a NAME="E2:68_2_2_15"/><i><font color="Green" title="E88">P52</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> )
 
<b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="txt" href="#E3:68"><i><font color="Green" title="E54">S1</font></i></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/>
<a NAME="E3:68_2_2_15"/><b>then </b><i><font color="Green" title="E89">P54</font></i>: 
for <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="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font> <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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="txt" href="#E5:68"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="#E1:68_2_2_15"><i><font color="Green" title="E87">P51</font></i></a></i>;<br/>
<a NAME="E4:68_2_2_15"/><i><font color="Green" title="E90">P55</font></i>: 
( <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="#E7:68"><i><font color="Green" title="E58">E0</font></i></a></i>;<br/>
<a NAME="E5:68_2_2_15"/>
( <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> &amp; <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> )
 
<b>by </b><i><a class="txt" href="#E4:68"><i><font color="Green" title="E55">A2</font></i></a></i>;<br/>
<a NAME="E6:68_2_2_15"/><b>then </b>
 <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:68_2_2_15"><i><font color="Green" title="E88">P52</font></i></a>, <a class="txt" href="#E4:68_2_2_15"><i><font color="Green" title="E90">P55</font></i></a>, <a class="txt" href="#E3:68_2_2_15"><i><font color="Green" title="E89">P54</font></i></a>, <a class="ref" href="mesfunc5.html#T91" title="MESFUNC5:th.91">MESFUNC5:91</a></i>;<br/>
<a NAME="E7:68_2_2_15"/><b>then </b>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K6" title="MESFUNC5:func.6">integral+</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:68_2_2_15"><i><font color="Green" title="E88">P52</font></i></a>, <a class="txt" href="#E4:68"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="#E4:68_2_2_15"><i><font color="Green" title="E90">P55</font></i></a>, <a class="ref" href="mesfunc5.html#T94" title="MESFUNC5:th.94">MESFUNC5:94</a></i>;<br/>
<a NAME="E8:68_2_2_15"/><b>then </b><i><font color="Green" title="E91">P56</font></i>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:68_2_2_15"><i><font color="Green" title="E88">P52</font></i></a>, <a class="txt" href="#E4:68"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="#E4:68_2_2_15"><i><font color="Green" title="E90">P55</font></i></a>, <a class="ref" href="mesfunc5.html#T94" title="MESFUNC5:th.94">MESFUNC5:94</a></i>;<br/>
<a NAME="E9:68_2_2_15"/>
<font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E47:68_2_2"><i><font color="Green" title="E86">P4</font></i></a>, <a class="txt" href="#E8:68_2_2_15"><i><font color="Green" title="E91">P56</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:68_2_2_15"/>
<font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c18">n</font>
 <b>by </b><i><a class="txt" href="#E47:68_2_2"><i><font color="Green" title="E86">P4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E49:68_2_2"/><b>then </b><i><font color="Green" title="E87">X1</font></i>: 
<font color="Maroon" title="c17">I</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>
 <b>by </b><i><a class="ref" href="rinfsup2.html#T7" title="RINFSUP2:th.7">RINFSUP2:7</a></i>;<br/><a NAME="E50:68_2_2"/><i><font color="Green" title="E88">R3</font></i>: 
for <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="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> holds <br/><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">x</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:()"><i><font color="Red">::  thesis: </font></i></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="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> implies <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> )</span><br/>

<b>assume </b><a NAME="E1:68_2_2_16"/><i><font color="Green" title="E89">Q0</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a></span><br/>

<a NAME="E2:68_2_2_16"/>
<font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">E</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<b>hence </b><a NAME="E3:68_2_2_16"/>
<font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="txt" href="#E6:68"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="#E1:68_2_2_16"><i><font color="Green" title="E89">Q0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E51:68_2_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c14">G</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E34:68_2_2"><i><font color="Green" title="E80">P2</font></i></a>, <a class="ref" href="mesfunc8.html#D10" title="MESFUNC8:def.10">MESFUNC8:def 10</a></i>;<br/><a NAME="E52:68_2_2"/><b>then </b><i><font color="Green" title="E89">R5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c14">G</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E21:68_2_2"><i><font color="Green" title="E73">G2</font></i></a></i>;<br/><a NAME="E53:68_2_2"/><i><font color="Green" title="E90">R2</font></i>: 
<font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E52">A0</font></i></a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E54:68_2_2"/>
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c15">J</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c15">J</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c18">n</font> )</span><br/>

<b>assume </b><a NAME="E1:68_2_2_17"/>
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c15">J</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c18">n</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c19">n1</font> = <font color="Maroon" title="c18">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>  ;<br/>
<a NAME="E3:68_2_2_17"/><i><font color="Green" title="E91">R64</font></i>: 
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" title="c15">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_2_17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c20">n</font></span>)</span></span><br/>

<a NAME="E1:68_2_2_17_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E31:68_2_2"><i><font color="Green" title="E77">Y1</font></i></a></i>;<br/>
<a NAME="E2:68_2_2_17_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p2">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c20">n</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/>
<b>hence </b><a NAME="E3:68_2_2_17_1"/>
<font color="Maroon" title="c15">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c20">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c20">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E43:68_2_2"><i><font color="Green" title="E84">Q2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:68_2_2_17"/>
<font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E31:68_2_2"><i><font color="Green" title="E77">Y1</font></i></a></i>;<br/>
<a NAME="E5:68_2_2_17"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c15">J</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c19">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c14">G</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E40:68_2_2"><i><font color="Green" title="E83">V3</font></i></a>, <a class="txt" href="#E3:68_2_2_17"><i><font color="Green" title="E91">R64</font></i></a>, <a class="txt" href="#E32:68_2_2"><i><font color="Green" title="E78">Z1</font></i></a>, <a class="ref" href="mesfunc9.html#T46" target="_self" title="MESFUNC9:th.46">FSUM3</a></i>;<br/>
<a NAME="E6:68_2_2_17"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c15">J</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c19">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E33:68_2_2"><i><font color="Green" title="E79">X0</font></i></a></i>;<br/>
<a NAME="E7:68_2_2_17"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c15">J</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c19">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">n1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c7">E0</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E20:68_2_2"><i><font color="Green" title="E72">G1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:68_2_2_17"/>
<font color="Maroon" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c15">J</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c18">n</font>
 <b>by </b><i><a class="txt" href="#E45:68_2_2"><i><font color="Green" title="E85">P3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E55:68_2_2"/><b>then </b>
<font color="Maroon" title="c17">I</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c15">J</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/><a NAME="E56:68_2_2"/><b>then </b>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c17">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c10">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E34:68_2_2"><i><font color="Green" title="E80">P2</font></i></a>, <a class="txt" href="#E52:68_2_2"><i><font color="Green" title="E89">R5</font></i></a>, <a class="txt" href="#E43:68_2_2"><i><font color="Green" title="E84">Q2</font></i></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/><b>hence </b><a NAME="E57:68_2_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" title="c17">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) &amp; <font color="Maroon" title="c17">I</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c17">I</font> )
 <b>by </b><i><a class="txt" href="#E47:68_2_2"><i><font color="Green" title="E86">P4</font></i></a>, <a class="txt" href="#E49:68_2_2"><i><font color="Green" title="E87">X1</font></i></a>, <a class="txt" href="#E16:68_2_2"><i><font color="Green" title="E69">LL1</font></i></a>, <a class="txt" href="#E53:68_2_2"><i><font color="Green" title="E90">R2</font></i></a>, <a class="txt" href="#E20:68_2_2"><i><font color="Green" title="E72">G1</font></i></a>, <a class="txt" href="#E50:68_2_2"><i><font color="Green" title="E88">R3</font></i></a>, <a class="ref" href="mesfunc9.html#T19" target="_self" title="MESFUNC9:th.19">Lm15</a>, <a class="ref" href="rinfsup2.html#T37" title="RINFSUP2:th.37">RINFSUP2:37</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
