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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> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</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 <br/> ( <font color="Olive" title="b4">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b5">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <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">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="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</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"><span class="p2">(<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> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">E</font></span>)</span> ) &amp; <font color="Olive" title="b5">I</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="Olive" title="b2">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="Olive" title="b4">F</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">E</font></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="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> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</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 <br/> ( <font color="Olive" title="b3">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b4">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <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">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="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</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"><span class="p2">(<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> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">E</font></span>)</span> ) &amp; <font color="Olive" title="b4">I</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="Olive" title="b1">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="Olive" title="b3">F</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">E</font></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="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> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</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 <br/> ( <font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <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">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="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</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"><span class="p2">(<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> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E</font></span>)</span> ) &amp; <font color="Olive" title="b3">I</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="Olive" title="b2">F</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">E</font></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="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> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</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 <br/> ( <font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <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">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="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</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"><span class="p2">(<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> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span> ) &amp; <font color="Olive" title="b2">I</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="Olive" title="b1">F</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></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="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> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</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 <br/> ( <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> &amp; <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">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="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</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"><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="Olive" title="b2">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span> ) &amp; <font color="Olive" title="b1">I</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="c5">F</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></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="Olive" title="b1">I</font> ) )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:66"/><i><font color="Green" title="E51">A1</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="E2:66"/><i><font color="Green" title="E52">S1</font></i>: 
<font color="Maroon" title="c5">F</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a>
 <b>and </b><br/><a NAME="E3:66"/><i><font color="Green" title="E53">S2</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="E4:66"/><i><font color="Green" title="E54">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 <br/> ( <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> &amp; <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="E5:66"/><i><font color="Green" title="E55">A3</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="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</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"><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="Olive" title="b2">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span> ) &amp; <font color="Olive" title="b1">I</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="c5">F</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></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="Olive" title="b1">I</font> )</span><br/>

<a NAME="E6:66"/><i><font color="Green" title="E56">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  <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="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a>
 
<b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="mesfunc5.html#T18" title="MESFUNC5:th.18">MESFUNC5:18</a></i>;<br/>
<a NAME="E7:66"/><b>then </b><i><font color="Green" title="E57">G3</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  <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <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>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="mesfunc9.html#T41" target="_self" title="MESFUNC9:th.41">ADD1</a></i>;<br/>
<a NAME="E8:66"/><b>then </b><i><font color="Green" title="E58">A4</font></i>: 
 <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="c5">F</font></span>)</span> <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="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E2:66"><i><font color="Green" title="E52">S1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="txt" href="#E5:66"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="mesfunc9.html#T44" target="_self" title="MESFUNC9:th.44">ADD2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">FF</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="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><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>)</span><b> such that </b><br/><a NAME="E10:66"/><i><font color="Green" title="E59">A5</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/> ( ( for <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</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> <font color="Olive" title="b1">n</font></span>)</span> ) ) &amp; ( for <font color="Olive" title="b2">m</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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> ) &amp; ( for <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</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>  <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="Olive" title="b1">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">j</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"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">k</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>  <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="Olive" title="b1">n</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Olive" title="b1">n</font></span>)</span> <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> &amp;  <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font></span>)</span> <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="Olive" title="b1">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">x</font> ) ) )
 <b>by </b><i><a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="txt" href="#E4:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="mesfunc9.html#T49" target="_self" title="MESFUNC9:th.49">Th131z</a></i>;<br/>
<a NAME="E11:66"/><i><font color="Green" title="E60">A6</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">x</font> 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>  <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="Olive" title="b1">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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"> 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="c7">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a></span><br/><b>let </b><font color="Maroon" title="c8">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="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="c7">n</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> )</span><br/>



<b>assume </b><a NAME="E1:66_1"/><i><font color="Green" title="E61">A61</font></i>: 
<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="c7">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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a></span><br/>

<b>let </b><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="valued_0.html#D15" title="VALUED_0:def.15">VALUED_0:def 15</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="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> or  not <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> or  not <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> )</span><br/>

<b>assume </b><a NAME="E2:66_1"/><i><font color="Green" title="E62">Z</font></i>: 
( <font color="Maroon" title="c9">i</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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> &amp; <font color="Maroon" title="c10">j</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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> &amp; <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</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"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c11">i</font> = <font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c12">j</font> = <font color="Maroon" title="c10">j</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E4:66_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">x</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="#E10:66"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="#E1:66_1"><i><font color="Green" title="E61">A61</font></i></a>, <a class="txt" href="#E2:66_1"><i><font color="Green" title="E62">Z</font></i></a></i>;<br/>
<a NAME="E5:66_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">i</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c8">x</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="E6:66_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</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="E12:66"/><i><font color="Green" title="E61">A7</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="c5">F</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <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="E13:66"/><b>then </b><i><font color="Green" title="E62">A72</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="c5">F</font></span>)</span></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="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="mesfunc9.html#D4" target="_self" title="MESFUNC9:def.4">Def0</a></i>;<br/>
<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="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>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="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 holds <br/>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font>;<br/>
<a NAME="E14:66"/><i><font color="Green" title="E63">Z1</font></i>: 
for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</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="b3">F1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font>,<font color="Olive" title="b3">F1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i1</font>, <font color="Maroon" title="c8">j1</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">F1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">j1</font>,<font color="Olive" title="b1">F1</font>]</span><br/>



<b>reconsider </b><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c7">i1</font>, <font color="Maroon" title="c10">j</font> = <font color="Maroon" title="c8">j1</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">F1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">F1</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="c7">i1</font>,<font color="Maroon" title="c8">j1</font>,<font color="Maroon" title="c11">F1</font>]</span><br/>

<b>thus </b><a NAME="E3:66_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>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j1</font> holds <br/><font color="Maroon" title="c11">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</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="c7">FF2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <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> such that </b><br/><a NAME="E16:66"/><i><font color="Green" title="E64">Z2</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</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">i</font>,<font color="Olive" title="b2">j</font>,<font color="Maroon" title="c7">FF2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="#E14:66"><i><font color="Green" title="E63">Z1</font></i></a>);<br/></i>
<a NAME="E17:66"/><i><font color="Green" title="E65">Q1</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/> ( ( for <font color="Olive" title="b2">m</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> &amp; <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> ) ) &amp;  <a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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"> ( ( for <font color="Olive" title="b1">m</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> &amp; <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> ) ) &amp;  <a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> )</span><br/>

<div><i><font color="Green" title="E66">T0</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> &amp; <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c10">n1</font> = <font color="Maroon" title="c8">n</font>, <font color="Maroon" title="c11">m1</font> = <font color="Maroon" title="c9">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E2:66_3_1"/><i><font color="Green" title="E67">T1</font></i>: 
( <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">FF2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c9">m</font> &amp; <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">FF2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">m</font>,<font color="Maroon" title="c8">n</font> )
 <b>by </b><i><a class="ref" href="mesfunc9.html#D8" target="_self" title="MESFUNC9:def.8">defproj1</a>, <a class="ref" href="mesfunc9.html#D9" target="_self" title="MESFUNC9:def.9">defproj2</a></i>;<br/><a NAME="E3:66_3_1"/><i><font color="Green" title="E68">T2</font></i>: 
( <font color="Maroon" title="c7">FF2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <font color="Maroon" title="c10">n1</font>,<font color="Maroon" title="c11">m1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c10">n1</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c7">FF2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <font color="Maroon" title="c11">m1</font>,<font color="Maroon" title="c10">n1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c11">m1</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font> )
 <b>by </b><i><a class="txt" href="#E16:66"><i><font color="Green" title="E64">Z2</font></i></a></i>;<br/><a NAME="E4:66_3_1"/>
(  <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">n1</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> <font color="Maroon" title="c11">m1</font></span>)</span> &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="c11">m1</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="txt" href="#E3:66"><i><font color="Green" title="E53">S2</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="E5:66_3_1"/>
(  <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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</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="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E2:66_3_1"><i><font color="Green" title="E67">T1</font></i></a>, <a class="txt" href="#E3:66_3_1"><i><font color="Green" title="E68">T2</font></i></a>, <a class="txt" href="#E10:66"><i><font color="Green" title="E59">A5</font></i></a></i>; <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="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )</span><br/><b>thus </b><a NAME="E6:66_3_1"/>
( <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )
 <b>by </b><i><a class="txt" href="#E10:66"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="#E2:66_3_1"><i><font color="Green" title="E67">T1</font></i></a>, <a class="txt" href="#E3:66_3_1"><i><font color="Green" title="E68">T2</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="E2:66_3"/>
for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i1</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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">j1</font></span>)</span> &amp;  <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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i1</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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">j1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i1</font>, <font color="Maroon" title="c10">j1</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">i1</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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">j1</font></span>)</span> &amp;  <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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">i1</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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">j1</font></span>)</span> )</span><br/>

<a NAME="E1:66_3_2"/>
(  <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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">i1</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">j1</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">i1</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">j1</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="#E1:66_3"><i><font color="Green" title="E66">T0</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:66_3_2"/>
(  <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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">i1</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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">j1</font></span>)</span> &amp;  <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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">i1</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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">j1</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>
<b>hence </b><a NAME="E3:66_3"/>
( ( for <font color="Olive" title="b1">m</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> &amp; <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> ) ) &amp;  <a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> )
 <b>by </b><i><a class="ref" href="mesfunc9.html#T35" target="_self" title="MESFUNC9:th.35">Lem10</a>, <a class="txt" href="#E1:66_3"><i><font color="Green" title="E66">T0</font></i></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</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="E18:66"/><i><font color="Green" title="E66">K2</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> holds <br/> ( <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</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="66_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">n</font>, <font color="Maroon" title="c9">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"> ( <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c10">n1</font> = <font color="Maroon" title="c8">n</font>, <font color="Maroon" title="c11">m1</font> = <font color="Maroon" title="c9">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:66_4"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">FF2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <font color="Maroon" title="c10">n1</font>,<font color="Maroon" title="c11">m1</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D8" target="_self" title="MESFUNC9:def.8">defproj1</a></i>;<br/>
<a NAME="E3:66_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font>
 
<b>by </b><i><a class="txt" href="#E16:66"><i><font color="Green" title="E64">Z2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:66_4"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="#E10:66"><i><font color="Green" title="E59">A5</font></i></a></i>; <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="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a></span><br/>

<a NAME="E5:66_4"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">FF2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <font color="Maroon" title="c11">m1</font>,<font color="Maroon" title="c10">n1</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D9" target="_self" title="MESFUNC9:def.9">defproj2</a></i>;<br/>
<a NAME="E6:66_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font>
 
<b>by </b><i><a class="txt" href="#E16:66"><i><font color="Green" title="E64">Z2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:66_4"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">m</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="#E10:66"><i><font color="Green" title="E59">A5</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="E19:66"/><i><font color="Green" title="E67">K3</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="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/>( <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font></span>)</span> <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> &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="Olive" title="b1">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_5"><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"> 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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <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> &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="c8">n</font></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>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c9">n1</font> = <font color="Maroon" title="c8">n</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E2:66_5"/><i><font color="Green" title="E68">KL0</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="c9">n1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/>
<b>let </b><font color="Maroon" title="c10">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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> implies ( <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &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="c8">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> ) )</span><br/>

<b>assume </b><a NAME="E3:66_5"/><i><font color="Green" title="E69">KK0</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &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="c8">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> )</span><br/>

<a NAME="E4:66_5"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">i</font>, <font color="Maroon" title="c12">j</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="valued_0.html#D15" title="VALUED_0:def.15">VALUED_0:def 15</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">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> or  not <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> or  not <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> )</span><br/>

<b>assume </b><a NAME="E1:66_5_1"/><i><font color="Green" title="E70">KK5</font></i>: 
( <font color="Maroon" title="c11">i</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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> &amp; <font color="Maroon" title="c12">j</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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c13">i1</font> = <font color="Maroon" title="c11">i</font>, <font color="Maroon" title="c14">j1</font> = <font color="Maroon" title="c12">j</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:66_5_1"/><i><font color="Green" title="E71">KK3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">i1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">x</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">j1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="txt" href="#E10:66"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="#E3:66_5"><i><font color="Green" title="E69">KK0</font></i></a>, <a class="txt" href="#E2:66_5"><i><font color="Green" title="E68">KL0</font></i></a>, <a class="txt" href="#E1:66_5_1"><i><font color="Green" title="E70">KK5</font></i></a></i>;<br/>
<a NAME="E4:66_5_1"/><i><font color="Green" title="E72">KK1</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">i1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">j1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">x</font> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E5:66_5_1"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">i1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">FF2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c13">i1</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D8" target="_self" title="MESFUNC9:def.8">defproj1</a></i>;<br/>
<a NAME="E6:66_5_1"/><b>then </b><i><font color="Green" title="E73">KK2</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">i1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">i1</font>
 
<b>by </b><i><a class="txt" href="#E16:66"><i><font color="Green" title="E64">Z2</font></i></a></i>;<br/>
<a NAME="E7:66_5_1"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">j1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">FF2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c14">j1</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D8" target="_self" title="MESFUNC9:def.8">defproj1</a></i>;<br/>
<a NAME="E8:66_5_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">i1</font> <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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">j1</font>
 
<b>by </b><i><a class="txt" href="#E16:66"><i><font color="Green" title="E64">Z2</font></i></a>, <a class="txt" href="#E4:66_5_1"><i><font color="Green" title="E72">KK1</font></i></a>, <a class="txt" href="#E6:66_5_1"><i><font color="Green" title="E73">KK2</font></i></a>, <a class="txt" href="#E3:66_5_1"><i><font color="Green" title="E71">KK3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:66_5_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</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>hence </b><a NAME="E5:66_5"/><i><font color="Green" title="E70">K31</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><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"> <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="c8">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span></span><br/>

<a NAME="E6:66_5"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  ex <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</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#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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">  ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )</span><br/>

<b>take </b><font color="Maroon" title="c12">m</font> = <font color="Maroon" title="c11">k</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">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )</span><br/>

<a NAME="E1:66_5_2"/><i><font color="Green" title="E71">K321</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E2:66_5_2"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">m</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="#E18:66"><i><font color="Green" title="E66">K2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:66_5_2"/>
( <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">m</font> )
 <b>by </b><i><a class="txt" href="#E1:66_5_2"><i><font color="Green" title="E71">K321</font></i></a>, <a class="ref" href="supinf_2.html#T58" title="SUPINF_2:th.58">SUPINF_2:58</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="E7:66_5"/><b>then </b><i><font color="Green" title="E71">K33</font></i>: 
not <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D10" title="MESFUNC5:def.10">MESFUNC5:def 10</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_5_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_5_3"/>
( <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> or <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a> )
 </span><b>by </b><i><a class="txt" href="#E5:66_5"><i><font color="Green" title="E70">K31</font></i></a>, <a class="txt" href="#E7:66_5"><i><font color="Green" title="E71">K33</font></i></a>, <a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_5_3_1"><b>suppose </b></a><a NAME="E1:66_5_3_1"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</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#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:66_5_3_1"/><b>then </b>
( not <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a> &amp; not <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T56" title="MESFUNC5:th.56">MESFUNC5:56</a>, <a class="ref" href="mesfunc5.html#T57" title="MESFUNC5:th.57">MESFUNC5:57</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">lP</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:66_5_3_1"/><i><font color="Green" title="E72">T2</font></i>: 
(  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">lP</font> &amp; ( for <font color="Olive" title="b1">p</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <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="Olive" title="b1">p</font> holds <br/> ex <font color="Olive" title="b2">nn</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b3">mm</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">mm</font> 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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b3">mm</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> ) &amp; <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> )
 <b>by </b><i><a class="txt" href="#E5:66_5"><i><font color="Green" title="E70">K31</font></i></a>, <a class="ref" href="mesfunc5.html#D12" title="MESFUNC5:def.12">MESFUNC5:def 12</a></i>;<br/><a NAME="E5:66_5_3_1"/><i><font color="Green" title="E73">T3</font></i>: 
for <font color="Olive" title="b1">p</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <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="Olive" title="b1">p</font> holds <br/> ex <font color="Olive" title="b2">nn</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b3">mm</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">mm</font> 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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b3">mm</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c11">lP</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_5_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">p</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">p</font> implies  ex <font color="Olive" title="b1">nn</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">mm</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">mm</font> 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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">mm</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c11">lP</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">p</font> )</span><br/>

<b>assume </b><a NAME="E1:66_5_3_1_1"/>
 <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="c12">p</font>
 ; <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">nn</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">mm</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">mm</font> 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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">mm</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c11">lP</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">p</font></span><br/>

<b>then consider </b><font color="Maroon" title="c13">nn</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:66_5_3_1_1"/><i><font color="Green" title="E74">T31</font></i>: 
for <font color="Olive" title="b1">mm</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c13">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">mm</font> 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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">mm</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="#E4:66_5_3_1"><i><font color="Green" title="E72">T2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">nn</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">mm</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c13">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">mm</font> 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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">mm</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c11">lP</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">p</font></span><br/>

<b>let </b><font color="Maroon" title="c14">mm</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="c13">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">mm</font> 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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">mm</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c11">lP</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">p</font> )</span><br/>

<b>assume </b><a NAME="E4:66_5_3_1_1"/><i><font color="Green" title="E75">T32</font></i>: 
<font color="Maroon" title="c13">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">mm</font>
 ; <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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">mm</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c11">lP</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">p</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c15">mm1</font> = <font color="Maroon" title="c14">mm</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="E6:66_5_3_1_1"/><i><font color="Green" title="E76">T33</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">mm</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E7:66_5_3_1_1"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">FF2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c14">mm</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D8" target="_self" title="MESFUNC9:def.8">defproj1</a></i>;<br/>
<a NAME="E8:66_5_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">mm</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">mm1</font>
 
<b>by </b><i><a class="txt" href="#E16:66"><i><font color="Green" title="E64">Z2</font></i></a></i>;<br/>
<a NAME="E9:66_5_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">mm</font>
 
<b>by </b><i><a class="txt" href="#E6:66_5_3_1_1"><i><font color="Green" title="E76">T33</font></i></a>, <a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<b>hence </b><a NAME="E10:66_5_3_1_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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">mm</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c11">lP</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="#E4:66_5_3_1_1"><i><font color="Green" title="E75">T32</font></i></a>, <a class="txt" href="#E3:66_5_3_1_1"><i><font color="Green" title="E74">T31</font></i></a>, <a class="txt" href="#E4:66_5_3_1"><i><font color="Green" title="E72">T2</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="E6:66_5_3_1"/><b>then </b><i><font color="Green" title="E74">T4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D8" title="MESFUNC5:def.8">MESFUNC5:def 8</a></i>;<br/><a NAME="E7:66_5_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</a></i>;<br/><a NAME="E8:66_5_3_1"/><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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c11">lP</font>
 <b>by </b><i><a class="txt" href="#E5:66_5_3_1"><i><font color="Green" title="E73">T3</font></i></a>, <a class="txt" href="#E6:66_5_3_1"><i><font color="Green" title="E74">T4</font></i></a>, <a class="ref" href="mesfunc5.html#D12" title="MESFUNC5:def.12">MESFUNC5:def 12</a></i>;<br/><b>hence </b><a NAME="E9:66_5_3_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="c8">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:66"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="#E4:66_5_3_1"><i><font color="Green" title="E72">T2</font></i></a>, <a class="txt" href="#E2:66_5"><i><font color="Green" title="E68">KL0</font></i></a>, <a class="txt" href="#E3:66_5"><i><font color="Green" title="E69">KK0</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_5_3_2"><b>suppose </b></a><a NAME="E1:66_5_3_2"/><i><font color="Green" title="E72">KK1</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</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#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:66_5_3_2"/><b>then </b><i><font color="Green" title="E73">KK5</font></i>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</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="#E5:66_5"><i><font color="Green" title="E70">K31</font></i></a>, <a class="ref" href="mesfunc5.html#D12" title="MESFUNC5:def.12">MESFUNC5:def 12</a></i>;<br/><a NAME="E3:66_5_3_2"/>
for <font color="Olive" title="b1">g</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <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="Olive" title="b1">g</font> holds <br/> ex <font color="Olive" title="b2">nn</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b3">mm</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">mm</font> holds <br/><font color="Olive" title="b1">g</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b3">mm</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_5_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">g</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">g</font> implies  ex <font color="Olive" title="b1">nn</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">mm</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">mm</font> holds <br/><font color="Maroon" title="c11">g</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">mm</font> )</span><br/>

<b>assume </b><a NAME="E1:66_5_3_2_1"/>
 <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="c11">g</font>
 ; <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">nn</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>for <font color="Olive" title="b2">mm</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">mm</font> holds <br/><font color="Maroon" title="c11">g</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">mm</font></span><br/>

<b>then consider </b><font color="Maroon" title="c12">nn</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:66_5_3_2_1"/><i><font color="Green" title="E74">KK8</font></i>: 
for <font color="Olive" title="b1">mm</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c12">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">mm</font> holds <br/><font color="Maroon" title="c11">g</font> <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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">mm</font>
 <b>by </b><i><a class="txt" href="#E1:66_5_3_2"><i><font color="Green" title="E72">KK1</font></i></a>, <a class="ref" href="mesfunc5.html#D9" title="MESFUNC5:def.9">MESFUNC5:def 9</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">nn</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">mm</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c12">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">mm</font> holds <br/><font color="Maroon" title="c11">g</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">mm</font></span><br/>

<b>let </b><font color="Maroon" title="c13">mm</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">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">mm</font> implies <font color="Maroon" title="c11">g</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">mm</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c14">mm1</font> = <font color="Maroon" title="c13">mm</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/>
<b>assume </b><a NAME="E5:66_5_3_2_1"/>
<font color="Maroon" title="c12">nn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">mm</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">g</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">mm</font></span><br/>

<a NAME="E6:66_5_3_2_1"/><b>then </b><i><font color="Green" title="E75">KK9</font></i>: 
<font color="Maroon" title="c11">g</font> <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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">mm</font>
 
<b>by </b><i><a class="txt" href="#E3:66_5_3_2_1"><i><font color="Green" title="E74">KK8</font></i></a></i>;<br/>
<a NAME="E7:66_5_3_2_1"/><i><font color="Green" title="E76">KK10</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">mm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">mm</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E8:66_5_3_2_1"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">mm</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">FF2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c14">mm1</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D8" target="_self" title="MESFUNC9:def.8">defproj1</a></i>;<br/>
<a NAME="E9:66_5_3_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">mm</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">mm</font>
 
<b>by </b><i><a class="txt" href="#E16:66"><i><font color="Green" title="E64">Z2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:66_5_3_2_1"/>
<font color="Maroon" title="c11">g</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">mm</font>
 <b>by </b><i><a class="txt" href="#E6:66_5_3_2_1"><i><font color="Green" title="E75">KK9</font></i></a>, <a class="txt" href="#E7:66_5_3_2_1"><i><font color="Green" title="E76">KK10</font></i></a>, <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="E4:66_5_3_2"/><b>then </b><i><font color="Green" title="E74">KK2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D9" title="MESFUNC5:def.9">MESFUNC5:def 9</a></i>;<br/><a NAME="E5:66_5_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</a></i>;<br/><a NAME="E6:66_5_3_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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</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="#E4:66_5_3_2"><i><font color="Green" title="E74">KK2</font></i></a>, <a class="ref" href="mesfunc5.html#D12" title="MESFUNC5:def.12">MESFUNC5:def 12</a></i>;<br/><b>hence </b><a NAME="E7:66_5_3_2"/>
<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="c8">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c10">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:66"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="#E3:66_5"><i><font color="Green" title="E69">KK0</font></i></a>, <a class="txt" href="#E2:66_5"><i><font color="Green" title="E68">KL0</font></i></a>, <a class="txt" href="#E2:66_5_3_2"><i><font color="Green" title="E73">KK5</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></div><b>end;</b></div>

</div><b>end;</b></div>
<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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,$1</span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c8">P</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="E21:66"/><i><font color="Green" title="E68">K5</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="c8">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">k</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">k</font>)
 <b>from </b><i><a class="ref" href="seqfunc.html#S1" title="SEQFUNC:sch.1">SEQFUNC:sch 1</a>();<br/></i>
<a NAME="E22:66"/>
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="c8">P</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="c8">P</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="66_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">n</font>, <font color="Maroon" title="c10">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="c8">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">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="c8">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span><br/>

<a NAME="E1:66_6"/><i><font color="Green" title="E69">K51</font></i>: 
(  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c9">n</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c9">n</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">m</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">m</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> )
 
<b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a></i>;<br/>
<a NAME="E2:66_6"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E21:66"><i><font color="Green" title="E68">K5</font></i></a></i>;<br/>
<a NAME="E3:66_6"/><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="c8">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c9">n</font></span>)</span> <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="#E1:66_6"><i><font color="Green" title="E69">K51</font></i></a>, <a class="ref" href="mesfunc9.html#T29" target="_self" title="MESFUNC9:th.29">ADD0</a></i>;<br/>
<a NAME="E4:66_6"/><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="c8">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">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="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a></i>;<br/>
<a NAME="E5:66_6"/><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="c8">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">m</font></span>)</span> <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="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a></i>;<br/>
<a NAME="E6:66_6"/><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="c8">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">m</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:66_6"><i><font color="Green" title="E69">K51</font></i></a>, <a class="ref" href="mesfunc9.html#T29" target="_self" title="MESFUNC9:th.29">ADD0</a></i>;<br/>
<b>hence </b><a NAME="E7:66_6"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c9">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="c8">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E21:66"><i><font color="Green" title="E68">K5</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="c9">P</font> = <font color="Maroon" title="c8">P</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/>
<a NAME="E24:66"/><i><font color="Green" title="E69">K6</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">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_7"><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">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E1:66_7"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a></i>;<br/>
<a NAME="E2:66_7"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#T35" target="_self" title="MESFUNC9:th.35">Lem10</a></i>;<br/>
<b>hence </b><a NAME="E3:66_7"/>
<font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c10">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="#E21:66"><i><font color="Green" title="E68">K5</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="E25:66"/><i><font color="Green" title="E70">K7</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">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">i</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b2">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b4">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_8"><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"> ( <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font> implies for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> 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/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">x</font> <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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">x</font> )</span><br/>

<b>assume </b><a NAME="E1:66_8"/><i><font color="Green" title="E71">J21</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</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">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> 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/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">x</font> <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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">x</font></span><br/>

<b>let </b><font color="Maroon" title="c12">i</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"> 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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><b>let </b><font color="Maroon" title="c13">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="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font> <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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font> )</span><br/>



<b>assume </b><a NAME="E2:66_8"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font> <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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font></span><br/>

<a NAME="E3:66_8"/><b>then </b><i><font color="Green" title="E72">JJ0</font></i>: 
<font color="Maroon" title="c13">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="c12">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <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">i1</font> = <font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c15">n1</font> = <font color="Maroon" title="c10">n</font>, <font color="Maroon" title="c16">m1</font> = <font color="Maroon" title="c11">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:66_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">FF2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <font color="Maroon" title="c14">i1</font>,<font color="Maroon" title="c15">n1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D9" target="_self" title="MESFUNC9:def.9">defproj2</a></i>;<br/>
<a NAME="E6:66_8"/><b>then </b><i><font color="Green" title="E73">JJ1</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <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="c13">x</font>
 
<b>by </b><i><a class="txt" href="#E16:66"><i><font color="Green" title="E64">Z2</font></i></a></i>;<br/>
<a NAME="E7:66_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">FF2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <font color="Maroon" title="c14">i1</font>,<font color="Maroon" title="c16">m1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D9" target="_self" title="MESFUNC9:def.9">defproj2</a></i>;<br/>
<a NAME="E8:66_8"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <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="Maroon" title="c13">x</font>
 
<b>by </b><i><a class="txt" href="#E16:66"><i><font color="Green" title="E64">Z2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:66_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font> <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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="#E1:66_8"><i><font color="Green" title="E71">J21</font></i></a>, <a class="txt" href="#E3:66_8"><i><font color="Green" title="E72">JJ0</font></i></a>, <a class="txt" href="#E6:66_8"><i><font color="Green" title="E73">JJ1</font></i></a>, <a class="txt" href="#E10:66"><i><font color="Green" title="E59">A5</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="E26:66"/><i><font color="Green" title="E71">K8</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">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">i</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b2">m</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b4">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_9"><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"> ( <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font> implies for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> 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/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">x</font> )</span><br/>

<b>assume </b><a NAME="E1:66_9"/><i><font color="Green" title="E72">K81</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</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">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> 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/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">x</font></span><br/>

<b>let </b><font color="Maroon" title="c12">i</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"> 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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><b>let </b><font color="Maroon" title="c13">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="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font> )</span><br/>



<b>assume </b><a NAME="E2:66_9"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font></span><br/>

<a NAME="E3:66_9"/><b>then </b>
( <font color="Maroon" title="c13">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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <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="c13">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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <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="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a></i>;<br/>
<a NAME="E4:66_9"/><b>then </b><i><font color="Green" title="E73">K83</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 
<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="E5:66_9"/><i><font color="Green" title="E74">K82</font></i>: 
(  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> )
 
<b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a></i>;<br/>
<a NAME="E6:66_9"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> 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> <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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">x</font> <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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</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"> 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> <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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">i</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><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> <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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">i</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">i</font></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="E1:66_9_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">i</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">x</font></span><br/>

<a NAME="E2:66_9_1"/><b>then </b>
<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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <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="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:66_9_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:66_9_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">i</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">i</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="#E25:66"><i><font color="Green" title="E70">K7</font></i></a>, <a class="txt" href="#E1:66_9"><i><font color="Green" title="E72">K81</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="E7:66_9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="#E5:66_9"><i><font color="Green" title="E74">K82</font></i></a>, <a class="txt" href="#E4:66_9"><i><font color="Green" title="E73">K83</font></i></a>, <a class="ref" href="mesfunc9.html#T42" target="_self" title="MESFUNC9:th.42">ADD4</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="E27:66"/><i><font color="Green" title="E72">K9</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="c9">P</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="c9">P</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_10"><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"> ( <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font> implies 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="c9">P</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</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> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c12">n1</font> = <font color="Maroon" title="c10">n</font>, <font color="Maroon" title="c13">m1</font> = <font color="Maroon" title="c11">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:66_10"/><i><font color="Green" title="E73">K91</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</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">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="c9">P</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</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></span><br/>

<b>let </b><font color="Maroon" title="c14">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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</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="c14">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="c9">P</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="Maroon" title="c14">x</font> )</span><br/>

<b>assume </b><a NAME="E3:66_10"/><i><font color="Green" title="E74">K92</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</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="c9">P</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="c14">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="c9">P</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="Maroon" title="c14">x</font></span><br/>

<a NAME="E4:66_10"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <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="c14">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <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="c14">x</font>
 
<b>by </b><i><a class="txt" href="#E2:66_10"><i><font color="Green" title="E73">K91</font></i></a>, <a class="txt" href="#E26:66"><i><font color="Green" title="E71">K8</font></i></a></i>;<br/>
<a NAME="E5:66_10"/><b>then </b><i><font color="Green" title="E75">K93</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</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="c14">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <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="c14">x</font>
 
<b>by </b><i><a class="txt" href="#E21:66"><i><font color="Green" title="E68">K5</font></i></a></i>;<br/>
<a NAME="E6:66_10"/><i><font color="Green" title="E76">K94</font></i>: 
(  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> )
 
<b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a></i>;<br/>
<a NAME="E7:66_10"/><i><font color="Green" title="E77">K95</font></i>: 
<font color="Maroon" title="c14">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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <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="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a>, <a class="txt" href="#E3:66_10"><i><font color="Green" title="E74">K92</font></i></a></i>;<br/>
<a NAME="E8:66_10"/>
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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span> <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>
 
<b>by </b><i><a class="txt" href="#E18:66"><i><font color="Green" title="E66">K2</font></i></a></i>;<br/>
<a NAME="E9:66_10"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>
 
<b>by </b><i><a class="txt" href="#E6:66_10"><i><font color="Green" title="E76">K94</font></i></a>, <a class="txt" href="#E7:66_10"><i><font color="Green" title="E77">K95</font></i></a>, <a class="ref" href="mesfunc9.html#T38" target="_self" title="MESFUNC9:th.38">ADD3</a></i>;<br/>
<a NAME="E10:66_10"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n1</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">m1</font>
 
<b>by </b><i><a class="txt" href="#E2:66_10"><i><font color="Green" title="E73">K91</font></i></a>, <a class="ref" href="rinfsup2.html#T7" title="RINFSUP2:th.7">RINFSUP2:7</a></i>;<br/>
<a NAME="E11:66_10"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <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="c14">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">m1</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E12:66_10"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <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="c14">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <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="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E13:66_10"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <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="c14">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="c9">P</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="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="txt" href="#E21:66"><i><font color="Green" title="E68">K5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:66_10"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</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="c14">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="c9">P</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="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="#E5:66_10"><i><font color="Green" title="E75">K93</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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="E28:66"/><i><font color="Green" title="E73">J1</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">p</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">n</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b2">p</font></span>)</span></span>)</span> <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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b3">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b2">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b3">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b2">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">p</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b3">x</font> &amp; <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">p</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"><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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b3">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">n</font>, <font color="Maroon" title="c11">p</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">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">n</font> implies 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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <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#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</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"><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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c12">p1</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c13">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/>
<b>assume </b><a NAME="E2:66_11"/><i><font color="Green" title="E74">BB1</font></i>: 
<font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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"> 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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <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#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</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"><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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font> )</span><br/>

<b>let </b><font color="Maroon" title="c14">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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> implies ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <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="c14">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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> &amp; <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> ) )</span><br/>

<b>assume </b><a NAME="E3:66_11"/><i><font color="Green" title="E75">BB0</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <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="c14">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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> &amp; <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> )</span><br/>

<a NAME="E4:66_11"/><b>then </b><i><font color="Green" title="E76">BB2</font></i>: 
<font color="Maroon" title="c14">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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span> <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="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a></i>;<br/>
<a NAME="E5:66_11"/>
(  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> ) )
 
<b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a>, <a class="txt" href="#E18:66"><i><font color="Green" title="E66">K2</font></i></a></i>;<br/>
<a NAME="E6:66_11"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>
 
<b>by </b><i><a class="txt" href="#E4:66_11"><i><font color="Green" title="E76">BB2</font></i></a>, <a class="ref" href="mesfunc9.html#T38" target="_self" title="MESFUNC9:th.38">ADD3</a></i>;<br/>
<a NAME="E7:66_11"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n1</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">p1</font>
 
<b>by </b><i><a class="txt" href="#E2:66_11"><i><font color="Green" title="E74">BB1</font></i></a>, <a class="ref" href="rinfsup2.html#T7" title="RINFSUP2:th.7">RINFSUP2:7</a></i>;<br/>
<a NAME="E8:66_11"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <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="c14">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">p</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E9:66_11"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <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="c14">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</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="E10:66_11"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <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="c14">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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="#E21:66"><i><font color="Green" title="E68">K5</font></i></a></i>; <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">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> &amp; <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> )</span><br/>

<b>thus </b><a NAME="E11:66_11"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="#E21:66"><i><font color="Green" title="E68">K5</font></i></a></i>; <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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> &amp; <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> )</span><br/>

<a NAME="E12:66_11"/><i><font color="Green" title="E77">ND31</font></i>: 
(  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> )
 
<b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a></i>;<br/>
<a NAME="E13:66_11"/><i><font color="Green" title="E78">ND32</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">x</font> 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> <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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="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></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span> <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="b2">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="b1">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_11_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"> 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> <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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="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></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">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#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="c15">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><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> <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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="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></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</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#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="c15">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font> )</span><br/>



<b>reconsider </b><font color="Maroon" title="c17">n1</font> = <font color="Maroon" title="c15">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/>
<b>assume </b><a NAME="E2:66_11_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="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></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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</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#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="c15">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">x</font></span><br/>

<a NAME="E3:66_11_1"/><b>then </b>
<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="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="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:66_11_1"/><b>then </b><i><font color="Green" title="E79">N322</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="c5">F</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="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/>
<a NAME="E5:66_11_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font></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"><font color="Maroon" title="c7">FF2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <font color="Maroon" title="c17">n1</font>,<font color="Maroon" title="c12">p1</font></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="mesfunc9.html#D9" target="_self" title="MESFUNC9:def.9">defproj2</a></i>;<br/>
<a NAME="E6:66_11_1"/><b>then </b><i><font color="Green" title="E80">N323</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font></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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></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="#E16:66"><i><font color="Green" title="E64">Z2</font></i></a></i>;<br/>
<a NAME="E7:66_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c16">x</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>
 
<b>by </b><i><a class="txt" href="#E4:66_11_1"><i><font color="Green" title="E79">N322</font></i></a>, <a class="txt" href="#E11:66"><i><font color="Green" title="E60">A6</font></i></a></i>;<br/>
<a NAME="E8:66_11_1"/><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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c16">x</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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c16">x</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="E9:66_11_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c16">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup2.html#T23" title="RINFSUP2:th.23">RINFSUP2:23</a></i>;<br/>
<a NAME="E10:66_11_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">p</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="c15">n</font></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="#E4:66_11_1"><i><font color="Green" title="E79">N322</font></i></a>, <a class="txt" href="#E10:66"><i><font color="Green" title="E59">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:66_11_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</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#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="c15">n</font></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="#E6:66_11_1"><i><font color="Green" title="E80">N323</font></i></a>, <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="E14:66_11"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="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></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:66_11"><i><font color="Green" title="E75">BB0</font></i></a>, <a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E4:66_11"><i><font color="Green" title="E76">BB2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E15:66_11"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="#E2:66"><i><font color="Green" title="E52">S1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="txt" href="#E12:66_11"><i><font color="Green" title="E77">ND31</font></i></a>, <a class="txt" href="#E13:66_11"><i><font color="Green" title="E78">ND32</font></i></a>, <a class="ref" href="mesfunc9.html#T42" target="_self" title="MESFUNC9:th.42">ADD4</a></i>; <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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font></span><br/>

<a NAME="E16:66_11"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>
 
<b>by </b><i><a class="txt" href="#E3:66_11"><i><font color="Green" title="E75">BB0</font></i></a>, <a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="txt" href="#E4:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="mesfunc9.html#T38" target="_self" title="MESFUNC9:th.38">ADD3</a></i>;<br/>
<a NAME="E17:66_11"/><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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</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="E18:66_11"/><b>then </b>
<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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup2.html#T23" title="RINFSUP2:th.23">RINFSUP2:23</a></i>;<br/>
<a NAME="E19:66_11"/><b>then </b>
<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="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x</font></span>)</span>
 
<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="E20:66_11"/>
<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="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="#E3:66_11"><i><font color="Green" title="E75">BB0</font></i></a>, <a class="txt" href="#E13:66"><i><font color="Green" title="E62">A72</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>
<a NAME="E29:66"/>
 <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="c9">P</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">P</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="E30:66"/><b>then </b>
 <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="c9">P</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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <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="#E21:66"><i><font color="Green" title="E68">K5</font></i></a></i>;<br/>
<a NAME="E31:66"/><b>then </b>
 <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="c9">P</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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <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="mesfunc9.html#D4" target="_self" title="MESFUNC9:def.4">Def0</a></i>;<br/>
<a NAME="E32:66"/><b>then </b>
 <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="c9">P</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="c7">FF2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D9" target="_self" title="MESFUNC9:def.9">defproj2</a></i>;<br/>
<a NAME="E33:66"/><b>then </b><i><font color="Green" title="E74">N1</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="c9">P</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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <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="#E16:66"><i><font color="Green" title="E64">Z2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">FF0</font> = <font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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>  ;<br/>
<a NAME="E35:66"/><i><font color="Green" title="E75">N21</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="c9">P</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="txt" href="#E10:66"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="#E33:66"><i><font color="Green" title="E74">N1</font></i></a></i>;<br/>
<a NAME="E36:66"/>
 <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="c5">F</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <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="E37:66"/><b>then </b><i><font color="Green" title="E76">N5</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="c5">F</font></span>)</span></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="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="mesfunc9.html#D4" target="_self" title="MESFUNC9:def.4">Def0</a></i>;<br/>
<a NAME="E38:66"/><i><font color="Green" title="E77">N8</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"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c9">P</font></span>)</span> holds <br/><font color="Maroon" title="c9">P</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="66_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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="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="c9">P</font></span>)</span> implies <font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> )</span><br/>

<b>assume </b><a NAME="E1:66_12"/><i><font color="Green" title="E78">N4</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="c9">P</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="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a></span><br/>

<a NAME="E2:66_12"/>
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="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</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="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</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="66_12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c13">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="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">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="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font> )</span><br/>

<b>assume </b><a NAME="E1:66_12_1"/>
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">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="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">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="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">n</font></span><br/>

<a NAME="E2:66_12_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">m</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">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="#E27:66"><i><font color="Green" title="E72">K9</font></i></a>, <a class="txt" href="#E1:66_12"><i><font color="Green" title="E78">N4</font></i></a>, <a class="txt" href="#E35:66"><i><font color="Green" title="E75">N21</font></i></a>, <a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a></i>;<br/>
<a NAME="E3:66_12_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">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="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<b>hence </b><a NAME="E4:66_12_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">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="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">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="E3:66_12"/><b>then </b>
<font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</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/>
<b>hence </b><a NAME="E4:66_12"/>
<font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><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>
<a NAME="E39:66"/><i><font color="Green" title="E78">HHH</font></i>: 
for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><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> <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">k</font></span>)</span> <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> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">k</font>, <font color="Maroon" title="c12">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"> 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> <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">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="c12">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><b>let </b><font color="Maroon" title="c13">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="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">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="c12">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font> )</span><br/>





<b>assume </b><a NAME="E1:66_13"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">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="c12">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font></span><br/>

<a NAME="E2:66_13"/><b>then </b>
<font color="Maroon" title="c13">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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<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="E3:66_13"/><b>then </b><i><font color="Green" title="E79">H1</font></i>: 
<font color="Maroon" title="c13">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="c12">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <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">m1</font> = <font color="Maroon" title="c12">m</font>, <font color="Maroon" title="c15">k1</font> = <font color="Maroon" title="c11">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:66_13"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">m</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">FF2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <font color="Maroon" title="c14">m1</font>,<font color="Maroon" title="c15">k1</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D9" target="_self" title="MESFUNC9:def.9">defproj2</a></i>;<br/>
<a NAME="E6:66_13"/><b>then </b><i><font color="Green" title="E80">H2</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">m</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c11">k</font>
 
<b>by </b><i><a class="txt" href="#E16:66"><i><font color="Green" title="E64">Z2</font></i></a></i>;<br/>
<a NAME="E7:66_13"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">j</font>, <font color="Maroon" title="c17">k</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="valued_0.html#D15" title="VALUED_0:def.15">VALUED_0:def 15</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="c16">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> or  not <font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> or  not <font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> )</span><br/>

<b>assume </b><a NAME="E1:66_13_1"/><i><font color="Green" title="E81">H3</font></i>: 
( <font color="Maroon" title="c16">j</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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> &amp; <font color="Maroon" title="c17">k</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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> &amp; <font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</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"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c16">j</font>, <font color="Maroon" title="c19">k</font> = <font color="Maroon" title="c17">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:66_13_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c18">j</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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c19">k</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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</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="E4:66_13_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font> <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font>
 <b>by </b><i><a class="txt" href="#E3:66_13"><i><font color="Green" title="E79">H1</font></i></a>, <a class="txt" href="#E1:66_13_1"><i><font color="Green" title="E81">H3</font></i></a>, <a class="txt" href="#E10:66"><i><font color="Green" title="E59">A5</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="E8:66_13"/><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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</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="E9:66_13"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rinfsup2.html#T23" title="RINFSUP2:th.23">RINFSUP2:23</a></i>;<br/>
<a NAME="E10:66_13"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">FF</font> <a href="mesfunc9.html#K10" title="MESFUNC9:func.10">.</a> <font color="Maroon" title="c14">m1</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">k</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="c14">m1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font>
 
<b>by </b><i><a class="txt" href="#E3:66_13"><i><font color="Green" title="E79">H1</font></i></a>, <a class="txt" href="#E10:66"><i><font color="Green" title="E59">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:66_13"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">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="c12">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="#E6:66_13"><i><font color="Green" title="E80">H2</font></i></a>, <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="E40:66"/><i><font color="Green" title="E79">U0</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><br/> for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">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="c9">P</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">k</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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"> for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="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="c9">P</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">k</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">k</font></span><br/><b>let </b><font color="Maroon" title="c12">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"> ( <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="c9">P</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">k</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">k</font> )</span><br/>



<b>assume </b><a NAME="E1:66_14"/><i><font color="Green" title="E80">N4</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="c9">P</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">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">k</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">k</font></span><br/>

<a NAME="E2:66_14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">k</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="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E3:66_14"/><b>then </b><i><font color="Green" title="E81">U1</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">k</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="#E21:66"><i><font color="Green" title="E68">K5</font></i></a></i>;<br/>
<a NAME="E4:66_14"/>
 <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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c12">k</font></span>)</span> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font>
 
<b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a></i>;<br/>
<a NAME="E5:66_14"/><b>then </b><i><font color="Green" title="E82">N73</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:66_14"><i><font color="Green" title="E80">N4</font></i></a>, <a class="txt" href="#E10:66"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="#E33:66"><i><font color="Green" title="E74">N1</font></i></a>, <a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a></i>;<br/>
<a NAME="E6:66_14"/>
(  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c12">k</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c12">k</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> 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> <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">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="b1">m</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">x</font> ) )
 
<b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a>, <a class="txt" href="#E39:66"><i><font color="Green" title="E78">HHH</font></i></a></i>;<br/>
<a NAME="E7:66_14"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">k</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c12">k</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="#E2:66"><i><font color="Green" title="E52">S1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="txt" href="#E5:66_14"><i><font color="Green" title="E82">N73</font></i></a>, <a class="ref" href="mesfunc9.html#T42" target="_self" title="MESFUNC9:th.42">ADD4</a></i>;<br/>
<b>hence </b><a NAME="E8:66_14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">k</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c12">k</font>
 <b>by </b><i><a class="txt" href="#E3:66_14"><i><font color="Green" title="E81">U1</font></i></a>, <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="E41:66"/><i><font color="Green" title="E80">LL7</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"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c9">P</font></span>)</span> holds <br/> <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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="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="c9">P</font></span>)</span> implies  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:66_15"/><i><font color="Green" title="E81">N4</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="c9">P</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span><br/>

<a NAME="E2:66_15"/><b>then </b><i><font color="Green" title="E82">NN8</font></i>: 
<font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 
<b>by </b><i><a class="txt" href="#E38:66"><i><font color="Green" title="E77">N8</font></i></a></i>;<br/>
<a NAME="E3:66_15"/><i><font color="Green" title="E83">N9</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 
<b>by </b><i><a class="txt" href="#E1:66_15"><i><font color="Green" title="E81">N4</font></i></a>, <a class="txt" href="#E35:66"><i><font color="Green" title="E75">N21</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="txt" href="#E4:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="mesfunc9.html#T38" target="_self" title="MESFUNC9:th.38">ADD3</a></i>;<br/>
<a NAME="E4:66_15"/>
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  <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">k</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<b>by </b><i><a class="txt" href="#E1:66_15"><i><font color="Green" title="E81">N4</font></i></a>, <a class="txt" href="#E40:66"><i><font color="Green" title="E79">U0</font></i></a></i>;<br/>
<a NAME="E5:66_15"/><b>then </b><i><font color="Green" title="E84">N10</font></i>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:66_15"><i><font color="Green" title="E82">NN8</font></i></a>, <a class="txt" href="#E3:66_15"><i><font color="Green" title="E83">N9</font></i></a>, <a class="ref" href="rinfsup2.html#T38" title="RINFSUP2:th.38">RINFSUP2:38</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="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>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="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 holds <br/>$3 <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> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</font>;<br/>
<a NAME="E6:66_15"/><i><font color="Green" title="E85">ZZ1</font></i>: 
for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</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="b3">PP2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font>,<font color="Olive" title="b3">PP2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">i1</font>, <font color="Maroon" title="c13">j1</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">PP2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">i1</font>,<font color="Maroon" title="c13">j1</font>,<font color="Olive" title="b1">PP2</font>]</span><br/>



<b>reconsider </b><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c12">i1</font>, <font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c13">j1</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">F1</font> = <span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c16">F1</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>2</sub></font>[<font color="Maroon" title="c12">i1</font>,<font color="Maroon" title="c13">j1</font>,<font color="Maroon" title="c16">F1</font>]</span><br/>

<b>thus </b><a NAME="E3:66_15_1"/>
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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i1</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j1</font> holds <br/><font color="Maroon" title="c16">F1</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> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</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="c12">PP2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <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> such that </b><br/><a NAME="E8:66_15"/><i><font color="Green" title="E86">ZZ2</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</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">i</font>,<font color="Olive" title="b2">j</font>,<font color="Maroon" title="c12">PP2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="#E6:66_15"><i><font color="Green" title="E85">ZZ1</font></i></a>);<br/></i>
<a NAME="E9:66_15"/><i><font color="Green" title="E87">ZZ3</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">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  <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Olive" title="b2">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">p</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> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Olive" title="b1">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c14">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"> <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c14">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">p</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> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c13">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c15">p1</font> = <font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c16">n1</font> = <font color="Maroon" title="c14">n</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E2:66_15_2"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c14">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c12">PP2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">n</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D9" target="_self" title="MESFUNC9:def.9">defproj2</a></i>;<br/>
<b>hence </b><a NAME="E3:66_15_2"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c14">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">p</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> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c13">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="txt" href="#E8:66_15"><i><font color="Green" title="E86">ZZ2</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="E10:66_15"/><i><font color="Green" title="E88">J2</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 <br/> ( <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Olive" title="b1">n</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Olive" title="b1">n</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_15_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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"> ( <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c13">n</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c13">n</font></span>)</span> )</span><br/>

<a NAME="E1:66_15_3"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_15_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">j</font>, <font color="Maroon" title="c15">k</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="valued_0.html#D15" title="VALUED_0:def.15">VALUED_0:def 15</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="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> or  not <font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> or  not <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> )</span><br/>

<b>assume </b><a NAME="E1:66_15_3_1"/><i><font color="Green" title="E89">J21</font></i>: 
( <font color="Maroon" title="c14">j</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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> &amp; <font color="Maroon" title="c15">k</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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> &amp; <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c16">j</font> = <font color="Maroon" title="c14">j</font>, <font color="Maroon" title="c17">k</font> = <font color="Maroon" title="c15">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:66_15_3_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">k</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="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E4:66_15_3_1"/><b>then </b><i><font color="Green" title="E90">J22</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c16">j</font></span>)</span></span>)</span> <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="c11">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <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="c11">x</font> )
 
<b>by </b><i><a class="txt" href="#E9:66_15"><i><font color="Green" title="E87">ZZ3</font></i></a></i>;<br/>
<a NAME="E5:66_15_3_1"/><i><font color="Green" title="E91">J23</font></i>: 
(  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c16">j</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c16">j</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c17">k</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c17">k</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> )
 
<b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a></i>;<br/>
<a NAME="E6:66_15_3_1"/><i><font color="Green" title="E92">N73</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c16">j</font></span>)</span> <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="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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c17">k</font></span>)</span> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> )
 
<b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a></i>;<br/>
<a NAME="E7:66_15_3_1"/><b>then </b>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">z</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">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c16">j</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c16">j</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">z</font> <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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b2">z</font>
 
<b>by </b><i><a class="txt" href="#E1:66_15_3_1"><i><font color="Green" title="E89">J21</font></i></a>, <a class="txt" href="#E25:66"><i><font color="Green" title="E70">K7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:66_15_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="txt" href="#E4:66_15_3_1"><i><font color="Green" title="E90">J22</font></i></a>, <a class="txt" href="#E5:66_15_3_1"><i><font color="Green" title="E91">J23</font></i></a>, <a class="ref" href="mesfunc9.html#T42" target="_self" title="MESFUNC9:th.42">ADD4</a>, <a class="txt" href="#E6:66_15_3_1"><i><font color="Green" title="E92">N73</font></i></a>, <a class="txt" href="#E35:66"><i><font color="Green" title="E75">N21</font></i></a>, <a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E1:66_15"><i><font color="Green" title="E81">N4</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="E2:66_15_3"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><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"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c13">n</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c13">n</font></span>)</span> )</span><br/>

<b>hence </b><a NAME="E3:66_15_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c13">n</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c13">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rinfsup2.html#T21" title="RINFSUP2:th.21">RINFSUP2:21</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="E11:66_15"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_15_4"><b>proof </b></a><div class="add">

<a NAME="E1:66_15_4"/>
<font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc9.html#V1" title="MESFUNC9:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="#E5:66"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="#E35:66"><i><font color="Green" title="E75">N21</font></i></a>, <a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E1:66_15"><i><font color="Green" title="E81">N4</font></i></a></i>;<br/>
<a NAME="E2:66_15_4"/><b>then </b><i><font color="Green" title="E89">R00</font></i>: 
 <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</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="c11">x</font></span>)</span> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D2" target="_self" title="MESFUNC9:def.2">Def2</a></i>;<br/>
<a NAME="E3:66_15_4"/><i><font color="Green" title="E90">R0</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_15_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_15_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_15_4_1_1"/>
(  <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</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="c11">x</font></span>)</span> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> or  <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</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="c11">x</font></span>)</span> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a> or  <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</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="c11">x</font></span>)</span> is <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a> )
 </span><b>by </b><i><a class="txt" href="#E2:66_15_4"><i><font color="Green" title="E89">R00</font></i></a>, <a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_15_4_1_1_1"><b>suppose </b></a><a NAME="E1:66_15_4_1_1_1"/>
 <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</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="c11">x</font></span>)</span> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a></span><br/><div class="add"><a NAME="E2:66_15_4_1_1_1"/><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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>
 <b>by </b><i><a class="txt" href="#E2:66"><i><font color="Green" title="E52">S1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="txt" href="#E35:66"><i><font color="Green" title="E75">N21</font></i></a>, <a class="txt" href="#E1:66_15"><i><font color="Green" title="E81">N4</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="E3:66_15_4_1_1_1"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</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="66_15_4_1_1_2"><b>suppose </b></a><a NAME="E1:66_15_4_1_1_2"/>
 <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</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="c11">x</font></span>)</span> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a></span><br/><div class="add"><a NAME="E2:66_15_4_1_1_2"/><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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>
 <b>by </b><i><a class="txt" href="#E2:66"><i><font color="Green" title="E52">S1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="txt" href="#E35:66"><i><font color="Green" title="E75">N21</font></i></a>, <a class="txt" href="#E1:66_15"><i><font color="Green" title="E81">N4</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="E3:66_15_4_1_1_2"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</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="66_15_4_1_1_3"><b>suppose </b></a><a NAME="E1:66_15_4_1_1_3"/>
 <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</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="c11">x</font></span>)</span> is <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a></span><br/><div class="add"><a NAME="E2:66_15_4_1_1_3"/><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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a>
 <b>by </b><i><a class="txt" href="#E2:66"><i><font color="Green" title="E52">S1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="txt" href="#E35:66"><i><font color="Green" title="E75">N21</font></i></a>, <a class="txt" href="#E1:66_15"><i><font color="Green" title="E81">N4</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="E3:66_15_4_1_1_3"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#D11" title="MESFUNC5:def.11">MESFUNC5:def 11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:66_15_4"/>
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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</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#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_15_4_2"><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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</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#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c14">n'</font> = <font color="Maroon" title="c13">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="E2:66_15_4_2"/><i><font color="Green" title="E91">R1</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E2:66_15"><i><font color="Green" title="E82">NN8</font></i></a>, <a class="ref" href="rinfsup2.html#T21" title="RINFSUP2:th.21">RINFSUP2:21</a></i>;<br/>
<a NAME="E3:66_15_4_2"/><i><font color="Green" title="E92">R2</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 
<b>by </b><i><a class="txt" href="#E10:66_15"><i><font color="Green" title="E88">J2</font></i></a></i>;<br/>
<a NAME="E4:66_15_4_2"/>
for <font color="Olive" title="b1">p</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  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">p</font> <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="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_15_4_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">p</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font> <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="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font></span><br/>

<a NAME="E1:66_15_4_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</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="E2:66_15_4_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font></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="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E3:66_15_4_2_1"/><b>then </b><i><font color="Green" title="E93">R3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span></span>)</span> <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="c11">x</font>
 
<b>by </b><i><a class="txt" href="#E9:66_15"><i><font color="Green" title="E87">ZZ3</font></i></a></i>;<br/>
<a NAME="E4:66_15_4_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</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="E5:66_15_4_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font></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="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E6:66_15_4_2_1"/><b>then </b><i><font color="Green" title="E94">R4</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font></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="#E21:66"><i><font color="Green" title="E68">K5</font></i></a></i>;<br/>
<a NAME="E7:66_15_4_2_1"/><i><font color="Green" title="E95">R51</font></i>: 
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E8:66_15_4_2_1"/><i><font color="Green" title="E96">R52</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span> <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="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a>, <a class="txt" href="#E35:66"><i><font color="Green" title="E75">N21</font></i></a>, <a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E1:66_15"><i><font color="Green" title="E81">N4</font></i></a></i>;<br/>
<a NAME="E9:66_15_4_2_1"/>
(  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">i</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> ) )
 
<b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a>, <a class="txt" href="#E18:66"><i><font color="Green" title="E66">K2</font></i></a></i>;<br/>
<a NAME="E10:66_15_4_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a>
 
<b>by </b><i><a class="txt" href="#E8:66_15_4_2_1"><i><font color="Green" title="E96">R52</font></i></a>, <a class="ref" href="mesfunc9.html#T38" target="_self" title="MESFUNC9:th.38">ADD3</a></i>;<br/>
<a NAME="E11:66_15_4_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">n'</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:66_15_4_2_1"><i><font color="Green" title="E95">R51</font></i></a>, <a class="ref" href="rinfsup2.html#T7" title="RINFSUP2:th.7">RINFSUP2:7</a></i>;<br/>
<a NAME="E12:66_15_4_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">n'</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font></span>)</span></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">p</font></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="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<b>hence </b><a NAME="E13:66_15_4_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font> <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="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font>
 <b>by </b><i><a class="txt" href="#E3:66_15_4_2_1"><i><font color="Green" title="E93">R3</font></i></a>, <a class="txt" href="#E6:66_15_4_2_1"><i><font color="Green" title="E94">R4</font></i></a>, <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="E5:66_15_4_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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c14">n'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:66_15_4_2"><i><font color="Green" title="E91">R1</font></i></a>, <a class="txt" href="#E3:66_15_4_2"><i><font color="Green" title="E92">R2</font></i></a>, <a class="ref" href="rinfsup2.html#T38" title="RINFSUP2:th.38">RINFSUP2:38</a></i>;<br/>
<a NAME="E6:66_15_4_2"/><b>then </b><i><font color="Green" title="E93">J3</font></i>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:66_15_4_2"><i><font color="Green" title="E91">R1</font></i></a>, <a class="txt" href="#E10:66_15"><i><font color="Green" title="E88">J2</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> <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,$1</span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> $1;<br/>
<a NAME="E7:66_15_4_2"/><i><font color="Green" title="E94">V1</font></i>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <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> <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="c11">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_15_4_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:66_15_4_2_2"/>
for <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">p</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/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_15_4_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">p</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font> )</span><br/>

<b>assume </b><a NAME="E1:66_15_4_2_2_1"/>
<font color="Maroon" title="c15">p</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c16">p'</font> = <font color="Maroon" title="c15">p</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="E3:66_15_4_2_2_1"/><i><font color="Green" title="E95">V74</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">p'</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="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E4:66_15_4_2_2_1"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">p'</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> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c16">p'</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E9:66_15"><i><font color="Green" title="E87">ZZ3</font></i></a></i>;<br/>
<a NAME="E5:66_15_4_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">p'</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c16">p'</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D4" target="_self" title="MESFUNC9:def.4">Def0</a></i>;<br/>
<a NAME="E6:66_15_4_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">p'</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">FF2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c16">p'</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D9" target="_self" title="MESFUNC9:def.9">defproj2</a></i>;<br/>
<a NAME="E7:66_15_4_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">p'</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">p'</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D8" target="_self" title="MESFUNC9:def.8">defproj1</a></i>;<br/>
<b>hence </b><a NAME="E8:66_15_4_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">p</font>
 <b>by </b><i><a class="txt" href="#E3:66_15_4_2_2_1"><i><font color="Green" title="E95">V74</font></i></a>, <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="E2:66_15_4_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>
<b>hence </b><a NAME="E3:66_15_4_2_2"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <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> <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="c11">x</font>
 <b>by </b><i><a class="txt" href="#E19:66"><i><font color="Green" title="E67">K3</font></i></a>, <a class="txt" href="#E1:66_15"><i><font color="Green" title="E81">N4</font></i></a>, <a class="txt" href="#E35:66"><i><font color="Green" title="E75">N21</font></i></a>, <a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</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="E8:66_15_4_2"/>
<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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <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="c11">x</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E9:66_15_4_2"/><b>then </b><i><font color="Green" title="E95">V2</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="#E7:66_15_4_2"><i><font color="Green" title="E94">V1</font></i></a>, <a class="ref" href="mesfunc9.html#D4" target="_self" title="MESFUNC9:def.4">Def0</a></i>;<br/>
<a NAME="E10:66_15_4_2"/><i><font color="Green" title="E96">V3</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="66_15_4_2_3"><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>reconsider </b><font color="Maroon" title="c16">k'</font> = <font color="Maroon" title="c15">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:66_15_4_2_3"/><i><font color="Green" title="E97">V31</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="E3:66_15_4_2_3"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c16">k'</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 
<b>by </b><i><a class="txt" href="#E10:66_15"><i><font color="Green" title="E88">J2</font></i></a></i>;<br/>
<a NAME="E4:66_15_4_2_3"/><b>then </b><i><font color="Green" title="E98">V41</font></i>: 
( <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp; <span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> )
 
<b>by </b><i><a class="txt" href="#E19:66"><i><font color="Green" title="E67">K3</font></i></a>, <a class="txt" href="#E1:66_15"><i><font color="Green" title="E81">N4</font></i></a>, <a class="txt" href="#E35:66"><i><font color="Green" title="E75">N21</font></i></a>, <a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_15_4_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">m</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="c17">m</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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> implies  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">m</font> )</span><br/><b>assume </b><a NAME="E1:66_15_4_2_3_1"/>
<font color="Maroon" title="c17">m</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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">m</font></span><br/><b>then reconsider </b><font color="Maroon" title="c18">m1</font> = <font color="Maroon" title="c17">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E3:66_15_4_2_3_1"/><i><font color="Green" title="E99">V413</font></i>: 
for <font color="Olive" title="b1">l</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c18">m1</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">l</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="#E18:66"><i><font color="Green" title="E66">K2</font></i></a></i>;<br/><a NAME="E4:66_15_4_2_3_1"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">m1</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> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c18">m1</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">k'</font>
 <b>by </b><i><a class="txt" href="#E9:66_15"><i><font color="Green" title="E87">ZZ3</font></i></a></i>;<br/><a NAME="E5:66_15_4_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">m1</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="#E3:66_15_4_2_3_1"><i><font color="Green" title="E99">V413</font></i></a>, <a class="ref" href="mesfunc9.html#T36" target="_self" title="MESFUNC9:th.36">ADD3C</a></i>;<br/><a NAME="E6:66_15_4_2_3_1"/><b>then </b>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">m1</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="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/><b>hence </b><a NAME="E7:66_15_4_2_3_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">m</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="E6:66_15_4_2_3"/><b>then </b><i><font color="Green" title="E99">V42</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</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>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_15_4_2_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">m</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="c17">m</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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> implies  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">m</font> )</span><br/><b>assume </b><a NAME="E1:66_15_4_2_3_2"/>
<font color="Maroon" title="c17">m</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"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">m</font></span><br/><b>then reconsider </b><font color="Maroon" title="c18">m1</font> = <font color="Maroon" title="c17">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E3:66_15_4_2_3_2"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">m1</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="txt" href="#E18:66"><i><font color="Green" title="E66">K2</font></i></a></i>;<br/><a NAME="E4:66_15_4_2_3_2"/><b>then </b>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">m1</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="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/><b>hence </b><a NAME="E5:66_15_4_2_3_2"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">m</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="E8:66_15_4_2_3"/><b>then </b><i><font color="Green" title="E100">V43</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</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>;<br/>
<a NAME="E9:66_15_4_2_3"/>
for <font color="Olive" title="b1">p</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">p</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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_15_4_2_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">p</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">p</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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">p</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c18">p'</font> = <font color="Maroon" title="c17">p</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="E2:66_15_4_2_3_3"/><i><font color="Green" title="E101">V32</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">p'</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="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E3:66_15_4_2_3_3"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">p'</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> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c18">p'</font></span>)</span></span>)</span> <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>
 
<b>by </b><i><a class="txt" href="#E9:66_15"><i><font color="Green" title="E87">ZZ3</font></i></a></i>;<br/>
<a NAME="E4:66_15_4_2_3_3"/><b>then </b><i><font color="Green" title="E102">V33</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">p'</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c18">p'</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c18">p'</font></span>)</span> <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>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D4" target="_self" title="MESFUNC9:def.4">Def0</a></i>;<br/>
<a NAME="E5:66_15_4_2_3_3"/><i><font color="Green" title="E103">V34</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c18">p'</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">k'</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">p'</font>
 
<b>by </b><i><a class="txt" href="#E9:66_15"><i><font color="Green" title="E87">ZZ3</font></i></a></i>;<br/>
<a NAME="E6:66_15_4_2_3_3"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c18">p'</font></span>)</span> <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> <font color="Maroon" title="c7">FF2</font> <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">.</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>,<font color="Maroon" title="c18">p'</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D9" target="_self" title="MESFUNC9:def.9">defproj2</a></i>;<br/>
<a NAME="E7:66_15_4_2_3_3"/><b>then </b><i><font color="Green" title="E104">V35</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">p'</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">p'</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">p'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:66_15_4_2_3_3"><i><font color="Green" title="E102">V33</font></i></a>, <a class="txt" href="#E5:66_15_4_2_3_3"><i><font color="Green" title="E103">V34</font></i></a>, <a class="ref" href="mesfunc9.html#D8" target="_self" title="MESFUNC9:def.8">defproj1</a></i>;<br/>
<a NAME="E8:66_15_4_2_3_3"/>
(  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c17">p</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c17">p</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> &amp;  <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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c17">p</font></span>)</span> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> )
 
<b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a></i>;<br/>
<a NAME="E9:66_15_4_2_3_3"/><b>then </b>
<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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c17">p</font></span>)</span></span>)</span> <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>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#T29" target="_self" title="MESFUNC9:th.29">ADD0</a></i>;<br/>
<a NAME="E10:66_15_4_2_3_3"/><b>then </b>
<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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">p'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E9:66_15"><i><font color="Green" title="E87">ZZ3</font></i></a></i>;<br/>
<a NAME="E11:66_15_4_2_3_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">p'</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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">p'</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">p'</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:66_15"><i><font color="Green" title="E81">N4</font></i></a>, <a class="txt" href="#E35:66"><i><font color="Green" title="E75">N21</font></i></a>, <a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E7:66_15_4_2_3_3"><i><font color="Green" title="E104">V35</font></i></a>, <a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a></i>;<br/>
<a NAME="E12:66_15_4_2_3_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">p'</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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">p</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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">p'</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
 
<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="E13:66_15_4_2_3_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">p</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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c17">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:66_15_4_2_3_3"><i><font color="Green" title="E101">V32</font></i></a>, <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="E10:66_15_4_2_3"/><b>then </b><i><font color="Green" title="E101">V44</font></i>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:66_15_4_2_3"><i><font color="Green" title="E98">V41</font></i></a>, <a class="txt" href="#E6:66_15_4_2_3"><i><font color="Green" title="E99">V42</font></i></a>, <a class="txt" href="#E8:66_15_4_2_3"><i><font color="Green" title="E100">V43</font></i></a>, <a class="ref" href="mesfunc9.html#T11" target="_self" title="MESFUNC9:th.11">LIM2</a></i>;<br/>
<a NAME="E11:66_15_4_2_3"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K8" title="MESFUNC9:func.8">ProjMap1</a> <font color="Maroon" title="c7">FF2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <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="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="c11">x</font>
 
<b>by </b><i><a class="txt" href="#E19:66"><i><font color="Green" title="E67">K3</font></i></a>, <a class="txt" href="#E1:66_15"><i><font color="Green" title="E81">N4</font></i></a>, <a class="txt" href="#E35:66"><i><font color="Green" title="E75">N21</font></i></a>, <a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a></i>;<br/>
<a NAME="E12:66_15_4_2_3"/><b>then </b><i><font color="Green" title="E102">V46</font></i>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <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="c11">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="c5">F</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="c11">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E10:66_15_4_2_3"><i><font color="Green" title="E101">V44</font></i></a>, <a class="txt" href="#E2:66_15_4_2_3"><i><font color="Green" title="E97">V31</font></i></a>, <a class="ref" href="mesfunc5.html#D13" title="MESFUNC5:def.13">MESFUNC5:def 13</a></i>;<br/>
<a NAME="E13:66_15_4_2_3"/>
<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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <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>
 
<b>by </b><i><a class="txt" href="#E2:66"><i><font color="Green" title="E52">S1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="txt" href="#E1:66_15"><i><font color="Green" title="E81">N4</font></i></a>, <a class="txt" href="#E35:66"><i><font color="Green" title="E75">N21</font></i></a>, <a class="ref" href="mesfunc9.html#T29" target="_self" title="MESFUNC9:th.29">ADD0</a></i>;<br/>
<a NAME="E14:66_15_4_2_3"/><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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <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="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D4" target="_self" title="MESFUNC9:def.4">Def0</a></i>;<br/>
<a NAME="E15:66_15_4_2_3"/><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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">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"><span class="p3">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">+</a> <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="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="#E12:66_15_4_2_3"><i><font color="Green" title="E102">V46</font></i></a>, <a class="ref" href="mesfunc1.html#D3" title="MESFUNC1:def.3">MESFUNC1:def 3</a></i>;<br/>
<a NAME="E16:66_15_4_2_3"/><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"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <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="c11">x</font>
 
<b>by </b><i><a class="ref" href="mesfunc9.html#D4" target="_self" title="MESFUNC9:def.4">Def0</a></i>;<br/>
<b>hence </b><a NAME="E17:66_15_4_2_3"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c12">PP2</font>,<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="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</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>
 <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="E11:66_15_4_2"/>
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="#E9:66_15_4_2"><i><font color="Green" title="E95">V2</font></i></a>, <a class="txt" href="#E10:66_15_4_2"><i><font color="Green" title="E96">V3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E12:66_15_4_2"/>
<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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</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#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:66_15_4_2"><i><font color="Green" title="E93">J3</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="E5:66_15_4"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:66_15_4"><i><font color="Green" title="E90">R0</font></i></a>, <a class="ref" href="mesfunc9.html#T9" target="_self" title="MESFUNC9:th.9">LIM3</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="E12:66_15"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:66_15"><i><font color="Green" title="E84">N10</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<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#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c11">J</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="E43:66"/><i><font color="Green" title="E81">JJ1</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="c11">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> <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="c12">J</font> = <font color="Maroon" title="c11">J</font> as   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> ;<br/>
<a NAME="E45:66"/><i><font color="Green" title="E82">T0</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">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#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</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="66_16"><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">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c14">n'</font> = <font color="Maroon" title="c13">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="E2:66_16"/>
<font color="Maroon" title="c12">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E43:66"><i><font color="Green" title="E81">JJ1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:66_16"/>
<font color="Maroon" title="c12">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">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="E46:66"/><i><font color="Green" title="E83">T1</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="c9">P</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"><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="c5">F</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_17"><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="c9">P</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"><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="c5">F</font></span>)</span></span>)</span></span><br/>

<a NAME="E1:66_17"/><i><font color="Green" title="E84">K51</font></i>: 
(  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c13">n</font> is <a href="mesfunc9.html#V2" title="MESFUNC9:attr.2">additive</a> &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c13">n</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> )
 
<b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a></i>;<br/>
<a NAME="E2:66_17"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p3">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E21:66"><i><font color="Green" title="E68">K5</font></i></a></i>;<br/>
<a NAME="E3:66_17"/><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="c9">P</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <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="#E1:66_17"><i><font color="Green" title="E84">K51</font></i></a>, <a class="ref" href="mesfunc9.html#T29" target="_self" title="MESFUNC9:th.29">ADD0</a></i>;<br/>
<b>hence </b><a NAME="E4:66_17"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</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"><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="c5">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a>, <a class="txt" href="#E13:66"><i><font color="Green" title="E62">A72</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="E47:66"/><i><font color="Green" title="E84">T2</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"><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="c5">F</font></span>)</span></span>)</span> holds <br/>( <font color="Maroon" title="c9">P</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> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">F</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="66_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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="c13">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> <span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span> implies ( <font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</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="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font> ) )</span><br/>

<b>assume </b><a NAME="E1:66_18"/><i><font color="Green" title="E85">T21</font></i>: 
<font color="Maroon" title="c13">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> <span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span></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="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</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="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font> )</span><br/>

<a NAME="E2:66_18"/><b>then </b>
<font color="Maroon" title="c13">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="c9">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E37:66"><i><font color="Green" title="E76">N5</font></i></a>, <a class="txt" href="#E10:66"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="#E33:66"><i><font color="Green" title="E74">N1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:66_18"/>
<font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><a class="txt" href="#E38:66"><i><font color="Green" title="E77">N8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font></span><br/>

<a NAME="E4:66_18"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E41:66"><i><font color="Green" title="E80">LL7</font></i></a>, <a class="txt" href="#E1:66_18"><i><font color="Green" title="E85">T21</font></i></a>, <a class="txt" href="#E35:66"><i><font color="Green" title="E75">N21</font></i></a>, <a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E37:66"><i><font color="Green" title="E76">N5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:66_18"/>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="#E1:66_18"><i><font color="Green" title="E85">T21</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>
<a NAME="E48:66"/><i><font color="Green" title="E85">T3</font></i>: 
(  <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="c5">F</font></span>)</span> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</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="c9">P</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> ) &amp; <font color="Maroon" title="c12">J</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_19"><b>proof </b></a><div class="add">

<a NAME="E1:66_19"/>
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> <span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span> holds <br/><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="c5">F</font></span>)</span></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="66_19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">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> <span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span> implies <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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">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="E1:66_19_1"/><i><font color="Green" title="E86">T31</font></i>: 
<font color="Maroon" title="c13">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> <span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></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"><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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">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="c14">x1</font> = <font color="Maroon" title="c13">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="E3:66_19_1"/>
<font color="Maroon" title="c13">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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:66_19_1"><i><font color="Green" title="E86">T31</font></i></a>, <a class="txt" href="#E12:66"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="mesfunc9.html#D4" target="_self" title="MESFUNC9:def.4">Def0</a></i>;<br/>
<a NAME="E4:66_19_1"/><b>then </b><i><font color="Green" title="E87">T32</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x1</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 
<b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="ref" href="mesfunc9.html#T38" target="_self" title="MESFUNC9:th.38">ADD3</a></i>;<br/>
<a NAME="E5:66_19_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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">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="66_19_1_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"> <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">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="E1:66_19_1_1"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="mesfunc9.html#T36" target="_self" title="MESFUNC9:th.36">ADD3C</a></i>;<br/>
<a NAME="E2:66_19_1_1"/><b>then </b>
<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="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">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="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/>
<b>hence </b><a NAME="E3:66_19_1_1"/>
<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="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">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="E6:66_19_1"/><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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c14">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="#E4:66_19_1"><i><font color="Green" title="E87">T32</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="E7:66_19_1"/>
<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="c5">F</font></span>)</span></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">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="#E1:66_19_1"><i><font color="Green" title="E86">T31</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="E2:66_19"/>
 <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="c5">F</font></span>)</span> 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"> ( ( 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">P</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> ) &amp; <font color="Maroon" title="c12">J</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> )</span><br/>

<b>thus </b><a NAME="E3:66_19"/><i><font color="Green" title="E86">T33</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">P</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>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">J</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_19_2"><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="c9">P</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="E1:66_19_2"/>
( ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">k</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> ) &amp;  <a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c13">n</font> is <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a> )
 
<b>by </b><i><a class="txt" href="#E17:66"><i><font color="Green" title="E65">Q1</font></i></a>, <a class="txt" href="#E18:66"><i><font color="Green" title="E66">K2</font></i></a></i>;<br/>
<a NAME="E2:66_19_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <span class="p2">(<span class="default"><a href="mesfunc9.html#K9" title="MESFUNC9:func.9">ProjMap2</a> <font color="Maroon" title="c7">FF2</font>,<font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <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="ref" href="mesfunc9.html#T36" target="_self" title="MESFUNC9:th.36">ADD3C</a></i>;<br/>
<b>hence </b><a NAME="E3:66_19_2"/>
<font color="Maroon" title="c9">P</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="#E21:66"><i><font color="Green" title="E68">K5</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="E4:66_19"/>
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="c12">J</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="c12">J</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="66_19_3"><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 <font color="Maroon" title="c12">J</font> <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> <font color="Maroon" title="c12">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font> )</span><br/>

<b>assume </b><a NAME="E1:66_19_3"/><i><font color="Green" title="E87">T34</font></i>: 
<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"> <font color="Maroon" title="c12">J</font> <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> <font color="Maroon" title="c12">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font></span><br/>

<a NAME="E2:66_19_3"/><i><font color="Green" title="E88">T35</font></i>: 
( <font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c9">P</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> &amp; <font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> )
 
<b>by </b><i><a class="txt" href="#E24:66"><i><font color="Green" title="E69">K6</font></i></a>, <a class="txt" href="#E3:66_19"><i><font color="Green" title="E86">T33</font></i></a></i>;<br/>
<a NAME="E3:66_19_3"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">l</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="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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">m</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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">l</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_19_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">n</font>, <font color="Maroon" title="c16">m</font>, <font color="Maroon" title="c17">l</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="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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">m</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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">l</font></span>)</span></span><br/>

<a NAME="E1:66_19_3_1"/>
( <font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">m</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="txt" href="#E24:66"><i><font color="Green" title="E69">K6</font></i></a></i>;<br/>
<a NAME="E2:66_19_3_1"/><b>then </b>
( <font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> &amp; <font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">m</font> is <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">without+infty</a> )
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T20" title="MESFUNC5:th.20">MESFUNC5:20</a></i>;<br/>
<a NAME="E3:66_19_3_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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">m</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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T23" title="MESFUNC5:th.23">MESFUNC5:23</a></i>;<br/>
<a NAME="E4:66_19_3_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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">m</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"><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="c5">F</font></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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E46:66"><i><font color="Green" title="E83">T1</font></i></a></i>;<br/>
<a NAME="E5:66_19_3_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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">m</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"><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="c5">F</font></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"><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="c5">F</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E46:66"><i><font color="Green" title="E83">T1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:66_19_3_1"/>
 <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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c16">m</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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E46:66"><i><font color="Green" title="E83">T1</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="E4:66_19_3"/><b>then </b><i><font color="Green" title="E89">T37</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> &amp;  <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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font></span>)</span> )
 
;<br/>
<a NAME="E5:66_19_3"/><b>then </b><i><font color="Green" title="E90">T36</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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"><span class="p3">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</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"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">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/>
<a NAME="E6:66_19_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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">P</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="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="c9">P</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="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_19_3_2"><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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</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="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="c9">P</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="c15">x</font> )</span><br/>

<b>assume </b><a NAME="E1:66_19_3_2"/>
<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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font></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="c9">P</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="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="c9">P</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="c15">x</font></span><br/>

<a NAME="E2:66_19_3_2"/><b>then </b>
<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"><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="c5">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E46:66"><i><font color="Green" title="E83">T1</font></i></a>, <a class="txt" href="#E4:66_19_3"><i><font color="Green" title="E89">T37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:66_19_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</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="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="c9">P</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="c15">x</font>
 <b>by </b><i><a class="txt" href="#E13:66"><i><font color="Green" title="E62">A72</font></i></a>, <a class="txt" href="#E1:66_19_3"><i><font color="Green" title="E87">T34</font></i></a>, <a class="txt" href="#E27:66"><i><font color="Green" title="E72">K9</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="E7:66_19_3"/><b>then </b>
 <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">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="c9">P</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="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="mesfunc1.html#K5" title="MESFUNC1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">m</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:66_19_3"><i><font color="Green" title="E88">T35</font></i></a>, <a class="ref" href="mesfunc5.html#T76" title="MESFUNC5:th.76">MESFUNC5:76</a></i>;<br/>
<a NAME="E8:66_19_3"/><b>then </b>
<font color="Maroon" title="c12">J</font> <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>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E45:66"><i><font color="Green" title="E82">T0</font></i></a>, <a class="txt" href="#E5:66_19_3"><i><font color="Green" title="E90">T36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:66_19_3"/>
<font color="Maroon" title="c12">J</font> <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> <font color="Maroon" title="c12">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font>
 <b>by </b><i><a class="txt" href="#E45:66"><i><font color="Green" title="E82">T0</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="E5:66_19"/><b>then </b>
<font color="Maroon" title="c12">J</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/>
<b>hence </b><a NAME="E6:66_19"/>
<font color="Maroon" title="c12">J</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 <b>by </b><i><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>
<a NAME="E49:66"/><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"><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="c5">F</font></span>)</span></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="c12">J</font>
 
<b>by </b><i><a class="txt" href="#E8:66"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="#E45:66"><i><font color="Green" title="E82">T0</font></i></a>, <a class="txt" href="#E46:66"><i><font color="Green" title="E83">T1</font></i></a>, <a class="txt" href="#E47:66"><i><font color="Green" title="E84">T2</font></i></a>, <a class="txt" href="#E13:66"><i><font color="Green" title="E62">A72</font></i></a>, <a class="txt" href="#E24:66"><i><font color="Green" title="E69">K6</font></i></a>, <a class="txt" href="#E27:66"><i><font color="Green" title="E72">K9</font></i></a>, <a class="ref" href="mesfunc5.html#D15" title="MESFUNC5:def.15">MESFUNC5:def 15</a></i>;<br/>
<a NAME="E50:66"/><b>then </b><i><font color="Green" title="E86">T4</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> <span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span></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="c12">J</font>
 
<b>by </b><i><a class="txt" href="#E8:66"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="#E48:66"><i><font color="Green" title="E85">T3</font></i></a>, <a class="txt" href="#E37:66"><i><font color="Green" title="E76">N5</font></i></a>, <a class="ref" href="mesfunc5.html#T94" title="MESFUNC5:th.94">MESFUNC5:94</a></i>;<br/>
<a NAME="E51:66"/><i><font color="Green" title="E87">T5</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">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="c9">P</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="66_20"><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">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span><br/>

<a NAME="E1:66_20"/><i><font color="Green" title="E88">T01</font></i>: 
( <font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c9">P</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="#E24:66"><i><font color="Green" title="E69">K6</font></i></a>, <a class="txt" href="#E48:66"><i><font color="Green" title="E85">T3</font></i></a></i>;<br/>
<a NAME="E2:66_20"/>
<font color="Maroon" title="c12">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K5" title="MESFUNC5:func.5">integral'</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E45:66"><i><font color="Green" title="E82">T0</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:66_20"/>
<font color="Maroon" title="c12">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c13">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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:66_20"><i><font color="Green" title="E88">T01</font></i></a>, <a class="ref" href="mesfunc5.html#T95" title="MESFUNC5:th.95">MESFUNC5:95</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>consider </b><font color="Maroon" title="c13">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E53:66"/><i><font color="Green" title="E88">T7</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="c13">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;  <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="c5">F</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c13">I</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="#E2:66"><i><font color="Green" title="E52">S1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E4:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="mesfunc9.html#T50" target="_self" title="MESFUNC9:th.50">FSUM2</a></i>;<br/>
<a NAME="E54:66"/>
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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">I</font> 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> <font color="Maroon" title="c13">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="66_21"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">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="c14">n</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> <font color="Maroon" title="c13">I</font> 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> <font color="Maroon" title="c13">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">n</font> )</span><br/>

<b>assume </b><a NAME="E1:66_21"/>
<font color="Maroon" title="c14">n</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> <font color="Maroon" title="c13">I</font>
 ; <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="c13">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">n</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c15">n1</font> = <font color="Maroon" title="c14">n</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E3:66_21"/>
( <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="c15">n1</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="c15">n1</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="c15">n1</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="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="txt" href="#E4:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/>
<a NAME="E4:66_21"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="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="c15">n1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T96" title="MESFUNC5:th.96">MESFUNC5:96</a></i>;<br/>
<b>hence </b><a NAME="E5:66_21"/>
 <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="c13">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="txt" href="#E53:66"><i><font color="Green" title="E88">T7</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="E55:66"/><b>then </b>
<font color="Maroon" title="c13">I</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>;<br/>
<a NAME="E56:66"/><b>then </b>
 <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c13">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="mesfunc9.html#T16" target="_self" title="MESFUNC9:th.16">ADD3a</a></i>;<br/>
<a NAME="E57:66"/><b>then </b><i><font color="Green" title="E89">Z1</font></i>: 
 <a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c13">I</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>
 
<b>by </b><i><a class="ref" href="rinfsup2.html#T37" title="RINFSUP2:th.37">RINFSUP2:37</a></i>;<br/>
<a NAME="E58:66"/>
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="c12">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c13">I</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="66_22"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">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"> <font color="Maroon" title="c12">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c13">I</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">n</font></span><br/>

<a NAME="E1:66_22"/><i><font color="Green" title="E90">C1</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E2:66"><i><font color="Green" title="E52">S1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="ref" href="mesfunc9.html#T29" target="_self" title="MESFUNC9:th.29">ADD0</a></i>;<br/>
<a NAME="E2:66_22"/><i><font color="Green" title="E91">V1</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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">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="#E37:66"><i><font color="Green" title="E76">N5</font></i></a>, <a class="txt" href="#E46:66"><i><font color="Green" title="E83">T1</font></i></a></i>;<br/>
<a NAME="E3:66_22"/><i><font color="Green" title="E92">C2</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">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="#E6:66"><i><font color="Green" title="E56">G2</font></i></a>, <a class="txt" href="#E4:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="mesfunc9.html#T41" target="_self" title="MESFUNC9:th.41">ADD1</a></i>;<br/>
<a NAME="E4:66_22"/><i><font color="Green" title="E93">C3</font></i>: 
<font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">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="#E24:66"><i><font color="Green" title="E69">K6</font></i></a>, <a class="ref" href="mesfunc2.html#T37" title="MESFUNC2:th.37">MESFUNC2:37</a></i>;<br/>
<a NAME="E5:66_22"/><i><font color="Green" title="E94">C5</font></i>: 
<font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="#E48:66"><i><font color="Green" title="E85">T3</font></i></a></i>;<br/>
<a NAME="E6:66_22"/><i><font color="Green" title="E95">C7</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="mesfunc9.html#T36" target="_self" title="MESFUNC9:th.36">ADD3C</a></i>;<br/>
<a NAME="E7:66_22"/>
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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">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#R1" title="XXREAL_0:pred.1">&lt;=</a> <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="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">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="66_22_1"><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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></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="E1:66_22_1"/>
<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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">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">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c15">x</font></span><br/>

<a NAME="E2:66_22_1"/><b>then </b>
<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"><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="c5">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E46:66"><i><font color="Green" title="E83">T1</font></i></a></i>;<br/>
<a NAME="E3:66_22_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">n</font> <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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">n</font>
 
<b>by </b><i><a class="txt" href="#E37:66"><i><font color="Green" title="E76">N5</font></i></a>, <a class="txt" href="#E40:66"><i><font color="Green" title="E79">U0</font></i></a>, <a class="txt" href="#E35:66"><i><font color="Green" title="E75">N21</font></i></a>, <a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a></i>;<br/>
<a NAME="E4:66_22_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">n</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="E5:66_22_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</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="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="E8:66_22"/><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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:66_22"><i><font color="Green" title="E90">C1</font></i></a>, <a class="txt" href="#E2:66_22"><i><font color="Green" title="E91">V1</font></i></a>, <a class="txt" href="#E3:66_22"><i><font color="Green" title="E92">C2</font></i></a>, <a class="txt" href="#E4:66_22"><i><font color="Green" title="E93">C3</font></i></a>, <a class="txt" href="#E5:66_22"><i><font color="Green" title="E94">C5</font></i></a>, <a class="txt" href="#E6:66_22"><i><font color="Green" title="E95">C7</font></i></a>, <a class="ref" href="mesfunc5.html#T91" title="MESFUNC5:th.91">MESFUNC5:91</a></i>;<br/>
<a NAME="E9:66_22"/><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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:66_22"><i><font color="Green" title="E91">V1</font></i></a>, <a class="txt" href="#E4:66_22"><i><font color="Green" title="E93">C3</font></i></a>, <a class="txt" href="#E48:66"><i><font color="Green" title="E85">T3</font></i></a>, <a class="ref" href="mesfunc5.html#T94" title="MESFUNC5:th.94">MESFUNC5:94</a></i>;<br/>
<a NAME="E10:66_22"/><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="c9">P</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:66_22"><i><font color="Green" title="E90">C1</font></i></a>, <a class="txt" href="#E7:66"><i><font color="Green" title="E57">G3</font></i></a>, <a class="txt" href="#E6:66_22"><i><font color="Green" title="E95">C7</font></i></a>, <a class="ref" href="mesfunc5.html#T94" title="MESFUNC5:th.94">MESFUNC5:94</a></i>;<br/>
<a NAME="E11:66_22"/><b>then </b>
<font color="Maroon" title="c12">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">n</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E51:66"><i><font color="Green" title="E87">T5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:66_22"/>
<font color="Maroon" title="c12">J</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c13">I</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="txt" href="#E53:66"><i><font color="Green" title="E88">T7</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="E59:66"/><b>then </b><i><font color="Green" title="E90">T8</font></i>: 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c12">J</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c13">I</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E48:66"><i><font color="Green" title="E85">T3</font></i></a>, <a class="txt" href="#E57:66"><i><font color="Green" title="E89">Z1</font></i></a>, <a class="ref" href="rinfsup2.html#T38" title="RINFSUP2:th.38">RINFSUP2:38</a></i>;<br/>
<a NAME="E60:66"/>
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"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c13">I</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#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"><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="c5">F</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_23"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">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"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c13">I</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">n</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"><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="c5">F</font></span>)</span></span>)</span></span><br/>

<a NAME="E1:66_23"/><i><font color="Green" title="E91">C1</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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E2:66"><i><font color="Green" title="E52">S1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="ref" href="mesfunc9.html#T29" target="_self" title="MESFUNC9:th.29">ADD0</a></i>;<br/>
<a NAME="E2:66_23"/><i><font color="Green" title="E92">C2</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">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="#E6:66"><i><font color="Green" title="E56">G2</font></i></a>, <a class="txt" href="#E4:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="mesfunc9.html#T41" target="_self" title="MESFUNC9:th.41">ADD1</a></i>;<br/>
<a NAME="E3:66_23"/><i><font color="Green" title="E93">C4</font></i>: 
 <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="c5">F</font></span>)</span> <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="#E1:66"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="#E2:66"><i><font color="Green" title="E52">S1</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="txt" href="#E5:66"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="#E7:66"><i><font color="Green" title="E57">G3</font></i></a>, <a class="ref" href="mesfunc9.html#T44" target="_self" title="MESFUNC9:th.44">ADD2</a></i>;<br/>
<a NAME="E4:66_23"/><i><font color="Green" title="E94">C7</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="mesfunc9.html#T36" target="_self" title="MESFUNC9:th.36">ADD3C</a></i>;<br/>
<a NAME="E5:66_23"/>
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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font></span>)</span> holds <br/><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="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">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#R1" title="XXREAL_0:pred.1">&lt;=</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="c5">F</font></span>)</span></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="#E1:66_23"><i><font color="Green" title="E91">C1</font></i></a>, <a class="txt" href="#E28:66"><i><font color="Green" title="E73">J1</font></i></a></i>;<br/>
<a NAME="E6:66_23"/><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"><span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</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"><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="c5">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:66_23"><i><font color="Green" title="E91">C1</font></i></a>, <a class="txt" href="#E2:66_23"><i><font color="Green" title="E92">C2</font></i></a>, <a class="txt" href="#E37:66"><i><font color="Green" title="E76">N5</font></i></a>, <a class="txt" href="#E3:66_23"><i><font color="Green" title="E93">C4</font></i></a>, <a class="txt" href="#E4:66_23"><i><font color="Green" title="E94">C7</font></i></a>, <a class="txt" href="#E48:66"><i><font color="Green" title="E85">T3</font></i></a>, <a class="ref" href="mesfunc5.html#T91" title="MESFUNC5:th.91">MESFUNC5:91</a></i>;<br/>
<a NAME="E7:66_23"/><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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</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"><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="c5">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:66_23"><i><font color="Green" title="E91">C1</font></i></a>, <a class="txt" href="#E7:66"><i><font color="Green" title="E57">G3</font></i></a>, <a class="txt" href="#E4:66_23"><i><font color="Green" title="E94">C7</font></i></a>, <a class="ref" href="mesfunc5.html#T94" title="MESFUNC5:th.94">MESFUNC5:94</a></i>;<br/>
<a NAME="E8:66_23"/><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"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</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"><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="c5">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E37:66"><i><font color="Green" title="E76">N5</font></i></a>, <a class="txt" href="#E3:66_23"><i><font color="Green" title="E93">C4</font></i></a>, <a class="txt" href="#E48:66"><i><font color="Green" title="E85">T3</font></i></a>, <a class="ref" href="mesfunc5.html#T94" title="MESFUNC5:th.94">MESFUNC5:94</a></i>;<br/>
<b>hence </b><a NAME="E9:66_23"/>
<span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c13">I</font></span>)</span> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">n</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"><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="c5">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E53:66"><i><font color="Green" title="E88">T7</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="E61:66"/><b>then </b>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><a href="mesfunc9.html#K2" title="MESFUNC9:func.2">Partial_Sums</a> <font color="Maroon" title="c13">I</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"><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="c5">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E57:66"><i><font color="Green" title="E89">Z1</font></i></a>, <a class="ref" href="mesfunc9.html#T9" target="_self" title="MESFUNC9:th.9">LIM3</a></i>;<br/>
<a NAME="E62:66"/><b>then </b><i><font color="Green" title="E91">P9</font></i>: 
 <a href="mesfunc9.html#K3" title="MESFUNC9:func.3">Sum</a> <font color="Maroon" title="c13">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> <span class="p2">(<span class="default"><a href="mesfunc9.html#K4" title="MESFUNC9:func.4">Partial_Sums</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E59:66"><i><font color="Green" title="E90">T8</font></i></a>, <a class="txt" href="#E50:66"><i><font color="Green" title="E86">T4</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">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="c13">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"><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="Olive" title="b1">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span> ) &amp; <font color="Maroon" title="c13">I</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="c5">F</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></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="c13">I</font> )</span><br/>

<b>thus </b><a NAME="E63:66"/>
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="c13">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"><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="Olive" title="b1">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</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="c13">I</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="c5">F</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></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="c13">I</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_24"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">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="c13">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">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="c14">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></span>)</span></span><br/>

<a NAME="E1:66_24"/>
 <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> <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="c14">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:66"><i><font color="Green" title="E53">S2</font></i></a>, <a class="ref" href="mesfunc8.html#D2" title="MESFUNC8:def.2">MESFUNC8:def 2</a></i>;<br/>
<a NAME="E2:66_24"/><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">n</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c14">n</font>
 
<b>by </b><i><a class="txt" href="#E1:66"><i><font color="Green" title="E51">A1</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="E3:66_24"/>
<font color="Maroon" title="c13">I</font> <a href="mesfunc9.html#K1" title="MESFUNC9:func.1">.</a> <font color="Maroon" title="c14">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="c14">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="#E53:66"><i><font color="Green" title="E88">T7</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>thus </b><a NAME="E64:66"/>
( <font color="Maroon" title="c13">I</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="c5">F</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">E</font></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="c13">I</font> )
 <b>by </b><i><a class="txt" href="#E62:66"><i><font color="Green" title="E91">P9</font></i></a>, <a class="txt" href="#E37:66"><i><font color="Green" title="E76">N5</font></i></a>, <a class="txt" href="#E57:66"><i><font color="Green" title="E89">Z1</font></i></a>, <a class="ref" href="mesfunc9.html#D2" target="_self" title="MESFUNC9:def.2">Def2</a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</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>
