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<b>let </b><font color="Maroon" title="c5">s1</font>, <font color="Maroon" title="c6">s2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:108_1_2"/><i><font color="Green" title="E103">A9</font></i>: 
 ex <font color="Olive" title="b1">F1</font> being   <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  ex <font color="Olive" title="b2">K1</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 <br/> ( <font color="Olive" title="b1">F1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</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"><font color="Olive" title="b1">F1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">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> <font color="Maroon" title="c4">f</font> ) ) &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">F1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> ) &amp; ( for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">m</font> holds <br/>for <font color="Olive" title="b5">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">F1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b4">m</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b5">x</font> ) &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/>( <font color="Olive" title="b1">F1</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F1</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b3">x</font> ) ) &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b2">K1</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b3">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="Olive" title="b1">F1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> ) &amp; <font color="Olive" title="b2">K1</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp; <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b2">K1</font> )
 <b>and </b><br/><a NAME="E2:108_1_2"/><i><font color="Green" title="E104">A10</font></i>: 
 ex <font color="Olive" title="b1">F2</font> being   <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  ex <font color="Olive" title="b2">K2</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 <br/> ( <font color="Olive" title="b1">F2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</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"><font color="Olive" title="b1">F2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">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> <font color="Maroon" title="c4">f</font> ) ) &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">F2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> ) &amp; ( for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">m</font> holds <br/>for <font color="Olive" title="b5">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">F2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b4">m</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b5">x</font> ) &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/>( <font color="Olive" title="b1">F2</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F2</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b3">x</font> ) ) &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b2">K2</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b3">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="Olive" title="b1">F2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> ) &amp; <font color="Olive" title="b2">K2</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp; <font color="Maroon" title="c6">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b2">K2</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">F1</font> being    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> , <font color="Maroon" title="c8">K1</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E4:108_1_2"/><i><font color="Green" title="E105">A11</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="c7">F1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</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> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</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> <font color="Maroon" title="c4">f</font> ) ) &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="c7">F1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> ) &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</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="c7">F1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b3">x</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/>( <font color="Maroon" title="c7">F1</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="c7">F1</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> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font> ) ) &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="c8">K1</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</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="c7">F1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) &amp; <font color="Maroon" title="c8">K1</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp; <font color="Maroon" title="c5">s1</font> <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="c8">K1</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:108_1_2"><i><font color="Green" title="E103">A9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">F2</font> being    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> , <font color="Maroon" title="c10">K2</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><b> such that </b><br/><a NAME="E6:108_1_2"/><i><font color="Green" title="E106">A12</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="c9">F2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</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> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</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> <font color="Maroon" title="c4">f</font> ) ) &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">F2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> ) &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</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">F2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b3">x</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/>( <font color="Maroon" title="c9">F2</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">F2</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> <font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Olive" title="b1">x</font> ) ) &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="c10">K2</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</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">F2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) &amp; <font color="Maroon" title="c10">K2</font> is <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> &amp; <font color="Maroon" title="c6">s2</font> <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="c10">K2</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E2:108_1_2"><i><font color="Green" title="E104">A10</font></i></a></i>;<br/>
<a NAME="E7:108_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/>( <font color="Maroon" title="c7">F1</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; <font color="Maroon" title="c9">F2</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="c7">F1</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"><font color="Maroon" title="c9">F2</font> <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="108_1_2_1"><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>;<br/>

<b>assume </b><a NAME="E1:108_1_2_1"/><i><font color="Green" title="E107">A13</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> <font color="Maroon" title="c4">f</font>
 ;<br/>

<b>then </b> <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c11">x</font></span>)</span> = 
<font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <font color="Maroon" title="c11">x</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E4:108_1_2"><i><font color="Green" title="E105">A11</font></i></a></i>
<br/>.= 
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</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="mesfunc5.html#E6:108_1_2"><i><font color="Green" title="E106">A12</font></i></a>, <a class="txt" href="mesfunc5.html#E1:108_1_2_1"><i><font color="Green" title="E107">A13</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E3:108_1_2_1"/>
( <font color="Maroon" title="c7">F1</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> &amp; <font color="Maroon" title="c9">F2</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> &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</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"><font color="Maroon" title="c9">F2</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="mesfunc5.html#E4:108_1_2"><i><font color="Green" title="E105">A11</font></i></a>, <a class="txt" href="mesfunc5.html#E6:108_1_2"><i><font color="Green" title="E106">A12</font></i></a>, <a class="txt" href="mesfunc5.html#E1:108_1_2_1"><i><font color="Green" title="E107">A13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:108_1_2"/>
<font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s2</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:108"><i><font color="Green" title="E101">A1</font></i></a>, <a class="txt" href="mesfunc5.html#E4:108_1_2"><i><font color="Green" title="E105">A11</font></i></a>, <a class="txt" href="mesfunc5.html#E6:108_1_2"><i><font color="Green" title="E106">A12</font></i></a>, <a class="ref" href="mesfunc5.html#T82" target="_self" title="MESFUNC5:th.82">Th82</a></i>;<br/>


</div>
