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<span class="kw">let </span><font color="Maroon" title="c1">D</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b3">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">H1</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b3">X</font> &amp; <font color="Olive" title="b2">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b3">X</font> holds <br/>( <font color="Olive" title="b1">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Olive" title="b2">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b3">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Olive" title="b2">H2</font></span>)</span>,<font color="Olive" title="b3">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b3">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Olive" title="b2">H2</font>,<font color="Olive" title="b3">X</font></span>)</span> &amp; <font color="Olive" title="b1">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Olive" title="b2">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b3">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Olive" title="b2">H2</font></span>)</span>,<font color="Olive" title="b3">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b3">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Olive" title="b2">H2</font>,<font color="Olive" title="b3">X</font></span>)</span> &amp; <font color="Olive" title="b1">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Olive" title="b2">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b3">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Olive" title="b2">H2</font></span>)</span>,<font color="Olive" title="b3">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b3">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Olive" title="b2">H2</font>,<font color="Olive" title="b3">X</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">H1</font>, <font color="Maroon" title="c3">H2</font> be   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b1">X</font> holds <br/>( <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b1">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Olive" title="b1">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Olive" title="b1">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Olive" title="b1">X</font></span>)</span> &amp; <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b1">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Olive" title="b1">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Olive" title="b1">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Olive" title="b1">X</font></span>)</span> &amp; <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Olive" title="b1">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Olive" title="b1">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Olive" title="b1">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Olive" title="b1">X</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> implies ( <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> &amp; <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> &amp; <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:54"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">and </span><br/><a NAME="E2:54"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> &amp; <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> &amp; <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> )</span><br/>

<div><span class="lab"><font color="Green" title="E38">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:54_1"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> </span><br/><a NAME="E2:54_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></span>;<br/><a NAME="E3:54_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#T19" title="SEQ_2:th.19">SEQ_2:19</a></span>;<br/><span class="kw">hence </span><a NAME="E4:54_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E1:54_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="seqfunc.html#T35" target="_self" title="SEQFUNC:th.35">Th35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E39">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:54_2"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> </span><br/><a NAME="E2:54_2"/><span class="kw">then </span>
( <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></span>;<br/><a NAME="E3:54_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#T25" title="SEQ_2:th.25">SEQ_2:25</a></span>;<br/><span class="kw">hence </span><a NAME="E4:54_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E1:54_2"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="seqfunc.html#T35" target="_self" title="SEQFUNC:th.35">Th35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:54"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
( <font color="Maroon" title="c4">X</font> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c2">H1</font> &amp; <font color="Maroon" title="c4">X</font> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c3">H2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></span>;<br/>
<a NAME="E6:54"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seqfunc.html#T38" target="_self" title="SEQFUNC:th.38">Th38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:54"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> &amp; <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> &amp; <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> )</span><br/>

<div><span class="lab"><font color="Green" title="E42">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:54_3"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span><br/><a NAME="E2:54_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A11</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/><a NAME="E3:54_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A12</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E4:54_3"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_3"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:54_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A14</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>;<br/><a NAME="E6:54_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A15</font></span>: 
( <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_3"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E4:54_3"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E3:54_3"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54_3"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="seq_2.html#T20" title="SEQ_2:th.20">SEQ_2:20</a></span>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E5:54_3"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="seqfunc.html#T35" target="_self" title="SEQFUNC:th.35">Th35</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E43">A16</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:54_4"/><span class="lab"><font color="Green" title="E44">A17</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> </span><br/><a NAME="E2:54_4"/><span class="kw">then </span>
( <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></span>;<br/><a NAME="E3:54_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></span>;<br/><span class="kw">hence </span><a NAME="E4:54_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E1:54_4"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="seqfunc.html#T35" target="_self" title="SEQFUNC:th.35">Th35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E44">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:54_5"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span><br/><a NAME="E2:54_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A19</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" title="VALUED_1:def.4">VALUED_1:def 4</a></span>;<br/><a NAME="E3:54_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A20</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E4:54_5"/><span class="lab"><font color="Green" title="E47">A21</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_5"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:54_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A22</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>;<br/><a NAME="E6:54_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A23</font></span>: 
( <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E4:54_5"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E3:54_5"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54_5"><span class="lab"><font color="Green" title="E49">A23</font></span></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></span>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E5:54_5"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="ref" href="seqfunc.html#T35" target="_self" title="SEQFUNC:th.35">Th35</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E45">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:54_6"/><span class="lab"><font color="Green" title="E46">A25</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span><br/><a NAME="E2:54_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A26</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></span>;<br/><a NAME="E3:54_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A27</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E4:54_6"/><span class="lab"><font color="Green" title="E49">A28</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_6"><span class="lab"><font color="Green" title="E47">A26</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:54_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A29</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>;<br/><a NAME="E6:54_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A30</font></span>: 
( <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_6"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E4:54_6"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E3:54_6"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54_6"><span class="lab"><font color="Green" title="E51">A30</font></span></a>, <a class="ref" href="seq_2.html#T26" title="SEQ_2:th.26">SEQ_2:26</a></span>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E5:54_6"><span class="lab"><font color="Green" title="E50">A29</font></span></a>, <a class="ref" href="seqfunc.html#T35" target="_self" title="SEQFUNC:th.35">Th35</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>
<br/>.= 
<font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>
<br/>.= 
<font color="Maroon" title="c4">X</font>

;<br/>
<span class="kw">hence </span><a NAME="E13:54"/>
 <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E8:54"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> &amp; <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> )</span><br/>

<a NAME="E14:54"/>
<font color="Maroon" title="c4">X</font> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="ref" href="seqfunc.html#T38" target="_self" title="SEQFUNC:th.38">Th38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:54"/><span class="lab"><font color="Green" title="E46">A31</font></span>: 
<font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> &amp; <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> )</span><br/>

 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></span>
<br/>.= 
<font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>
<br/>.= 
<font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>
<br/>.= 
<font color="Maroon" title="c4">X</font>

;<br/>
<span class="kw">hence </span><a NAME="E17:54"/>
 <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:54"><span class="lab"><font color="Green" title="E46">A31</font></span></a>, <a class="txt" href="#E11:54"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span> )</span><br/>

<a NAME="E18:54"/>
<font color="Maroon" title="c4">X</font> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="ref" href="seqfunc.html#T38" target="_self" title="SEQFUNC:th.38">Th38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:54"/><span class="lab"><font color="Green" title="E47">A32</font></span>: 
<font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="ref" href="seqfunc.html#T21" target="_self" title="SEQFUNC:th.21">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span><br/>

 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</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#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" title="VALUED_1:def.4">VALUED_1:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>
<br/>.= 
<font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">X</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>
<br/>.= 
<font color="Maroon" title="c4">X</font>

;<br/>
<span class="kw">hence </span><a NAME="E21:54"/>
 <a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:54"><span class="lab"><font color="Green" title="E47">A32</font></span></a>, <a class="txt" href="#E10:54"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="ref" href="seqfunc.html#D14" target="_self" title="SEQFUNC:def.14">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
