<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">FT1</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</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">FT2</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">filled</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <br/> for <font color="Olive" title="b2">n0</font>, <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">FT1</font>,<font color="Olive" title="b1">FT2</font>  st <font color="Olive" title="b4">f</font> <a href="fintopo4.html#R2" title="FINTOPO4:pred.2">is_continuous</a> <font color="Olive" title="b2">n0</font> &amp; <font color="Olive" title="b2">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> holds <br/><font color="Olive" title="b4">f</font> <a href="fintopo4.html#R2" title="FINTOPO4:pred.2">is_continuous</a> <font color="Olive" title="b3">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">FT2</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">filled</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</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">n0</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">FT1</font>,<font color="Maroon" title="c2">FT2</font>  st <font color="Olive" title="b3">f</font> <a href="fintopo4.html#R2" title="FINTOPO4:pred.2">is_continuous</a> <font color="Olive" title="b1">n0</font> &amp; <font color="Olive" title="b1">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Olive" title="b3">f</font> <a href="fintopo4.html#R2" title="FINTOPO4:pred.2">is_continuous</a> <font color="Olive" title="b2">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">n0</font>, <font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">FT1</font>,<font color="Maroon" title="c2">FT2</font>  st <font color="Olive" title="b1">f</font> <a href="fintopo4.html#R2" title="FINTOPO4:pred.2">is_continuous</a> <font color="Maroon" title="c3">n0</font> &amp; <font color="Maroon" title="c3">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> holds <br/><font color="Olive" title="b1">f</font> <a href="fintopo4.html#R2" title="FINTOPO4:pred.2">is_continuous</a> <font color="Maroon" title="c4">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">FT1</font>,<font color="Maroon" title="c2">FT2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="fintopo4.html#R2" title="FINTOPO4:pred.2">is_continuous</a> <font color="Maroon" title="c3">n0</font> &amp; <font color="Maroon" title="c3">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> implies <font color="Maroon" title="c5">f</font> <a href="fintopo4.html#R2" title="FINTOPO4:pred.2">is_continuous</a> <font color="Maroon" title="c4">n</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="fintopo4.html#R2" title="FINTOPO4:pred.2">is_continuous</a> <font color="Maroon" title="c3">n0</font>
 <span class="kw">and </span><br/><a NAME="E2:15"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c3">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="fintopo4.html#R2" title="FINTOPO4:pred.2">is_continuous</a> <font color="Maroon" title="c4">n</font></span><br/>

<a NAME="E3:15"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT1</font><br/> for <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">FT2</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">FT1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Olive" title="b1">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Olive" title="b2">y</font>,<font color="Maroon" title="c4">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FT1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">FT2</font>  st <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">FT1</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c6">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c4">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">FT2</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">FT1</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> implies <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c6">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">n</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">FT1</font>
 <span class="kw">and </span><br/><a NAME="E2:15_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">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="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c6">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">n</font></span><br/>

<a NAME="E3:15_1"/>
(  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c3">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo3.html#K7" title="FINTOPO3:func.7">Finf</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c7">y</font></span>)</span>,<font color="Maroon" title="c3">n0</font> &amp;  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo3.html#K7" title="FINTOPO3:func.7">Finf</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c7">y</font></span>)</span>,<font color="Maroon" title="c4">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo3.html#D10" title="FINTOPO3:def.10">FINTOPO3:def 10</a></span>;<br/>
<a NAME="E4:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A4</font></span>: 
 <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c3">n0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="fintopo4.html#T1" target="_self" title="FINTOPO4:th.1">Th1</a></span>;<br/>
<a NAME="E5:15_1"/>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c6">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c3">n0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E2:15_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="fintopo4.html#D2" target="_self" title="FINTOPO4:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:15_1"/>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c6">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:15"/>
<font color="Maroon" title="c5">f</font> <a href="fintopo4.html#R2" title="FINTOPO4:pred.2">is_continuous</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo4.html#D2" target="_self" title="FINTOPO4:def.2">Def2</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>
