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<b>let </b><font color="Maroon" title="c1">Y</font>, <font color="Maroon" title="c2">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X1</font>, <font color="Olive" title="b2">X0</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">X0</font>,<font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b2">X0</font> holds <br/>for <font color="Olive" title="b4">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b5">x0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b2">X0</font><br/> for <font color="Olive" title="b6">x1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">X1</font>  st <font color="Olive" title="b4">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">A</font> &amp; <font color="Olive" title="b4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">X1</font> &amp; <font color="Olive" title="b5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">x1</font> holds <br/>( <font color="Olive" title="b3">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b5">x0</font> iff <font color="Olive" title="b3">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Olive" title="b1">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b6">x1</font> )</span><br/><b>let </b><font color="Maroon" title="c3">X1</font>, <font color="Maroon" title="c4">X0</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">X0</font>,<font color="Maroon" title="c1">Y</font>  st <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X0</font> holds <br/>for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b3">x0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X0</font><br/> for <font color="Olive" title="b4">x1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font>  st <font color="Olive" title="b2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A</font> &amp; <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x1</font> holds <br/>( <font color="Olive" title="b1">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b3">x0</font> iff <font color="Olive" title="b1">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b4">x1</font> )</span><br/><b>let </b><font color="Maroon" title="c5">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">X0</font>,<font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X0</font> implies for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b2">x0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X0</font><br/> for <font color="Olive" title="b3">x1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font>  st <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font> &amp; <font color="Olive" title="b2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x1</font> holds <br/>( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b2">x0</font> iff <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b3">x1</font> ) )</span><br/>









<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E60">A1</font></i>: 
<font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b2">x0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X0</font><br/> for <font color="Olive" title="b3">x1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font>  st <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font> &amp; <font color="Olive" title="b2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x1</font> holds <br/>( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b2">x0</font> iff <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b3">x1</font> )</span><br/>

<b>let </b><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X0</font><br/> for <font color="Olive" title="b2">x1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font>  st <font color="Maroon" title="c6">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font> &amp; <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x1</font> holds <br/>( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b1">x0</font> iff <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b2">x1</font> )</span><br/><b>let </b><font color="Maroon" title="c7">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font>  st <font color="Maroon" title="c6">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font> &amp; <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x1</font> holds <br/>( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x0</font> iff <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b1">x1</font> )</span><br/><b>let </b><font color="Maroon" title="c8">x1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font> &amp; <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x1</font> implies ( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x0</font> iff <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c8">x1</font> ) )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E2:81"/><i><font color="Green" title="E61">A2</font></i>: 
( <font color="Maroon" title="c6">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> )
 <b>and </b><br/><a NAME="E3:81"/><i><font color="Green" title="E62">A3</font></i>: 
( <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font> &amp; <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x0</font> iff <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c8">x1</font> )</span><br/>

<b>thus </b><a NAME="E4:81"/>
( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x0</font> implies <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c8">x1</font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:81"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="tmap_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="tmap_1.html#T81" target="_self" title="TMAP_1:th.81">Th81</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c8">x1</font> implies <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x0</font> )</span><br/>

<b>thus </b><a NAME="E5:81"/>
( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c8">x1</font> implies <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x0</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<a NAME="E1:81_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X0</font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:81"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">B</font> = <font color="Maroon" title="c6">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">X0</font> <b>by </b><i><a class="txt" href="tmap_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:81_1"/><i><font color="Green" title="E63">A4</font></i>: 
( <font color="Maroon" title="c9">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> &amp; <font color="Maroon" title="c9">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font> )
 
<b>by </b><i><a class="txt" href="tmap_1.html#E2:81"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="tmap_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="tops_2.html#T33" title="TOPS_2:th.33">TOPS_2:33</a></i>;<br/>
<b>assume </b><a NAME="E4:81_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c8">x1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x0</font></span><br/>

<b>hence </b><a NAME="E5:81_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x0</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:81"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="tmap_1.html#E3:81"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="tmap_1.html#E3:81_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="tmap_1.html#T84" target="_self" title="TMAP_1:th.84">Th84</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div>
